22.2: Apgar Scoring
By the end of this section, you will be able to:
- Describe the purpose of the Apgar scoring assessment
- Define the questions the neonatal nurse needs to ask before an infant is born in the birth room
- Demonstrate all steps to complete an Apgar assessment
Immediately after birth, assessment of the newborn's overall physical condition is imperative to provide the most appropriate care for them. The Apgar assessment is performed by a member of the team caring for the newborn. The results of the assessment completed by the nurse will be shared with the provider and/or team. Though the newborn’s Apgar cannot predict morbidity and mortality, it can describe their condition and provide a record and evaluation of their fetal-to- neonatal transition .
In 1953, Virginia Apgar, MD, created a tool for assessing the newborn at 1 minute and again at 5 minutes post birth. This assessment included heart rate, respiration, reflex irritability, muscle tone, and color, all of which evaluate the response of the neonate when transitioning from life inside the uterus, or intrauterine life , to life outside the uterus, or extrauterine life . The tool is called the Apgar score (American College of Obstetricians and Gynecologists [ACOG], 2015) (Figure 22.2). This assessment score does not predict any neurologic outcome of the neonate, nor does it predict mortality or morbidity. However, if the score is less than 7 at 5 minutes post birth, both the American College of Obstetricians and Gynecologists (ACOG) and the American Academy of Pediatrics (AAP) recommend that another Apgar assessment be performed at 10 minutes (ACOG, 2015). A low score reflects the need for additional resuscitation effort.
The nurse who assesses the newborn in the labor and delivery room is often referred to as the baby nurse or the nurse responsible for the neonate. The labor nurse must continue to assist the birthing person after birth; therefore, an additional nurse comes into the room at the time of birth and is responsible for the newborn after birth, a second patient who requires individualized care. The neonatal nurse’s role includes ensuring a fully stocked, correctly calibrated radiant warmer set to a safe temperature with all items necessary to facilitate resuscitation if needed and to assist with the transition to extrauterine life. Prior to delivery, the neonatal nurse needs to know the answers to the following questions:
- How many babies are expected?
- What is the gestational age of the newborn(s)?
- What is the status (ruptured or intact) of the amniotic membranes, and what is the color of the amniotic fluid?
- Were there any complications during pregnancy, labor, and/or birth?
- Are there any untreated maternal risk factors, including group B streptococcus (GBS)–positive status, lack of prenatal care, hypertension, or hyperglycemia?
- What, if any, medications is the birth parent taking or receiving prior to delivery?
- Were there any congenital findings that would put the newborn at an increased risk at delivery?
The answers to those questions will prepare the nurse for what to expect when the newborn is delivered.
- If there are multiple newborns, each will need a neonatal nurse.
- If this is a preterm newborn, the nurse will need to notify the neonatal intensive care ( NICU ) team to join the room. Not all facilities offer this level of support and instead will have supportive personnel present to assist with resuscitative efforts. Up to four additional trained staff may be needed for complex birthing situations.
- In some cases, the amniotic fluid shows evidence of meconium , the neonate’s first bowel movement consisting of amniotic fluid, mucus, lanugo, and bile. Meconium staining indicates the newborn passed the stool while in utero and is therefore at risk for aspirating. The nurse will need to monitor the newborn’s airway more closely at birth.
- If other complications occurred during labor and birth, the nurse will need to adjust the plans for postdelivery care based on the complications.
These items must be taken into consideration as the nurse prepares to evaluate the neonate after the birth.
Newborn Muscle Tone (Activity)
The tension of healthy muscles that contribute a slight resistance to passive displacement of a limb is called muscle tone (Simon et al., 2023). The nurse assesses the degree to which the newborn is moving their extremities from flexion to extension. To receive the full 2 points for muscle tone/activity in this category, the newborn will have spontaneous, active movements in all extremities. If their arms and legs are flexed with little movement, the score will be a 1. If there is no movement and the newborn appears floppy with very little muscle tone, they will be given a score of zero for this category. This and every category will be reassessed in 5 minutes for any improvements.
This video explains the use of the Apgar score and reinforces the content with NCLEX practice questions.
A healthy term newborn will have elbows and hips that are flexed and have knees flexed up toward the abdomen, thus scoring 2 points on the Apgar assessment. A preterm or ill newborn may have more flaccid extremities and will therefore have points deducted from the Apgar score because they demonstrate less flexion in their muscle tone.
Pulse or Heart Rate
The nurse will auscultate the newborn’s heart rate at the apex or palpate at the junction of the umbilical cord and the skin. The average heart rate of a newborn is 110 to 160 beats per minute (bpm) (Children's Hospital of Philadelphia, 2022). If the heart rate is greater than 100 bpm, the nurse will score the newborn 2 points on the Apgar and go on to the next step of the assessment. However, if the newborn does not have a heartbeat or if the heart rate is less than 100 bpm, the nurse will need to initiate neonatal resuscitation immediately and call for more personnel, including the neonatal intensive care team, if the health system offers that level of support. A heart rate of less than 100 bpm receives a score of 1 on the Apgar assessment, whereas a score of zero indicates no heartbeat was found while auscultating or palpating.
Newborn Reflex Irritability or Grimace
The grimace response, or reflex irritability , also known as grimace response, describes the newborn’s response to stimulation from the nurse. For example, the nurse will assess the response from the newborn when they rub the soles of the newborn’s feet or back with a baby blanket to stimulate them. A normal newborn will cry or become agitated, while a compromised newborn won’t have much response. A cry or agitated response from a newborn to stimulation will receive the full 2 points; newborns who have mild reactions will receive a 1 on the Apgar assessment. Newborns who have an absent response to stimulation will receive a zero score.
Appearance of Newborn Skin Color
The nurse will assess the newborn for cyanosis, specifically on the trunk, and for pallor of the skin. Newborns will generally have acrocyanosis , cyanosis found only on the hands and feet, which is an expected finding that may last up to a week. Observation of acrocyanosis requires a score of 1 on the Apgar assessment. A newborn who is born pale or completely cyanotic is scored zero. The rare newborn who is completely pink is scored with a 2.
Respiration
The nurse must pay careful attention to how the newborn is breathing, observing particularly for any adventitious breath sounds, any retractions or grunting, or a complete absence of breath sounds. Ideally, the nurse wants to see a newborn crying vigorously, and the nurse will work to dry and stimulate a newborn until the vigorous cry is achieved. Lack of spontaneous respiratory effort requires immediate intervention. (See Table 22.1 for a summary of the Apgar chart.)
| Indicator | 0 Points | 1 Point | 2 Points | |
|---|---|---|---|---|
| A | Activity (muscle tone) | Absent, loose, flaccid without activity; floppy tone | Arms and legs flexed with little movement | Spontaneous, active motion with flexed muscle tone resisting extension |
| P | Pulse | Absent | Less than 100 beats per minute | Greater than 100 beats per minute |
| G | Grimace (reflex or irritability in response to stimulation) | Zero response to stimulation | Limited response to stimulation | Crying, movement, pulling away upon stimulation |
| A | Appearance (skin color) | Pale or blue | Pink, but extremities are blue | Entirely pink |
| R | Respirations | Not breathing | Slow and irregular, weak or gasping | Crying vigorously |
Apgar Scoring and Escalating the Level of Care
If a newborn has a 1-minute Apgar score of 6 or below, care is escalated as needed. A 5-minute Apgar score may be improved because of earlier assistance or, if it is under 7, may require the nurse to continue escalated care. Immediate care often means the 5-minute Apgar shows an improved newborn transition response. The nurse will need to continue resuscitative efforts by adding a 10-minute Apgar score and possibly initiating respiration therapy (more on that topic follows). If a neonate, whether term or preterm, is having difficulty breathing and is without proper muscle tone, they should be brought to the radiant warmer immediately post birth and given additional assessment (Hammer, 2021). Neonates who are having difficulties early in the transitional period need to have a thermometer attached to them to assess thermoregulation . These neonates need to stay between 36.5° C and 37.4° C (97.7° F and 99.3° F), as the risk for hypothermia is increased in newborns who are already struggling with the transition to life.
This video shows the original creator of the Apgar assessment, Dr. Virginia Apgar, teaching a new nurse to complete the assessment on a newborn using all five assessment criteria.
The newborn found to have insufficient respiratory effort or increased work of breathing can be supported with nursing interventions to assist during transition. These interventions include nasal suctioning with a bulb syringe , monitoring their oxygen saturations with a pulse oximeter , and repositioning to best maintain an open airway. If these supportive measures do not result in improved respiratory effort, positive pressure can be provided.
Newborn Care: Part 1
Brianne delivered a male infant, Marcus, vaginally. The delivery was complicated by shoulder dystocia. Apgar score was 5 at one minute and 9 at five minutes; birth weight, 3995 g (8 lbs 14 oz); length, 21 inches (52.5 cm).
| PMH |
Maternal Medical History
Parents live in a two-bedroom apartment on the second floor in a building with only a freight elevator. They have a small dog named Candy. Prenatal History BP range 118-128/64-80 Fundal Height at 36 weeks 40 cm Gestational Diabetes 36 years old, G1 P1 No known allergies |
| Nursing Notes |
Delivery Summary
After 20 hours of labor, Brianne had a vaginal delivery of a male infant with mild shoulder dystocia. Infant birth weight is 3995 g (8 lbs 14 oz). Placenta delivered spontaneously and intact by inspection. A second-degree vaginal laceration was repaired. Quantitative blood loss 450 mL Father, Trey, present for labor and birth, very supportive Marcus was dried off immediately after birth with warm blankets, and a hat was placed on his head. After the umbilical cord was clamped and cut, Marcus was placed directly on Brianne’s chest to begin skin to skin. Marcus and Brianne were covered with a warmed blanket. |
| Flow Chart |
Newborn assessment data at 30 minutes of age
Temp: 97.8º F (ax) Heart rate: 160 bpm Resp: 66 breaths Pulse oximetry: 92% Color: pink with acrocyanosis Respirations: shallow, irregular Nasal flaring Marcus has not been interested in nursing. Capillary glucose: 42 |
| Provider’s Orders |
Observe in Labor and Birth unit
VS every 30 minutes ×2, then hourly ×2, then every 4 hours Erythromycin ointment to both eyes within 1 hour of birth Phytonadione 1 mg IM anterior thigh within 1 hour of birth Encourage skin-to-skin contact until temperature stable Breast feed on demand Monitor intake and output until discharge Initial bath at 4 hours of age |
Positive pressure ventilation is a form of respiratory therapy that involves the delivery of room air or a mixture of oxygen and room air by bag-valve mask into the lungs (Potchileev et al., 2022). Use of the mask requires the nurse to form a tight seal around the newborn’s mouth and nose. This is done by making a “C” shape with the thumb and first finger around the mask of the bag-valve mask and applying firm but gentle pressure to the newborn’s face while delivering breaths to the newborn at a rate of 30 to 60 per minute (Figure 22.3). The bag-valve mask should be connected to the oxygen flowmeter using 100% oxygen or blended oxygen and a self-inflating bag (Hammer, 2021). Positive pressure ventilation is initiated if the neonate is apneic, is gasping for air, or has a heart rate of less than 100 bpm. The nurse provides supportive care by way of thermoregulation, stimulation, and oxygenation. After effective respirations and circulation, thermoregulation is the most critical factor for successful transition to extrauterine life. Immature physiologic and anatomic states of the newborn put them at increased risk for hypothermia because they are unable to maintain their own safe body temperature. Additionally, the nurse will need to call a specially trained team, whether that is a neonatal resuscitation team or a pediatric provider, to assess the newborn. All nurses who choose to work in an obstetrics unit will eventually be trained in neonatal resuscitation and know these steps, so that if a newborn needs additional assistance with transition, the nurse will be qualified to help while waiting for the NICU team to arrive. Figure 22.4 shows the steps in the neonatal resuscitation algorithm.
This video shows what neonatal resuscitation (NRP) looks like in action. Every nurse who works on a labor and delivery or a neonatal intensive care unit (NICU) will be certified by the unit in NRP.


6.1: Final Exam
Final Exam (50 %)
Instructions
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Answer question one and any other two
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Question one carries 30 marks
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Other questions carry 20 marks each
Questions
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(a). What is Cache memory- Explain working of a Cache memory. (10 marks)
(b). What is pipelining? Explain instruction pipelining. (10 marks)
(c). Describe interrupt driven I/O. (10 marks) -
Explain working of DMA data transfer. Compare it with programmed I/O and interrupt driven data transfer. (20 Marks)
3. Explain the difference between hardwired control and micro programmed control. Is it possible to have a hardwired control associated with a control memory? (20 Marks)
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Explain the block diagram of an I/O interface unit. (20 marks)
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Using a diagram, explain the following steps executing of a program
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Fetch
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Decode
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Execute (20 marks)
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Fetch
Grading Scheme
Marks to be distributed as indicated against each question in the answers students will provide
Feedback
1.(a). Cache memory
Is memory that stores program instructions that are frequently re-referenced by software during operation. Fast access to these instructions increases the overall speed of the software program.
(b). Pipelining
A form of computer organization in which successive steps of an instruction sequence are executed in turn by a sequence of modules able to operate concurrently, so that another instruction can be begun before the previous one is finished.
Instruction pipelining is a technique that implements a form of parallelism called instruction- level parallelism within a single processor. It therefore allows faster CPU throughput (the number of instructions that can be executed in a unit of time) than would otherwise be possible at a given clock rate.
(c) . Interrupt driven I/O
The CPU works on its given tasks continuously. When an input is available, such as when someone types a key on the keyboard, then the CPU is interrupted from its work to take care of the input data.
2. Working of DMA data transfer
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First the CPU programs the DMA controller by setting its registers so it knows what to transfer where
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It also issues a command to the disk controller telling it to read data from the disk into its internal buffer and verify the checksum.
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When valid data are in the disk controller’s buffer, DMA can begin. The DMA controller initiates the transfer by issuing a read request over the bus to the disk controller . This read request looks like any other read request, and the disk controller does not know (or care) whether it came from the CPU or from a DMA controller. Typically, the memory address to write to is on the bus’ address lines, so when the disk controller fetches the next word from its internal buffer, it knows where to write it. The write to memory is another standard bus cycle.
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When the write is complete, the disk controller sends an acknowledgement signal to the DMA controller, also over the bus. The DMA controller then increments the memory address to use and decrements the byte count. If the byte count is still greater than 0, steps 2 through 4 are repeated until the count reaches 0.
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At that time, the DMA controller interrupts the CPU to let it know that the transfer is now complete. When the operating system starts up, it does not have to copy the disk block to memory; it is already there.
Comparing DMA with programmed I/O and interrupt driven data transfer
Programmed I/O (PIO) refers to data transfers initiated by a CPU under driver software control to access registers or memory on a device. while the device interrupts the CPU when new data has arrived and is ready to be retrieved by the system processor. The actual actions to perform depend on whether the device uses I/O ports, memory mapping.
3. Micro programmed control is a control mechanism to generate control signals by using a memory called control storage (CS), which contains the control signals. Although micro programmed control seems to be advantageous to CISC machines, since CISC requires systematic development of sophisticated control signals, there is no intrinsic difference between these 2 control mechanisms.
Hardwired control is a control mechanism to generate control signals by using appropriate finite state machine (FSM). The pair of “microinstruction-register” and “control storage address register” can be regarded as a “state register” for the hardwired control. Note that the control storage can be regarded as a kind of combinational logic circuit. We can assign any 0, 1 values to each output corresponding to each address, which can be regarded as the input for a combinational logic circuit.
4.
5. Is the process by which a computer or a virtual machine performs the instructions of a computer program. The instructions in the program trigger sequences of simple actions on the executing machine.
Fetch; first step the CPU carries out is to fetch some data and instructions (program) from main memory then store them in its own internal temporary memory areas. These memory areas are called ‘registers’. The computer fetches the instruction from its memory and then executes it. This is done repeatedly from when the computer is booted up to when it is shut down. When the instruction has been decoded, the CPU can carry out the action that is needed. This is called executing the instruction. The CPU is designed to understand a set of instructions - the instruction set.
Decode; The next step is for the CPU to make sense of the instruction it has just fetched. This process is called ‘decode’. The CPU is designed to understand a specific set of commands.
These are called the ‘instruction set’ of the CPU. Each make of CPU has a different instruction set. The CPU decodes the instruction and prepares various areas within the chip in readiness of the next step.
Execute; This is the part of the cycle when data processing actually takes place. The instruction is carried out upon the data (executed). The result of this processing is stored in yet another register. Once the execute stage is complete, the CPU sets itself up to begin another cycle once more.


6.1: Antipodal Points
Recall, the unit \(2\)-sphere \(\mathbb{S}^2\) consists of all points \((a,b,c)\) in \(\mathbb{R}^3\) for which \(a^2 + b^2 + c^2 = 1\text{,}\) and \(\mathbb{S}^2\) may be mapped onto the extended plane by the stereographic projection map \(\phi: \mathbb{S}^2 \to \mathbb{C}^+\) defined by
\[ \phi(a,b,c) = \begin{cases}\dfrac{a}{1-c}+\dfrac{b}{1-c}i & \text{ if \(c \neq 1\); } \\ \infty & \text{ if \(c=1\) } \end{cases}\text{.} \]
Two distinct points on a sphere are called diametrically opposed points if they are on the same line through the center of the sphere. Diametrically opposed points on the sphere are also called antipodal points. If \(P = (a,b,c)\) is on \(\mathbb{S}^2\) then the point diametrically opposed to it is \(-P=(-a,-b,-c)\text{.}\) It turns out that \(\phi\) maps diametrically opposed points of the sphere to points in the extended plane that satisfy a particular equation.
Definition: Antipodal Points
Two points \(z\) and \(w\) in \(\mathbb{C}^+\) are called antipodal points if they satisfy the equation
\[ z \cdot \overline{w} = -1\text{.} \]
Furthermore, we set \(0\) and \(\infty\) to be antipodal points in \(\mathbb{C}^+\text{.}\) If \(z\) and \(w\) are antipodal points, we say \(w\) is antipodal to \(z\text{,}\) and vice versa.
Each \(z\) in \(\mathbb{C}^+\) has a unique antipodal point, \(z_a\text{,}\) given as follows:
\[ z_a =\begin{cases}-1/\overline{z} & \text{ if \(z \neq 0, \infty\); } \\ \infty & \text{ if \(z = 0\); } \\ 0 & \text{ if \(z=\infty\) } \end{cases} \text{.} \]
Since \(-\dfrac{1}{\overline{z}} = -\dfrac{1}{|z|^2}z\text{,}\) we note that \(z_a\) is a scaled version of \(z\text{,}\) so that \(z\) and \(z_a\) live on the same Euclidean line through the origin. Again, since \(0\) and \(\infty\) are antipodal points, this notion extends to all points \(z \in \mathbb{C}^+\text{.}\)
Given two diametrically opposed points on the unit sphere, their image points under stereographic projection are antipodal points in \(\mathbb{C}^+\text{.}\)
- Proof
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First note that the north pole \(N = (0,0,1)\) and the south pole \(S = (0,0,-1)\) are diametrically opposed points and they get sent by \(\phi\) to \(\infty\) and \(0\), respectively, in \(\mathbb{C}^+\text{;}\) so the lemma holds in this case.
Now suppose \(P = (a,b,c)\) is a point on the sphere with \(|c| \neq 1\text{,}\) and \(Q = (-a,-b,-c)\) is diametrically opposed to \(P\text{.}\) The images of these two points under stereographic projection are
\[ \phi(P) = \dfrac{a}{1-c} + \dfrac{b}{1-c} i~~\text{and}~~\phi(Q) = \dfrac{-a}{1+c} - \dfrac{b}{1+c} i\text{.} \]
If we expand the following product
\[ \phi(P) \cdot \overline{\phi(Q)} = \left[\dfrac{a}{1-c} +\dfrac{b}{1-c} i\right]\cdot \left[\dfrac{-a}{1+c} + \dfrac{b}{1+c} i\right]\text{,} \]
we obtain
\[ \phi(P) \cdot \overline{\phi(Q)} = -\dfrac{a^2 + b^2}{1-c^2} \]
which reduces to \(-1\) since \(a^2+b^2+c^2=1\text{.}\)
Thus, diametrically opposed points on the sphere get mapped via stereographic projection to antipodal points in the extended plane.
A cline \(C\) in \(\mathbb{C}^+\) is called a great circle if whenever \(z\) is on \(C\text{,}\) so is its antipodal point \(z_a\text{.}\) Some great circles are drawn in Figure \(6.1.1\). We note that the unit circle is a great circle in \(\mathbb{C}^+\) as is any line through the origin.
To construct a great circle in the extended plane, it is enough to ensure that it passes through one particular pair of antipodal points. This can be proved with the aid of stereographic projection, but an alternative proof is given below, one that does not leave the plane, but rather uses a proposition from Book III of Euclid's Elements .
If a cline in \(\mathbb{C}^+\) contains two antipodal points then it is a great circle.
- Proof
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Suppose \(C\) is a cline in \(\mathbb{C}^+\) containing antipodal points \(p\) and \(p_a\text{,}\) and suppose \(q\) is any other point on \(C\text{.}\) We show \(q_a\) is also on \(C\text{.}\)
If \(C\) is a line, it must go through the origin since \(p\) and \(p_a\) are on the same line through the origin. Since \(q\) and \(q_a\) are also on the same line through the origin, if \(q\) is on \(C\) then \(q_a\) is too.
If \(C\) is a circle, then the origin of the plane is in the interior of \(C\) and the chord \(pp_a\) contains the origin, as pictured in Figure 6.1.5. The line through \(q\) and the origin intersects \(C\) at another point, say \(w\text{.}\) We show \(w=q_a\text{.}\)
The intersecting chords theorem (Book III, Proposition 35 of Euclid's Elements ) applied to this figure tells us that \(|p|\cdot |p_a| = |q|\cdot |w|\text{.}\) (We leave the proof to Exercise \(6.1.7\). The proof essentially follows that of Lemma \(3.2.1\), except we consider a point inside the given circle.) As antipodal points, \(|p|\cdot|p_a| = 1\text{,}\) and it follows that \(|w|=\dfrac{1}{|q|}\text{.}\) Since the segment \(wq\) contains the origin, it follows that \(w\) is the point antipodal to \(q\text{.}\)
The following theorem tells us that inversions will play a central role in elliptic geometry, just as they do in hyperbolic geometry.
Theorem \(6.1.1\)
Reflection of \(\mathbb{S}^2\) about a great circle corresponds via stereographic projection to inversion about a great circle in \(\mathbb{C}^+\text{.}\)
We work through the relationship in one case, and refer the interested reader to [10] for the general proof.
Example \(6.1.1\): Reflection of \(\mathbb{S}^2\) about the Equator.
We argue that reflection of \(\mathbb{S}^2\) about the equator corresponds to inversion about the unit circle.
First of all, stereographic projection sends the equator of \(\mathbb{S}^2\) to the unit circle in \(\mathbb{C}^+\text{.}\) Now, reflection of \(\mathbb{S}^2\) across the equator sends the point \(P = (a,b,c)\) to the point \(P^* = (a,b,-c)\text{.}\) We must argue that \(\phi(P)\) and \(\phi(P^*)\) are symmetric with respect to the unit circle.
If \(|c| = 1\) then \(P\) and \(P^*\) correspond to the north and south poles, and their image points are \(0\) and \(\infty\text{,}\) and these points are symmetric with respect to the unit circle. So, assume \(|c| \neq 1\text{.}\) Notice that
\[ \phi(P) = \dfrac{a}{1-c}+\dfrac{b}{1-c}i ~~\text{and }~~\phi(P^*) = \dfrac{a}{1+c}+\dfrac{b}{1+c}i \]
are on the same ray beginning at the origin. Indeed, one is the positive scalar multiple of the other:
\[ \phi(P^*) = \dfrac{1-c}{1+c}\phi(P)\text{.} \]
Moreover,
\begin{align*} |\phi(P)|\cdot|\phi(P^*)| & = \dfrac{1-c}{1+c}\cdot |\phi(P)|^2\\ & = \dfrac{1-c}{1+c}\cdot \dfrac{a^2+b^2}{(1-c)^2}\\ & = \dfrac{a^2+b^2}{1-c^2}\\ & = 1\text{.} \end{align*}
Again, the last equality holds because \(a^2 + b^2 + c^2 = 1\text{.}\) Thus, \(\phi(P)\) and \(\phi(P^*)\) are symmetric with respect to the unit circle. It follows that inversion in the unit circle, \(i_{\mathbb{S}^1}\text{,}\) corresponds to reflection of \(\mathbb{S}^2\) across the equator, call this map \(R\text{,}\) by the equation
\[ i_{\mathbb{S}^1} \circ \phi = \phi \circ R\text{.} \]
We end the section with one more feature of the stereographic projection map. The proof can be found in [10] .
Theorem \(6.1.2\)
The image of a circle on \(\mathbb{S}^2\) via stereographic projection is a cline in \(\mathbb{C}^+\text{.}\) Moreover, the pre-image of a circle in \(\mathbb{C}^+\) is a circle on \(\mathbb{S}^2\text{.}\) The pre-image of a line in \(\mathbb{C}^+\) is a circle on \(\mathbb{S}^2\) that goes through \(N= (0,0,1)\text{.}\)
In fact, one can offer a constructive proof of this theorem. A circle on \(\mathbb{S}^2\) can be represented as the intersection of \(\mathbb{S}^2\) with a plane \(Ax + By + Cz + D = 0\) in 3-dimensional space. One can show that the circle in \(\mathbb{S}^2\) defined by \(Ax + By + Cz + D = 0\) gets mapped by \(\phi\) to the cline \((C+D)(u^2 + v^2)+2Au + 2Bv + (D-C)=0\) in the plane (described via \(u,v\) cartesian coordinates); conversely the circle \(|w - w_0| = r\) in \(\mathbb{C}\) gets mapped by the inverse function \(\phi^{-1}\) to the circle in \(\mathbb{S}^2\) given by the plane \(-2x_0 x - 2y_o y + (1 - |w_0|^2+r^2)z + (1 + |w_0|^2 - r^2) = 0\) where \(w_0 = x_0 + y_0i\text{.}\)
Example \(6.1.2\): The Image of a Circle Under \(\phi\).
Consider the circle on \(\mathbb{S}^2\) defined by the vertical plane \(x = -\dfrac{1}{2}\text{.}\) In standard form, this plane has constants \(A = 2, B = C = 0\text{,}\) and \(D = 1\text{,}\) so the image under \(\phi\) is the circle
\[ (u^2+v^2)+4u+1 = 0\text{.} \]
Completing the square we obtain the circle
\[ (u+2)^2 + v^2 = 3 \]
having center \((-2,0)\) and radius \(\sqrt{3}\text{.}\)
Example \(6.1.3\): The Pre-Image of a Circle Under \(\phi\).
The pre-image of the circle \(|w - (3 + 2i)| = 4\) in \(\mathbb{C}\) is the circle on \(\mathbb{S}^2\) defined by the plane
\[ -2 \cdot 3 x - 2\cdot 2 y + (1 - 13+4) z + (1 + 13 - 4) = 0 \]
or
\[ -3x - 2y - 4z + 5 = 0\text{.} \]
Exercises
Constructing an antipodal point . Suppose \(z\) is a point inside the unit circle. Prove that the following construction, which is depicted in Figure 6.1.11, gives \(z_a\text{,}\) the point antipodal to \(z\text{:}\) (1) Draw the line through \(z\) and the origin; (2) draw the line through the origin perpendicular to line (1), and let \(T\) be on line (2) and the unit circle; (3) construct the segment \(zT\text{;}\) (4) construct the perpendicular to segment (3) at point \(T\text{.}\) Line (4) intersects line (1) at the point \(z_a\text{.}\) Use similar triangles to prove that \(z_a = -\dfrac{1}{|z|^2}z.\)
Explain why any great circle in \(\mathbb{C}^+\) either contains \(0\) or has \(0\) in its interior.
Characterize those great circles in \(\mathbb{C}^+\) that are actually Euclidean lines.
Prove that reflection of \(\mathbb{S}^2\) across the great circle through \((0,0,1)\text{,}\) \((0,0,-1)\text{,}\) and \((1,0,0)\) corresponds via stereographic projection to reflection of \(\mathbb{C}^+\) across the real axis.
Determine the image under \(\phi\) of the circle \(z = \dfrac{1}{2}\) on the unit sphere.
Explain why reflection of \(\mathbb{S}^2\) across any longitudinal great circle (i.e. a great circle through the north and south poles) corresponds to reflection of \(\mathbb{C}^+\) across a line through the origin.
Prove the intersecting chords theorem in two parts.
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Suppose \(C\) is a circle with radius \(r\) centered at \(o\text{.}\) Suppose \(p\) is a point inside \(C\) and a line through \(p\) intersects \(C\) at points \(m\) and \(n\text{,}\) as pictured below. If we let \(s = |p-o|\) prove that \(|p-m|\cdot|p-n|=r^2-s^2\text{.}\)
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Prove the intersecting chords theorem: If \(mn\) and \(ab\) are any two chords of \(C\) passing through a given interior point \(p\text{,}\) then
\[ |m-p|\cdot|a-p|=|m-p|\cdot|b-p|\text{.} \]


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bar
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a force of 100,000 Newtons acting on a surface area of 1 square meter; the average pressure of Earth’s atmosphere at sea level is 1.013 bars
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basalt
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igneous rock produced by the cooling of lava; makes up most of Earth’s oceanic crust and is found on other planets that have experienced extensive volcanic activity
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convection
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movement caused within a gas or liquid by the tendency of hotter, and therefore less dense material, to rise and colder, denser material to sink under the influence of gravity, which consequently results in transfer of heat
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core
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the central part of the planet; consists of higher density material
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crust
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the outer layer of a terrestrial planet
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fault
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in geology, a crack or break in the crust of a planet along which slippage or movement can take place, accompanied by seismic activity
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granite
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a type of igneous silicate rock that makes up most of Earth’s continental crust
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greenhouse effect
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the blanketing (absorption) of infrared radiation near the surface of a planet—for example, by CO
2
in its atmosphere
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greenhouse gas
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a gas in an atmosphere that absorbs and emits radiation within the thermal infrared range; on Earth, these atmospheric gases primarily include carbon dioxide, methane, and water vapor
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igneous rock
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rock produced by cooling from a molten state
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magnetosphere
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the region around a planet in which its intrinsic magnetic field dominates the interplanetary field carried by the solar wind; hence, the region within which charged particles can be trapped by the planetary magnetic field
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mantle
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the largest part of Earth’s interior; lies between the crust and the core
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mass extinction
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the sudden disappearance in the fossil record of a large number of species of life, to be replaced by fossils of new species in subsequent layers; mass extinctions are indicators of catastrophic changes in the environment, such as might be produced by a large impact on Earth
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metamorphic rock
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rock produced by physical and chemical alteration (without melting) under high temperature and pressure
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ozone
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(O
3
) a heavy molecule of oxygen that contains three atoms rather than the more normal two
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photosynthesis
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a complex sequence of chemical reactions through which some living things can use sunlight to manufacture products that store energy (such as carbohydrates), releasing oxygen as one by-product
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plate tectonics
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the motion of segments or plates of the outer layer of a planet over the underlying mantle
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primitive rock
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rock that has not experienced great heat or pressure and therefore remains representative of the original condensed materials from the solar nebula
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rift zone
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in geology, a place where the crust is being torn apart by internal forces generally associated with the injection of new material from the mantle and with the slow separation of tectonic plates
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sedimentary rock
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rock formed by the deposition and cementing of fine grains of material, such as pieces of igneous rock or the shells of living things
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seismic wave
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a vibration that travels through the interior of Earth or any other object; on Earth, these are generally caused by earthquakes
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stratosphere
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the layer of Earth’s atmosphere above the troposphere and below the ionosphere
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subduction
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the sideways and downward movement of the edge of a plate of Earth’s crust into the mantle beneath another plate
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troposphere
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the lowest level of Earth’s atmosphere, where most weather takes place
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volcano
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a place where material from a planet’s mantle erupts on its surface


13.1: Introduction to the Somatic Nervous System
Chapter Learning Objectives
After studying this chapter, you will be able to:
- Describe the components of the somatic nervous system
- Describe the structures of general and special senses
- Describe regions of the central nervous system that contribute to somatic functions
- Explain the stimulus-response motor pathway
- Describe the components of a basic reflex arc and distinguish between different types of reflexes
The somatic nervous system includes all the afferent, sensory receptors from the skin, skeletal muscles, joints and special senses which are responsible for our conscious perception of the environment. It also includes the efferent, motor division which reaches the skeletal muscles and is responsible for our voluntary motor response. In contrast, the autonomic nervous system, which is discussed in the next chapter, includes afferent sensory receptors that are from visceral organs and are responsible for unconscious perception as well as an efferent motor division that reaches cardiac muscle, smooth muscle or glands for involuntary motor responses.
Our somatic senses make us aware of our surroundings or state of our body through peripheral sensory receptors. These receptors send sensory information through cranial and spinal nerves to the central nervous system. The neurons of the central nervous system need to process the information, a step called integration. Once processed, if a motor response is required, the neurons of the central nervous system will send a signal to skeletal muscles to move. Let's take the example of touching a hot stove (Figure \(\PageIndex{1}\)). Sensory receptors in the skin sense extreme temperature and the early signs of tissue damage. This triggers an action potential (a change in electrical properties of a neuron), which travels along the sensory fiber from the skin through the dorsal spinal root to the spinal cord, and directly activates a ventral horn motor neuron. That neuron sends a signal along its axon to excite the biceps brachii, causing contraction of the muscle and flexion of the forearm at the elbow to withdraw the hand from the hot stove. The basic withdrawal reflex explained above includes afferent somatosensory input (the painful stimulus), central processing (the synapse in the spinal cord), and efferent motor output (activation of a ventral motor neuron that causes contraction of the biceps brachii).
In this chapter, we will examine all the components of the somatic nervous system. For the sensory input, we will look at sensory receptors of general and special senses and how the sensory information is sent to the CNS. Once the information is received in the appropriate region of the CNS, central processing takes place to integrate the various information. Lastly, the CNS will send a motor response to the periphery.
Contributors and Attributions
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OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology


4.2: The Middle Ground
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The thinking of the English school is often viewed as a middle ground between liberal and realist theories. Its theory involves the idea of a society of states existing at the international level. Hedley Bull, one of the core figures of the English school, agreed with the traditional theories that the international system was anarchic. However, he insisted that this does not imply there are no norms (expected behaviours), thus claiming there is a societal aspect to international politics. In this sense, states form an ‘Anarchical Society’ (Bull 1977) where a type of order does exist, based on shared norms and behaviours. Due to its central premise, the English school is often characterised as having an international society approach to IR. This describes a world that is not quite realist and not quite liberal – but rather a world that has elements of both.
Constructivism is another theory commonly viewed as a middle ground, but this time between mainstream theories and the critical theories that we will explore later. It also has some familial links with the English school. Unlike scholars from other perspectives, constructivists highlight the importance of values and shared interests between individuals who interact on the global stage. Alexander Wendt, a prominent constructivist, described the relationship between agents (individuals) and structures (such as the state) as one in which structures not only constrain agents but also construct their identities and interests. His famous phrase ‘anarchy is what states make of it’ (Wendt 1992) sums this up well. Another way to explain this, and to explain the core of constructivism, is that the essence of international relations exists in the interactions between people. After all, states do not interact; it is agents of those states, such as politicians and diplomats, who interact. As those interacting on the world stage have accepted international anarchy as the defining principle, it has become part of our reality. However, if anarchy is what we make of it, then different states can perceive anarchy differently and the qualities of anarchy can even change over time. International anarchy could even be replaced with a different system if a critical mass of other individuals (and by proxy the states they represent) accepted the idea. To understand constructivism is to understand that ideas, or ‘norms’ as they are often called, have power. IR is, then, a never-ending journey of change chronicling the accumulation of the accepted norms of the past and the emerging norms of the future. As such, constructivists seek to study this process.


5.6: Memory and Attention
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If we want to remember something tomorrow, we have to consolidate it into long-term memory today. Long-term memory is the final, semi-permanent stage of memory. Unlike sensory and short-term memory, long-term memory has a theoretically infinite capacity, and information can remain there indefinitely. Long-term memory has also been called reference memory, because an individual must refer to the information in long-term memory when performing almost any task. Long-term memory can be broken down into two categories: explicit and implicit memory.
Explicit Memory
Explicit memory , also known as conscious or declarative memory , involves memory of facts, concepts, and events that require conscious recall of the information. In other words, the individual must actively think about retrieving the information from memory. This type of information is explicitly stored and retrieved—hence its name. Explicit memory can be further subdivided into semantic memory , which concerns facts, and episodic memory, which concerns primarily personal or autobiographical information.
Episodic Memory
Episodic memory is used for more contextualized memories. They are generally memories of specific moments, or episodes, in one's life. As such, they include sensations and emotions associated with the event, in addition to the who, what, where, and when of what happened. An example of an episodic memory would be recalling your family's trip to the beach. Autobiographical memory (memory for particular events in one's own life) is generally viewed as either equivalent to, or a subset of, episodic memory. One specific type of autobiographical memory is a flashbulb memory, which is a highly detailed, exceptionally vivid "snapshot" of the moment and circumstances in which a piece of surprising and consequential (or emotionally arousing) news was heard. For example, many people remember exactly where they were and what they were doing when they heard of the terrorist attacks on September 11, 2001. This is because it is a flashbulb memory.
Semantic and episodic memory are closely related; memory for facts can be enhanced with episodic memories associated with the fact, and vice versa. For example, the answer to the factual question "Are all apples red?" might be recalled by remembering the time you saw someone eating a green apple. Likewise, semantic memories about certain topics, such as football, can contribute to more detailed episodic memories of a particular personal event, like watching a football game. A person that barely knows the rules of football will remember the various plays and outcomes of the game in much less detail than a football expert.
Implicit Memory
In contrast to explicit (conscious) memory, implicit (also called "unconscious" or "procedural") memory involves procedures for completing actions. These actions develop with practice over time. Athletic skills are one example of implicit memory. You learn the fundamentals of a sport, practice them over and over, and then they flow naturally during a game. Rehearsing for a dance or musical performance is another example of implicit memory. Everyday examples include remembering how to tie your shoes, drive a car, or ride a bicycle. These memories are accessed without conscious awareness—they are automatically translated into actions without us even realizing it. As such, they can often be difficult to teach or explain to other people. Implicit memories differ from the semantic scripts described above in that they are usually actions that involve movement and motor coordination, whereas scripts tend to emphasize social norms or behaviors.
Short-Term Memory Storage
Short-term memory is the ability to hold information for a short duration of time (on the order of seconds). In the process of encoding, information enters the brain and can be quickly forgotten if it is not stored further in the short-term memory. George A. Miller suggested that the capacity of short-term memory storage is approximately seven items plus or minus two, but modern researchers are showing that this can vary depending on variables like the stored items' phonological properties. When several elements (such as digits, words, or pictures) are held in short-term memory simultaneously, their representations compete with each other for recall, or degrade each other. Thereby, new content gradually pushes out older content, unless the older content is actively protected against interference by rehearsal or by directing attention to it.
Information in the short-term memory is readily accessible, but for only a short time. It continuously decays, so in the absence of rehearsal (keeping information in short-term memory by mentally repeating it) it can be forgotten.
Long-Term Memory Storage
In contrast to short-term memory, long-term memory is the ability to hold semantic information for a prolonged period of time. Items stored in short-term memory move to long-term memory through rehearsal, processing, and use. The capacity of long-term memory storage is much greater than that of short-term memory, and perhaps unlimited. However, the duration of long-term memories is not permanent; unless a memory is occasionally recalled, it may fail to be recalled on later occasions. This is known as forgetting.
Long-term memory storage can be affected by traumatic brain injury or lesions. Amnesia, a deficit in memory, can be caused by brain damage. Anterograde amnesia is the inability to store new memories; retrograde amnesia is the inability to retrieve old memories. These types of amnesia indicate that memory does have a storage process. 25


38.11: Joints and Skeletal Movement - Movement at Synovial Joints
- Differentiate among the types of movements possible at synovial joints
Movement at Synovial Joints
The range of movement allowed by synovial joints is fairly wide. These movements can be classified as: gliding, angular, rotational, or special movement.
Gliding Movement
Gliding movements occur as relatively flat bone surfaces move past each other. They produce very little rotation or angular movement of the bones. The joints of the carpal and tarsal bones are examples of joints that produce gliding movements.
Angular Movement
Angular movements are produced by changing the angle between the bones of a joint. There are several different types of angular movements, including flexion, extension, hyperextension, abduction, adduction, and circumduction. Flexion, or bending, occurs when the angle between the bones decreases. Moving the forearm upward at the elbow or moving the wrist to move the hand toward the forearm are examples of flexion. In extension, the opposite of flexion, the angle between the bones of a joint increases. Straightening a limb after flexion is an example of extension. Extension past the normal anatomical position is referred to as hyperextension. This includes moving the neck back to look upward or bending the wrist so that the hand moves away from the forearm.
Abduction occurs when a bone moves away from the midline of the body. Examples of abduction include moving the arms or legs laterally to lift them straight out to the side. Adduction is the movement of a bone toward the midline of the body. Movement of the limbs inward after abduction is an example of adduction. Circumduction is the movement of a limb in a circular motion, as in swinging an arm around.
Rotational Movement
Rotational movement is the movement of a bone as it rotates around its longitudinal axis. Rotation can be toward the midline of the body, which is referred to as medial rotation, or away from the midline of the body, which is referred to as lateral rotation. Movement of the head from side to side is an example of rotation.
Special Movements
Some movements that cannot be classified as gliding, angular, or rotational are called special movements. Inversion involves moving the soles of the feet inward, toward the midline of the body. Eversion, the opposite of inversion, involves moving of the sole of the foot outward, away from the midline of the body. Protraction is the anterior movement of a bone in the horizontal plane. Retraction occurs as a joint moves back into position after protraction. Protraction and retraction can be seen in the movement of the mandible as the jaw is thrust outwards and then back inwards. Elevation is the movement of a bone upward, such as shrugging the shoulders, lifting the scapulae. Depression is the opposite of elevation and involves moving the bone downward, such as after the shoulders are shrugged and the scapulae return to their normal position from an elevated position. Dorsiflexion is a bending at the ankle such that the toes are lifted toward the knee. Plantarflexion is a bending at the ankle when the heel is lifted, such as when standing on the toes. Supination is the movement of the radius and ulna bones of the forearm so that the palm faces forward or up. Pronation is the opposite movement, in which the palm faces backward or down. Opposition is the movement of the thumb toward the fingers of the same hand, making it possible to grasp and hold objects.
Key Points
- Gliding movements occur as relatively flat bone surfaces move past each other, but they produce very little movement of the bones.
- Angular movements are produced when the angle between the bones of a joint changes; they include flexion, extension, hyperextension, abduction, adduction, and circumduction.
- Rotational movement involves moving the bone around its longitudinal axis; this can be movement toward the midline of the body (medial rotation) or away from the midline of the body (lateral rotation).
- Special movements are all the other movements that cannot be classified as gliding, angular, or rotational; these movements include inversion, eversion, protraction, and retraction.
- Other special movements include elevation, depression, supination, and pronation.
Key Terms
- adduction : the movement of a bone toward the midline of the body
- abduction : moving a bone away from the midline of the body
- supination : the action of rotating the forearm so that the palm of the hand is turned up or forward
- pronation : the action of rotating the forearm so that the palm of the hand is turned down or back


4.8: Respiratory System Learning Activities
Interactive Learning Activity: Label the anatomy of the respiratory system using this drag and drop activity.
Query \(\PageIndex{1}\)
Interactive Learning Activity: Study terms discussed in this chapter using these flashcards.
Query \(\PageIndex{2}\)
Interactive Learning Activity: Test your comprehension of respiratory system terms and concepts.
Query \(\PageIndex{3}\)
Interactive Learning Activity: Using the sample documentation provided, drag the terms on the right to their appropriate spaces within the documentation.
Query \(\PageIndex{4}\)
Interactive Learning Activity: Using the sample documentation provided, drag the terms on the right to their appropriate spaces within the documentation.
Query \(\PageIndex{5}\)
Interactive Learning Activity: Using the sample documentation provided, drag the terms on the right to their appropriate spaces within the documentation.
Query \(\PageIndex{6}\)
Interactive Learning Activity: Practice identifying and defining word parts for terms discussed in this chapter:
Query \(\PageIndex{7}\)
You can also print this as a Chapter 4 Practice Worksheet and check your answers with this Answer Key PDF .


20.5: Transmutation and Nuclear Energy
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Learning Objectives
By the end of this section, you will be able to:
- Describe the synthesis of transuranium nuclides
- Explain nuclear fission and fusion processes
- Relate the concepts of critical mass and nuclear chain reactions
- Summarize basic requirements for nuclear fission and fusion reactors
After the discovery of radioactivity, the field of nuclear chemistry was created and developed rapidly during the early twentieth century. A slew of new discoveries in the 1930s and 1940s, along with World War II, combined to usher in the Nuclear Age in the mid-twentieth century. Scientists learned how to create new substances, and certain isotopes of certain elements were found to possess the capacity to produce unprecedented amounts of energy, with the potential to cause tremendous damage during war, as well as produce enormous amounts of power for society’s needs during peace.
Synthesis of Nuclides
Nuclear transmutation is the conversion of one nuclide into another. It can occur by the radioactive decay of a nucleus, or the reaction of a nucleus with another particle. The first manmade nucleus was produced in Ernest Rutherford’s laboratory in 1919 by a transmutation reaction, the bombardment of one type of nuclei with other nuclei or with neutrons. Rutherford bombarded nitrogen atoms with high-speed α particles from a natural radioactive isotope of radium and observed protons resulting from the reaction:
\[\ce{_7^{14} N + _2^4 He -> _8^{17} O + _1^1H} \nonumber\]
The \(\ce{_8^{17}O}\) and \(\ce{_7^{14}N}\) nuclei that are produced are stable, so no further (nuclear) changes occur.
To reach the kinetic energies necessary to produce transmutation reactions, devices called particle accelerators are used. These devices use magnetic and electric fields to increase the speeds of nuclear particles. In all accelerators, the particles move in a vacuum to avoid collisions with gas molecules.
When neutrons are required for transmutation reactions, they are usually obtained from radioactive decay reactions or from various nuclear reactions occurring in nuclear reactors. The Chemistry in Everyday Life feature that follows discusses a famous particle accelerator that made worldwide news.
Chemistry in Everyday Life: CERN Particle Accelerator
Located near Geneva, the CERN (“Conseil Européen pour la Recherche Nucléaire,” or European Council for Nuclear Research) Laboratory is the world’s premier center for the investigations of the fundamental particles that make up matter. It contains the 27-kilometer (17 mile) long, circular Large Hadron Collider (LHC), the largest particle accelerator in the world (Figure \(\PageIndex{1}\)). In the LHC, particles are boosted to high energies and are then made to collide with each other or with stationary targets at nearly the speed of light. Superconducting electromagnets are used to produce a strong magnetic field that guides the particles around the ring. Specialized, purpose-built detectors observe and record the results of these collisions, which are then analyzed by CERN scientists using powerful computers.
In 2012, CERN announced that experiments at the LHC showed the first observations of the Higgs boson, an elementary particle that helps explain the origin of mass in fundamental particles. This long-anticipated discovery made worldwide news and resulted in the awarding of the 2013 Nobel Prize in Physics to François Englert and Peter Higgs, who had predicted the existence of this particle almost 50 years previously.
Link to Learning
Famous physicist Brian Cox talks about his work on the Large Hadron Collider at CERN, providing an entertaining and engaging tour of this massive project and the physics behind it.
View a short video from CERN, describing the basics of how its particle accelerators work.
Prior to 1940, the heaviest-known element was uranium, whose atomic number is 92. Now, many artificial elements have been synthesized and isolated, including several on such a large scale that they have had a profound effect on society. One of these—element 93, neptunium (Np)—was first made in 1940 by McMillan and Abelson by bombarding uranium-238 with neutrons. The reaction creates unstable uranium-239, with a half-life of 23.5 minutes, which then decays into neptunium-239. Neptunium-239 is also radioactive, with a half-life of 2.36 days, and it decays into plutonium-239. The nuclear reactions are:
\[\ce{_{92}^{238} U + _0^1 n -> _{92}^{239} U} \nonumber\]
\[\begin{aligned}
& { }_{92}^{239} \mathrm{U} \longrightarrow{ }_{93}^{239} \mathrm{~Np}+{ }_{-1}^0 \mathrm{e} \quad \text { half-life }=23.5 \mathrm{~min} \\
& { }_{93}^{239} \mathrm{~Np} \longrightarrow{ }_{94}^{239} \mathrm{Pu}+{ }_{-1}^0 \mathrm{e} \quad \text { half-life }=2.36 \text { days }
\end{aligned}\]
Plutonium is now mostly formed in nuclear reactors as a byproduct during the fission of U-235. Additional neutrons are released during this fission process (see the next section), some of which combine with U-238 nuclei to form uranium-239; this undergoes β decay to form neptunium-239, which in turn undergoes β decay to form plutonium-239 as illustrated in the preceding three equations. These processes are summarized in the equation:
\[{ }_{92}^{238} \mathrm{U}+{ }_0^1 \mathrm{n} \longrightarrow{ }_{92}^{239} \mathrm{U} \xrightarrow{\beta^{-}}{ }_{93}^{239} \mathrm{~Np} \xrightarrow{\beta^{-}}{ }_{94}^{239} \mathrm{Pu}\]
Heavier isotopes of plutonium—Pu-240, Pu-241, and Pu-242—are also produced when lighter plutonium nuclei capture neutrons. Some of this highly radioactive plutonium is used to produce military weapons, and the rest presents a serious storage problem because they have half-lives from thousands to hundreds of thousands of years.
Although they have not been prepared in the same quantity as plutonium, many other synthetic nuclei have been produced. Nuclear medicine has developed from the ability to convert atoms of one type into other types of atoms. Radioactive isotopes of several dozen elements are currently used for medical applications. The radiation produced by their decay is used to image or treat various organs or portions of the body, among other uses.
The elements beyond element 92 (uranium) are called transuranium elements . As of this writing, 22 transuranium elements have been produced and officially recognized by IUPAC; several other elements have formation claims that are waiting for approval. Some of these elements are shown in Table 21.3.
| Name | Symbol | Atomic Number | Reaction |
|---|---|---|---|
| americium | Am | 95 | \({ }_{94}^{239} \mathrm{Pu}+{ }_0^1 \mathrm{n} \longrightarrow{ }_{95}^{240} \mathrm{Am}+{ }_{-1}^0 \mathrm{e}\) |
| curium | Cm | 96 | \({ }_{94}^{239} \mathrm{Pu}+{ }_2^4 \mathrm{He} \longrightarrow{ }_{96}^{242} \mathrm{Cm}+{ }_0^1 \mathrm{n}\) |
| californium | Cf | 98 | \({ }_{96}^{242} \mathrm{Cm}+{ }_2^4 \mathrm{He} \longrightarrow{ }_{98}^{245} \mathrm{Cf}+{ }_0^1 \mathrm{n}\) |
| einsteinium | Es | 99 | \({ }_{96}^{242} \mathrm{Cm}+{ }_2^4 \mathrm{He} \longrightarrow{ }_{98}^{245} \mathrm{Cf}+{ }_0^1 \mathrm{n}\) |
| mendelevium | Md | 101 | \({ }_{99}^{253} \mathrm{Es}+{ }_2^4 \mathrm{He} \longrightarrow{ }_{101}^{256} \mathrm{Md}+{ }_0^1 \mathrm{n}\) |
| nobelium | No | 102 | \({ }_{96}^{246} \mathrm{Cm}+{ }_6^{12} \mathrm{C} \longrightarrow{ }_{102}^{254} \mathrm{No}+4{ }_0^1 \mathrm{n}\) |
| rutherfordium | Rf | 104 | \({ }_{98}^{249} \mathrm{Cf}+{ }_6^{12} \mathrm{C} \longrightarrow{ }_{104}^{257} \mathrm{Rf}+4{ }_0^1 \mathrm{n}\) \({ }_{98}^{249} \mathrm{Cf}+{ }_8^{18} \mathrm{O} \longrightarrow{ }_{106}^{263} \mathrm{Sg}+4{ }_0^1 \mathrm{n}\) |
| seaborgium | Sg | 106 | \({ }_{82}^{206} \mathrm{~Pb}+{ }_{24}^{54} \mathrm{Cr} \longrightarrow{ }_{106}^{257} \mathrm{Sg}+3{ }_0^1 \mathrm{n}\) |
| meitnerium | Mt | 107 | \({ }_{83}^{209} \mathrm{Bi}+{ }_{26}^{58} \mathrm{Fe} \longrightarrow{ }_{109}^{266} \mathrm{Mt}+{ }_0^1 \mathrm{n}\) |
Nuclear Fission
Many heavier elements with smaller binding energies per nucleon can decompose into more stable elements that have intermediate mass numbers and larger binding energies per nucleon—that is, mass numbers and binding energies per nucleon that are closer to the “peak” of the binding energy graph near 56 (see Figure 21.3). Sometimes neutrons are also produced. This decomposition is called fission , the breaking of a large nucleus into smaller pieces. The breaking is rather random with the formation of a large number of different products. Fission usually does not occur naturally, but is induced by bombardment with neutrons. The first reported nuclear fission occurred in 1939 when three German scientists, Lise Meitner, Otto Hahn, and Fritz Strassman, bombarded uranium-235 atoms with slow-moving neutrons that split the U-238 nuclei into smaller fragments that consisted of several neutrons and elements near the middle of the periodic table. Since then, fission has been observed in many other isotopes, including most actinide isotopes that have an odd number of neutrons. A typical nuclear fission reaction is shown in Figure \(\PageIndex{2}\).
Among the products of Meitner, Hahn, and Strassman’s fission reaction were barium, krypton, lanthanum, and cerium, all of which have nuclei that are more stable than uranium-235. Since then, hundreds of different isotopes have been observed among the products of fissionable substances. A few of the many reactions that occur for U-235, and a graph showing the distribution of its fission products and their yields, are shown in Figure \(\PageIndex{3}\). Similar fission reactions have been observed with other uranium isotopes, as well as with a variety of other isotopes such as those of plutonium.
A tremendous amount of energy is produced by the fission of heavy elements. For instance, when one mole of U-235 undergoes fission, the products weigh about 0.2 grams less than the reactants; this “lost” mass is converted into a very large amount of energy, about 1.8 10 10 kJ per mole of U-235. Nuclear fission reactions produce incredibly large amounts of energy compared to chemical reactions. The fission of 1 kilogram of uranium-235, for example, produces about 2.5 million times as much energy as is produced by burning 1 kilogram of coal.
As described earlier, when undergoing fission U-235 produces two “medium-sized” nuclei, and two or three neutrons. These neutrons may then cause the fission of other uranium-235 atoms, which in turn provide more neutrons that can cause fission of even more nuclei, and so on. If this occurs, we have a nuclear chain reaction (see Figure \(\PageIndex{4}\)). On the other hand, if too many neutrons escape the bulk material without interacting with a nucleus, then no chain reaction will occur.
Material that can sustain a nuclear fission chain reaction is said to be fissile or fissionable . (Technically, fissile material can undergo fission with neutrons of any energy, whereas fissionable material requires high-energy neutrons.) Nuclear fission becomes self-sustaining when the number of neutrons produced by fission equals or exceeds the number of neutrons absorbed by splitting nuclei plus the number that escape into the surroundings. The amount of a fissionable material that will support a self-sustaining chain reaction is a critical mass . An amount of fissionable material that cannot sustain a chain reaction is a subcritical mass . An amount of material in which there is an increasing rate of fission is known as a supercritical mass . The critical mass depends on the type of material: its purity, the temperature, the shape of the sample, and how the neutron reactions are controlled (Figure \(\PageIndex{5}\)).
An atomic bomb (Figure \(\PageIndex{6}\)) contains several pounds of fissionable material, \({ }_{92}^{235} \mathrm{U}\) or \({ }_{94}^{239} \mathrm{Pu}\) a source of neutrons, and an explosive device for compressing it quickly into a small volume. When fissionable material is in small pieces, the proportion of neutrons that escape through the relatively large surface area is great, and a chain reaction does not take place. When the small pieces of fissionable material are brought together quickly to form a body with a mass larger than the critical mass, the relative number of escaping neutrons decreases, and a chain reaction and explosion result.
Fission Reactors
Chain reactions of fissionable materials can be controlled and sustained without an explosion in a nuclear reactor (Figure \(\PageIndex{7}\)). Any nuclear reactor that produces power via the fission of uranium or plutonium by bombardment with neutrons must have at least five components: nuclear fuel consisting of fissionable material, a nuclear moderator, reactor coolant, control rods, and a shield and containment system. We will discuss these components in greater detail later in the section. The reactor works by separating the fissionable nuclear material such that a critical mass cannot be formed, controlling both the flux and absorption of neutrons to allow shutting down the fission reactions. In a nuclear reactor used for the production of electricity, the energy released by fission reactions is trapped as thermal energy and used to boil water and produce steam. The steam is used to turn a turbine, which powers a generator for the production of electricity.
Nuclear Fuels
Nuclear fuel consists of a fissionable isotope, such as uranium-235, which must be present in sufficient quantity to provide a self-sustaining chain reaction. In the United States, uranium ores contain from 0.05–0.3% of the uranium oxide U 3 O 8 ; the uranium in the ore is about 99.3% nonfissionable U-238 with only 0.7% fissionable U-235. Nuclear reactors require a fuel with a higher concentration of U-235 than is found in nature; it is normally enriched to have about 5% of uranium mass as U-235. At this concentration, it is not possible to achieve the supercritical mass necessary for a nuclear explosion. Uranium can be enriched by gaseous diffusion (the only method currently used in the US), using a gas centrifuge, or by laser separation.
In the gaseous diffusion enrichment plant where U-235 fuel is prepared, UF 6 (uranium hexafluoride) gas at low pressure moves through barriers that have holes just barely large enough for UF 6 to pass through. The slightly lighter 235 UF 6 molecules diffuse through the barrier slightly faster than the heavier 238 UF 6 molecules. This process is repeated through hundreds of barriers, gradually increasing the concentration of 235 UF 6 to the level needed by the nuclear reactor. The basis for this process, Graham’s law, is described in the chapter on gases. The enriched UF 6 gas is collected, cooled until it solidifies, and then taken to a fabrication facility where it is made into fuel assemblies. Each fuel assembly consists of fuel rods that contain many thimble-sized, ceramic-encased, enriched uranium (usually UO 2 ) fuel pellets. Modern nuclear reactors may contain as many as 10 million fuel pellets. The amount of energy in each of these pellets is equal to that in almost a ton of coal or 150 gallons of oil.
Nuclear Moderators
Neutrons produced by nuclear reactions move too fast to cause fission (refer back to Figure \(\PageIndex{5}\)). They must first be slowed to be absorbed by the fuel and produce additional nuclear reactions.  Early reactors used high-purity graphite as a moderator. Modern reactors in the US exclusively use heavy water or light water (ordinary H 2 O), whereas some reactors in other countries use other materials, such as carbon dioxide, beryllium, or graphite.
Reactor Coolants
A nuclear reactor coolant is used to carry the heat produced by the fission reaction to an external boiler and turbine, where it is transformed into electricity. Two overlapping coolant loops are often used; this counteracts the transfer of radioactivity from the reactor to the primary coolant loop. All nuclear power plants in the US use water as a coolant. Other coolants include molten sodium, lead, a lead-bismuth mixture, or molten salts.
Control Rods
Nuclear reactors use control rods (Figure \(\PageIndex{8}\)) to control the fission rate of the nuclear fuel by adjusting the number of slow neutrons present to keep the rate of the chain reaction at a safe level. Control rods are made of boron, cadmium, hafnium, or other elements that are able to absorb neutrons. Boron-10, for example, absorbs neutrons by a reaction that produces lithium-7 and alpha particles:
\[{ }_5^{10} \mathrm{~B}+{ }_0^1 \mathrm{n} \longrightarrow{ }_3^7 \mathrm{Li}+{ }_2^4 \mathrm{He}\]
When control rod assemblies are inserted into the fuel element in the reactor core, they absorb a larger fraction of the slow neutrons, thereby slowing the rate of the fission reaction and decreasing the power produced. Conversely, if the control rods are removed, fewer neutrons are absorbed, and the fission rate and energy production increase. In an emergency, the chain reaction can be shut down by fully inserting all of the control rods into the nuclear core between the fuel rods.
Shield and Containment System
During its operation, a nuclear reactor produces neutrons and other radiation. Even when shut down, the decay products are radioactive. In addition, an operating reactor is thermally very hot, and high pressures result from the circulation of water or another coolant through it. Thus, a reactor must withstand high temperatures and pressures, and must protect operating personnel from the radiation. Reactors are equipped with a containment system (or shield) that consists of three parts:
- The reactor vessel, a steel shell that is 3–20-centimeters thick and, with the moderator, absorbs much of the radiation produced by the reactor
- A main shield of 1–3 meters of high-density concrete
- A personnel shield of lighter materials that protects operators from γ rays and X-rays
In addition, reactors are often covered with a steel or concrete dome that is designed to contain any radioactive materials might be released by a reactor accident.
Link to Learning
Click here to watch a 3-minute video from the Nuclear Energy Institute on how nuclear reactors work.
Nuclear power plants are designed in such a way that they cannot form a supercritical mass of fissionable material and therefore cannot create a nuclear explosion. But as history has shown, failures of systems and safeguards can cause catastrophic accidents, including chemical explosions and nuclear meltdowns (damage to the reactor core from overheating). The following Chemistry in Everyday Life feature explores three infamous meltdown incidents.
Chemistry in Everyday Life: Nuclear Accidents
The importance of cooling and containment are amply illustrated by three major accidents that occurred with the nuclear reactors at nuclear power generating stations in the United States (Three Mile Island), the former Soviet Union (Chernobyl), and Japan (Fukushima).
In March 1979, the cooling system of the Unit 2 reactor at Three Mile Island Nuclear Generating Station in Pennsylvania failed, and the cooling water spilled from the reactor onto the floor of the containment building. After the pumps stopped, the reactors overheated due to the high radioactive decay heat produced in the first few days after the nuclear reactor shut down. The temperature of the core climbed to at least 2200 °C, and the upper portion of the core began to melt. In addition, the zirconium alloy cladding of the fuel rods began to react with steam and produced hydrogen:
\[ \ce{ Zr(s) + 2H2O (g) -> ZrO2(s) + 2 H2(g)} \nonumber\]
The hydrogen accumulated in the confinement building, and it was feared that there was danger of an explosion of the mixture of hydrogen and air in the building. Consequently, hydrogen gas and radioactive gases (primarily krypton and xenon) were vented from the building. Within a week, cooling water circulation was restored and the core began to cool. The plant was closed for nearly 10 years during the cleanup process.
Although zero discharge of radioactive material is desirable, the discharge of radioactive krypton and xenon, such as occurred at the Three Mile Island plant, is among the most tolerable. These gases readily disperse in the atmosphere and thus do not produce highly radioactive areas. Moreover, they are noble gases and are not incorporated into plant and animal matter in the food chain. Effectively none of the heavy elements of the core of the reactor were released into the environment, and no cleanup of the area outside of the containment building was necessary (Figure \(\PageIndex{9}\)).
Another major nuclear accident involving a reactor occurred in April 1986, at the Chernobyl Nuclear Power Plant in Ukraine, which was still a part of the former Soviet Union. While operating at low power during an unauthorized experiment with some of its safety devices shut off, one of the reactors at the plant became unstable. Its chain reaction became uncontrollable and increased to a level far beyond what the reactor was designed for. The steam pressure in the reactor rose to between 100 and 500 times the full power pressure and ruptured the reactor. Because the reactor was not enclosed in a containment building, a large amount of radioactive material spewed out, and additional fission products were released, as the graphite (carbon) moderator of the core ignited and burned. The fire was controlled, but over 200 plant workers and firefighters developed acute radiation sickness and at least 32 soon died from the effects of the radiation. It is predicted that about 4000 more deaths will occur among emergency workers and former Chernobyl residents from radiation-induced cancer and leukemia. The reactor has since been encapsulated in steel and concrete, a now-decaying structure known as the sarcophagus. Almost 30 years later, significant radiation problems still persist in the area, and Chernobyl largely remains a wasteland.
In 2011, the Fukushima Daiichi Nuclear Power Plant in Japan was badly damaged by a 9.0-magnitude earthquake and resulting tsunami. Three reactors up and running at the time were shut down automatically, and emergency generators came online to power electronics and coolant systems. However, the tsunami quickly flooded the emergency generators and cut power to the pumps that circulated coolant water through the reactors. High-temperature steam in the reactors reacted with zirconium alloy to produce hydrogen gas. The gas escaped into the containment building, and the mixture of hydrogen and air exploded. Radioactive material was released from the containment vessels as the result of deliberate venting to reduce the hydrogen pressure, deliberate discharge of coolant water into the sea, and accidental or uncontrolled events.
An evacuation zone around the damaged plant extended over 12.4 miles away, and an estimated 200,000 people were evacuated from the area. All 48 of Japan’s nuclear power plants were subsequently shut down, remaining shuttered as of December 2014. Since the disaster, public opinion has shifted from largely favoring to largely opposing increasing the use of nuclear power plants, and a restart of Japan’s atomic energy program is still stalled (Figure \(\PageIndex{10}\)).
The energy produced by a reactor fueled with enriched uranium results from the fission of uranium as well as from the fission of plutonium produced as the reactor operates. As discussed previously, the plutonium forms from the combination of neutrons and the uranium in the fuel. In any nuclear reactor, only about 0.1% of the mass of the fuel is converted into energy. The other 99.9% remains in the fuel rods as fission products and unused fuel. All of the fission products absorb neutrons, and after a period of several months to a few years, depending on the reactor, the fission products must be removed by changing the fuel rods. Otherwise, the concentration of these fission products would increase and absorb more neutrons until the reactor could no longer operate.
Spent fuel rods contain a variety of products, consisting of unstable nuclei ranging in atomic number from 25 to 60, some transuranium elements, including plutonium and americium, and unreacted uranium isotopes. The unstable nuclei and the transuranium isotopes give the spent fuel a dangerously high level of radioactivity. The long-lived isotopes require thousands of years to decay to a safe level. The ultimate fate of the nuclear reactor as a significant source of energy in the United States probably rests on whether or not a politically and scientifically satisfactory technique for processing and storing the components of spent fuel rods can be developed.
Link to Learning
Explore the information in this link to learn about the approaches to nuclear waste management.
Nuclear Fusion and Fusion Reactors
The process of converting very light nuclei into heavier nuclei is also accompanied by the conversion of mass into large amounts of energy, a process called fusion . The principal source of energy in the sun is a net fusion reaction in which four hydrogen nuclei fuse and produce one helium nucleus and two positrons. This is a net reaction of a more complicated series of events:
\[4_1^1 \mathrm{H} \longrightarrow{ }_2^4 \mathrm{He}+2{ }_{+1}^0 \mathrm{e}^{+} \nonumber\]
A helium nucleus has a mass that is 0.7% less than that of four hydrogen nuclei; this lost mass is converted into energy during the fusion. This reaction produces about 3.6 10 11 kJ of energy per mole of \({ }_2^4 \mathrm{He}) produced. This is somewhat larger than the energy produced by the nuclear fission of one mole of U-235 (1.8 10 10 kJ), and over 3 million times larger than the energy produced by the (chemical) combustion of one mole of octane (5471 kJ).
It has been determined that the nuclei of the heavy isotopes of hydrogen, a deuteron, \({ }_1^2 \mathrm{H}\), and a triton, \({ }_1^3 \mathrm{H}\), undergo fusion at extremely high temperatures (thermonuclear fusion). They form a helium nucleus and a neutron:
\[{ }_1^2 \mathrm{H}+{ }_1^3 \mathrm{H} \longrightarrow{ }_2^4 \mathrm{He}+{ }_0^1 \mathrm{n}\]
This change proceeds with a mass loss of 0.0188 amu, corresponding to the release of 1.69 10 9 kilojoules per mole of \({ }_2^4 \mathrm{He}) formed. The very high temperature is necessary to give the nuclei enough kinetic energy to overcome the very strong repulsive forces resulting from the positive charges on their nuclei so they can collide.
Useful fusion reactions require very high temperatures for their initiation—about 15,000,000 K or more. At these temperatures, all molecules dissociate into atoms, and the atoms ionize, forming plasma. These conditions occur in an extremely large number of locations throughout the universe—stars are powered by fusion. Humans have already figured out how to create temperatures high enough to achieve fusion on a large scale in thermonuclear weapons. A thermonuclear weapon such as a hydrogen bomb contains a nuclear fission bomb that, when exploded, gives off enough energy to produce the extremely high temperatures necessary for fusion to occur.
Another much more beneficial way to create fusion reactions is in a fusion reactor , a nuclear reactor in which fusion reactions of light nuclei are controlled. Because no solid materials are stable at such high temperatures, mechanical devices cannot contain the plasma in which fusion reactions occur. Two techniques to contain plasma at the density and temperature necessary for a fusion reaction are currently the focus of intensive research efforts: containment by a magnetic field and by the use of focused laser beams (Figure \(\PageIndex{11}\)). A number of large projects are working to attain one of the biggest goals in science: getting hydrogen fuel to ignite and produce more energy than the amount supplied to achieve the extremely high temperatures and pressures that are required for fusion. At the time of this writing, there are no self-sustaining fusion reactors operating in the world, although small-scale controlled fusion reactions have been run for very brief periods.


9.5: Fractals
Fractals are mathematical sets, usually obtained through recursion, that exhibit interesting dimensional properties. We’ll explore what that sentence means through the rest of the section. For now, we can begin with the idea of self-similarity, a characteristic of most fractals.
A shape is self-similar when it looks essentially the same from a distance as it does closer up.
Self-similarity can often be found in nature. In the Romanesco broccoli pictured below [1], if we zoom in on part of the image, the piece remaining looks similar to the whole.
Likewise, in the fern frond below [2], one piece of the frond looks similar to the whole.
Similarly, if we zoom in on the coastline of Portugal [3], each zoom reveals previously hidden detail, and the coastline, while not identical to the view from further way, does exhibit similar characteristics.
This self-similar behavior can be replicated through recursion: repeating a process over and over.
Suppose that we start with a filled-in triangle. We connect the midpoints of each side and remove the middle triangle. We then repeat this process.
If we repeat this process, the shape that emerges is called the Sierpinski gasket. Notice that it exhibits self-similarity – any piece of the gasket will look identical to the whole. In fact, we can say that the Sierpinski gasket contains three copies of itself, each half as tall and wide as the original. Of course, each of those copies also contains three copies of itself.
We can construct other fractals using a similar approach. To formalize this a bit, we’re going to introduce the idea of initiators and generators.
- An initiator is a starting shape
- A generator is an arranged collection of scaled copies of the initiator
To generate fractals from initiators and generators, we follow a simple rule:
At each step, replace every copy of the initiator with a scaled copy of the generator, rotating as necessary
This process is easiest to understand through example.
Use the initiator and generator shown to create the iterated fractal.
Solution
This tells us to, at each step, replace each line segment with the spiked shape shown in the generator. Notice that the generator itself is made up of 4 copies of the initiator. In step 1, the single line segment in the initiator is replaced with the generator. For step 2, each of the four line segments of step 1 is replaced with a scaled copy of the generator:
This process is repeated to form Step 3. Again, each line segment is replaced with a scaled copy of the generator.
Notice that since Step 0 only had 1 line segment, Step 1 only required one copy of Step 0.
Since Step 1 had 4 line segments, Step 2 required 4 copies of the generator.
Step 2 then had 16 line segments, so Step 3 required 16 copies of the generator.
Step 4, then, would require \(16 \times 4 = 64\) copies of the generator.
The shape resulting from iterating this process is called the Koch curve , named for Helge von Koch who first explored it in 1904.
Notice that the Sierpinski gasket can also be described using the initiator-generator approach
Use the initiator and generator below, however only iterate on the “branches.” Sketch several steps of the iteration.
Solution
We begin by replacing the initiator with the generator. We then replace each “branch” of Step 1 with a scaled copy of the generator to create Step 2.
We can repeat this process to create later steps. Repeating this process can create intricate tree shapes [4].
Use the initiator and generator shown to produce the next two stages
- Answer
-
Using iteration processes like those above can create a variety of beautiful images evocative of nature [5][6].
More natural shapes can be created by adding in randomness to the steps.
Create a variation on the Sierpinski gasket by randomly skewing the corner points each time an iteration is made.
Solution
Suppose we start with the triangle below. We begin, as before, by removing the middle triangle. We then add in some randomness.
We then repeat this process.
Continuing this process can create mountain-like structures.
The landscape [7] below was created using fractals, then colored and textured.
In addition to visual self-similarity, fractals exhibit other interesting properties. For example, notice that each step of the Sierpinski gasket iteration removes one quarter of the remaining area. If this process is continued indefinitely, we would end up essentially removing all the area, meaning we started with a 2-dimensional area, and somehow end up with something less than that, but seemingly more than just a 1-dimensional line.
To explore this idea, we need to discuss dimension. Something like a line is 1-dimensional; it only has length. Any curve is 1-dimensional. Things like boxes and circles are 2-dimensional, since they have length and width, describing an area. Objects like boxes and cylinders have length, width, and height, describing a volume, and are 3-dimensional.
Certain rules apply for scaling objects, related to their dimension.
If I had a line with length 1, and wanted scale its length by 2, I would need two copies of the original line. If I had a line of length 1, and wanted to scale its length by 3, I would need three copies of the original.
If I had a rectangle with length 2 and height 1, and wanted to scale its length and width by 2, I would need four copies of the original rectangle. If I wanted to scale the length and width by 3, I would need nine copies of the original rectangle.
If I had a cubical box with sides of length 1, and wanted to scale its length, its width, and its height by 2, I would need eight copies of the original cube. If I wanted to scale the length, width, and height by 3, I would need 27 copies of the original cube.
Notice that in the 1-dimensional case, copies needed = scale.
- In the 2-dimensional case, copies needed \(=\) scale\(^{2}\).
- In the 3-dimensional case, copies needed \(=\) scale\(^{3}\).
From these examples, we might infer a pattern.
To scale a \(D\)-dimensional shape by a scaling factor \(S,\) the number of copies \(C\) of the original shape needed will be given by:
Copies \(=\) Scale \({}^{\text{Dimension}}, \) or \(C=S^{D}\)
Use the scaling-dimension relation to determine the dimension of the Sierpinski gasket.
Solution
Suppose we define the original gasket to have side length 1. The larger gasket shown is twice as wide and twice as tall, so has been scaled by a factor of 2.
Notice that to construct the larger gasket, 3 copies of the original gasket are needed.
Using the scaling-dimension relation \(C=S^{D},\) we obtain the equation \(3=2^{D}\)
since \(2^{1}=2\) and \(2^{2}=4,\) we can immediately see that \(D\) is somewhere between 1 and \(2 ;\) the gasket is more than a 1 -dimensional shape, but we've taken away so much area its now less than 2-dimensional.
Solving the equation \(3=2^{D}\) requires logarithms. If you studied logarithms earlier, you may recall how to solve this equation (if not, just skip to the box below and use that formula):
\[\begin{array}{ll}
3=2^{D} & \text{Take the logarithm of both sides} \\
\log (3)=\log \left(2^{D}\right) & \text{Use the exponent property of logs} \\
\log (3)=D \log (2) & \text{Divide by log(2)} \\
D=\frac{\log (3)}{\log (2)} \approx 1.585 & \text{The dimension of the gasket is about 1.585}
\end{array}\]
To find the dimension \(D\) of a fractal, determine the scaling factor \(S\) and the number of copies \(C\) of the original shape needed, then use the formula
\[D=\frac{\log (C)}{\log (S)} \nonumber\]
Determine the fractal dimension of the fractal produced using the initiator and generator
- Answer
-
Scaling the fractal by a factor of 3 requires 5 copies of the original. \(D=\frac{\log (5)}{\log (3)} \approx 1.465\)
[1] en.Wikipedia.org/wiki/File:Ca...ractal_AVM.JPG
[2] http://www.flickr.com/photos/cjewel/3261398909/
[3] Openstreetmap.org, CC-BY-SA
[4] http://www.flickr.com/photos/visualarts/5436068969/
[5] en.Wikipedia.org/wiki/File:Fr...e_b_-_2%29.jpg
[6] en.Wikipedia.org/wiki/File:Ba...-_4_states.PNG
[7] en.Wikipedia.org/wiki/File:Fr...lLandscape.jpg


25.12.2: Thought Questions
Thought Questions
Suppose the Milky Way was a band of light extending only halfway around the sky (that is, in a semicircle). What, then, would you conclude about the Sun’s location in the Galaxy? Give your reasoning.
9.
Suppose somebody proposed that rather than invoking dark matter to explain the increased orbital velocities of stars beyond the Sun’s orbit, the problem could be solved by assuming that the Milky Way’s central black hole was much more massive. Does simply increasing the assumed mass of the Milky Way’s central supermassive black hole correctly resolve the issue of unexpectedly high orbital velocities in the Galaxy? Why or why not?
10.
The globular clusters revolve around the Galaxy in highly elliptical orbits. Where would you expect the clusters to spend most of their time? (Think of Kepler’s laws.) At any given time, would you expect most globular clusters to be moving at high or low speeds with respect to the center of the Galaxy? Why?
11.
Shapley used the positions of globular clusters to determine the location of the galactic center. Could he have used open clusters? Why or why not?
12.
Consider the following five kinds of objects: open cluster, giant molecular cloud, globular cluster, group of O and B stars, and planetary nebulae.
- Which occur only in spiral arms?
- Which occur only in the parts of the Galaxy other than the spiral arms?
- Which are thought to be very young?
- Which are thought to be very old?
- Which have the hottest stars?
13.
The dwarf galaxy in Sagittarius is the one closest to the Milky Way, yet it was discovered only in 1994. Can you think of a reason it was not discovered earlier? (Hint: Think about what else is in its constellation.)
14.
Suppose three stars lie in the disk of the Galaxy at distances of 20,000 light-years, 25,000 light-years, and 30,000 light-years from the galactic center, and suppose that right now all three are lined up in such a way that it is possible to draw a straight line through them and on to the center of the Galaxy. How will the relative positions of these three stars change with time? Assume that their orbits are all circular and lie in the plane of the disk.
15.
Why does star formation occur primarily in the disk of the Galaxy?
16.
Where in the Galaxy would you expect to find Type II supernovae, which are the explosions of massive stars that go through their lives very quickly? Where would you expect to find Type I supernovae, which involve the explosions of white dwarfs?
17.
Suppose that stars evolved without losing mass—that once matter was incorporated into a star, it remained there forever. How would the appearance of the Galaxy be different from what it is now? Would there be population I and population II stars? What other differences would there be?


8.2.2: Keeping Track of All Possible Outcomes
Lesson
Lets explore sample spaces for experiments with multiple parts.
Exercise \(\PageIndex{1}\): How Many Different Meals?
How many different meals are possible if each meal includes one main course, one side dish, and one drink?
| main courses | side dishes | drinks |
|---|---|---|
| grilled chicken | salad | milk |
| turkey sandwich | applesauce | juice |
| pasta salad | \(\underline{\quad}\) | water |
Exercise \(\PageIndex{2}\): Lists, Tables, and Trees
Consider the experiment: Flip a coin, and then roll a number cube.
Elena, Kiran, and Priya each use a different method for finding the sample space of this experiment.
- Elena carefully writes a list of all the options: Heads 1, Heads 2, Heads 3, Heads 4, Heads 5, Heads 6, Tails 1, Tails 2, Tails 3, Tails 4, Tails 5, Tails 6.
-
Kiran makes a table:
1 2 3 4 5 6 H H1 H2 H3 H4 H5 H6 T T1 T2 T3 T4 T5 T6 Table \(\PageIndex{2}\) - Priya draws a tree with branches in which each pathway represents a different outcome:
- Compare the three methods. What is the same about each method? What is different? Be prepared to explain why each method produces all the different outcomes without repeating any.
-
Which method do you prefer for this situation?
Pause here so your teacher can review your work. -
Find the sample space for each of these experiments using any method. Make sure you list every possible outcome without repeating any.
- Flip a dime, then flip a nickel, and then flip a penny. Record whether each lands heads or tails up.
 He must select one shirt and one pair of pants to wear for the day.
- Spin a color, and then spin a number.
d. Spin the hour hand on an analog clock, and then choose a.m. or p.m.
Exercise \(\PageIndex{3}\): How Many Sandwiches?
1. A submarine sandwich shop makes sandwiches with one kind of bread, one protein, one choice of cheese, and two vegetables. How many different sandwiches are possible? Explain your reasoning. You do not need to write out the sample space.
- Breads: Italian, white, wheat
- Proteins: Tuna, ham, turkey, beans
- Cheese: Provolone, Swiss, American, none
- Vegetables: Lettuce, tomatoes, peppers, onions, pickles
2. Andre knows he wants a sandwich that has ham, lettuce, and tomatoes on it. He doesn’t care about the type of bread or cheese. How many of the different sandwiches would make Andre happy?
3. If a sandwich is made by randomly choosing each of the options, what is the probability it will be a sandwich that Andre would be happy with?
Are you ready for more?
Describe a situation that involves three parts and has a total of 24 outcomes in the sample space.
Summary
Sometimes we need a systematic way to count the number of outcomes that are possible in a given situation. For example, suppose there are 3 people (A, B, and C) who want to run for the president of a club and 4 different people (1, 2, 3, and 4) who want to run for vice president of the club. We can use a tree , a table , or an ordered list to count how many different combinations are possible for a president to be paired with a vice president.
With a tree, we can start with a branch for each of the people who want to be president. Then for each possible president, we add a branch for each possible vice president, for a total of \(3\cdot 4=12\) possible pairs. We can also start by counting vice presidents first and then adding a branch for each possible president, for a total of \(3\cdot 4=12\) possible pairs.
A table can show the same result:
| 1 | 2 | 3 | 4 | |
|---|---|---|---|---|
| A | \(A, 1\) | \(A, 2\) | \(A, 3\) | \(A, 4\) |
| B | \(B, 1\) | \(B, 2\) | \(B, 3\) | \(B, 4\) |
| C | \(C, 1\) | \(C, 2\) | \(C, 3\) | \(C, 4\) |
So does this ordered list:
A1, A2, A3, A4, B1, B2, B3, B4, C1, C2, C3, C4
Practice
Exercise \(\PageIndex{4}\)
Noah is planning his birthday party. Here is a tree showing all of the possible themes, locations, and days of the week that Noah is considering.
- How many themes is Noah considering?
- How many locations is Noah considering?
- How many days of the week is Noah considering?
- One possibility that Noah is considering is a party with a space theme at the skating rink on Sunday. Write two other possible parties Noah is considering.
- How many different possible outcomes are in the sample space?
Exercise \(\PageIndex{5}\)
For each event, write the sample space and tell how many outcomes there are.
-
Lin selects one type of lettuce and one dressing to make a salad.
Lettuce types: iceberg, romaine
Dressings: ranch, Italian, French - Diego chooses rock, paper, or scissors, and Jada chooses rock, paper, or scissors.
- Spin these 3 spinners.
Exercise \(\PageIndex{6}\)
A simulation is done to represent kicking 5 field goals in a single game with a 72% probability of making each one. A 1 represents making the kick and a 0 represents missing the kick.
| trial | result |
|---|---|
| 1 | 10101 |
| 2 | 11010 |
| 3 | 00011 |
| 4 | 11111 |
| 5 | 10011 |
(From Unit 8.2.1)
Exercise \(\PageIndex{7}\)
There is a bag of 50 marbles.
- Andre takes out a marble, records its color, and puts it back in. In 4 trials, he gets a green marble 1 time.
- Jada takes out a marble, records its color, and puts it back in. In 12 trials, she gets a green marble 5 times.
- Noah takes out a marble, records its color, and puts it back in. In 9 trials, he gets a green marble 3 times.
Estimate the probability of getting a green marble from this bag. Explain your reasoning.
(From Unit 8.1.4)