2: Polar Covalent Bonds; Acids and Bases
This chapter provides a review of the more advanced material covered in a standard introductory chemistry course through a discussion of the following topics:
- the use of electronegativity to determine bond polarity, and the application of this knowledge to determine whether a given molecule possesses a dipole moment.
- the drawing and interpretation of organic chemical structures.
- the concept and determination of formal charge.
- resonance and drawing of resonance forms
- the Brønsted-Lowry and Lewis definitions of acids and bases, acidity constants and acid-base reactions.
- intermolecular forces
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- 2.0: Polar Covalent Bonds - Electronegativity
- Because the tendency of an element to gain or lose electrons is so important in determining its chemistry, various methods have been developed to quantitatively describe this tendency. The most important method uses a measurement called electronegativity, defined as the relative ability of an atom to attract electrons to itself in a chemical compound.
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- 2.1: Polar Covalent Bonds - Dipole Moments
- Mathematically, dipole moments are vectors; they possess both a magnitude and a direction. The dipole moment of a molecule is therefore the vector sum of the dipole moments of the individual bonds in the molecule. If the individual bond dipole moments cancel one another, there is no net dipole moment.
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- 2.3: Resonance
- Resonance structures are a set of two or more Lewis Structures that collectively describe the electronic bonding a single polyatomic species including fractional bonds and fractional charges. Resonance structure are capable of describing delocalized electrons that cannot be expressed by a single Lewis formula with an integer number of covalent bonds.
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- 2.4: Rules for Resonance Forms
- The above resonance structures show that the electrons are delocalized within the molecule and through this process the molecule gains extra stability. Ozone with both of its opposite formal charges creates a neutral molecule and through resonance it is a stable molecule. The extra electron that created the negative charge on one terminal oxygen can be delocalized by resonance through the other terminal oxygen.
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- 2.5: Drawing Resonance Forms
- Resonance structures are used when one Lewis structure for a single molecule cannot fully describe the bonding that takes place between neighboring atoms relative to the empirical data for the actual bond lengths between those atoms. The net sum of valid resonance structures is defined as a resonance hybrid, which represents the overall delocalization of electrons within the molecule. A molecule that has several resonance structures is more stable than one with fewer.
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- 2.6: Acids and Bases - The Brønsted-Lowry Definition
- In 1923, chemists Johannes Brønsted and Martin Lowry independently developed definitions of acids and bases based on compounds abilities to either donate or accept protons (H+ ions). Here, acids are defined as being able to donate protons in the form of hydrogen ions; whereas bases are defined as being able to accept protons. This took the Arrhenius definition one step further as water is no longer required to be present in the solution for acid and base reactions to occur.
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- 2.7: Acid and Base Strength
- The relative acidity of different compounds or functional groups – in other words, their relative capacity to donate a proton to a common base under identical conditions – is quantified by a number called the dissociation constant, abbreviated Ka. The common base chosen for comparison is water.
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- 2.9: Organic Acids and Organic Bases
- In the absence of pKa values, the relative strength of an organic acid can be predicted based on the stability of the conjugate base that it forms. The acid that forms the more stable conjugate base will be the stronger acid. The common factors that affect the conjugate base's stability are 1) the size and electronegativity of the the atom that has lost the proton, 2) resonance effects, 3) inductive effects, and 4) solvation effects.
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- 2.10: Acids and Bases - The Lewis Definition
- A broader definition is provided by the Lewis theory of acids and bases, in which a Lewis acid is an electron-pair acceptor and a Lewis base is an electron-pair donor. This definition covers Brønsted-Lowry proton transfer reactions, but also includes reactions in which no proton transfer is involved.
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- 2.11: Noncovalent Interactions Between Molecules
- In contrast to intramolecular forces, such as the covalent bonds that hold atoms together in molecules and polyatomic ions, intermolecular forces hold molecules together in a liquid or solid. Intermolecular forces are generally much weaker than covalent bonds. The most common intermolecular forces in organic chemistry are from strongest to weakest are hydrogen bonds, dipole-dipole interactions, and London Dispersion (van der Waals) forces.


16.25: Assignment- Four Personal Budgets in Excel
For this activity, you will complete four budgets – one Business budget, one Forecasts (projects) budget, one Home Budget, and one Customized Personal Budget.
Directions:
1. Review the module content related to Microsoft Excel 2010.
2. In Microsoft Excel, click on File, New, and Budgets.
3. Select one template out of each of the following categories:
- 1 template from Business Budgets
- 1 template from Forecasts (projects)
- 1 template from Home Budgets
4. Plan a budget from each of the categories for:
- your future business using the Business Budget template
- your future project(s) using the Forecasts template
- your future home expenses using the Home Budget template
5. Next, complete a customized personal budget based on your future income.
- To assist you with this, visit http://www.salary.com (Links to an external site.) and research your future salary based on your major and the area that you are planning to live in.
- Research the area where you would like to live after graduating from Wiley College.
- Be sure to consider the cost of housing, food, etc. in the area using, for example, a Google search related to the budget item and the area.
6. Save all four budgets:
- Business Budget using Excel Templates
- Forecasts (projects) Budget using Excel Templates
- Home Budget using Excel Templates
- Customized Personal Budget
7. Upload your budgets to the submission.
- Budget Assignment. Authored by : Veronica McEachin. Provided by : Wiley College . Project : Kaleidoscope Open Course Initiative. License : CC BY: Attribution


8.5: Exploring Mk - the "total garbage" test
One problem that arises sometimes in maximum likelihood optimization happens when instead of a peak, the likelihood surface has a long flat “ridge” of equally likely parameter values. In the case of the Mk model, it is common to find that all values of q greater than a certain value have the same likelihood. This is because above a certain rate, evolution has been so rapid that all traces of the history of evolution of that character have been obliterated. After this point, character states of each lineage are random, and have no relationship to the shape of the phylogenetic tree. Our optimization techniques will not work in this case because there is no value of q that has a higher likelihood than other values. Once we get onto the ridge, all values of q have the same likelihood.
For Mk models, there is a simple test that allows us to recognize when the likelihood surface has a long ridge, and q values cannot be estimated. I like to call this test the “total garbage” test because it can tell you if your data are “garbage” with respect to historical inference – that is, your data have no information about historical patterns of trait change. One can predict states just as well by choosing each species at random.
 That is, each species has some probability p of having character state 0, and some probability ( 1 − p ) of having state 1 (one can also generalize this test to multi-state models). This likelihood is easy to write down. For a tree of size \(n\), the probability of drawing n 0 species with state 0 is:
\[L_{garbage} = p^{n_0}(1 − p)^{n − n_0} \label{8.2}\]
This equation gives the likelihood of the “total garbage” model for any value of p . Equation 8.1 is related to a binomial distribution (lacking only the factorial term). We also know from probability theory that the ML estimate of p is n 0 / n , with likelihood given by the above formula.
Now consider the likelihood surface of the Mk model. When Mk likelihood surfaces have long ridges, they are nearly always for high values of q – and when the transition rate of character changes is high, this model converges to our “drawing from a hat” (or “garbage”) model. The likelihood ridge lies at the value that is exactly taken from equation 8.10 above.
Thus, one can compare the likelihood of our Mk model to the total garbage model. If the maximum likelihood value of q has the same likelihood as our garbage model, then we know that we are on a ridge of the likelihood surface and q cannot be estimated. We also have no ability to make any statements about the past evolution of our character – in particular, we cannot estimate ancestral character state with any precision. By contrast, if the likelihood of the Mk model is greater than the total garbage model, then our data contains some historical information. We can also make this comparison using AIC, considering the total garbage model as having a single parameter p.
For the squamates, we have n = 258 and n 0 = 207. We calculate p = n 0 / n = 207/258 = 0.8023256. So the likelihood of our garbage model is
L g a r b a g e = p n 0 (1 − p ) n − n 0 = 0.8023256 207 (1 − 0.8023256) 51 = 1.968142 e − 56.
This calculation is both easier and more useful, though, on a natural-log scale:
l n L g a r b a g e = n 0 ⋅ l n ( p )+( n − n 0 )⋅ l n (1 − p )=207 ⋅ l n (0.8023256)+51 ⋅ l n (1 − 0.8023256)= − 128.2677.
Compare this to the log-likelihood of our Mk model, l n L = −80.487176, and you will see that the garbage model is a terrible fit to these data. There is, in fact, some historical information about species' traits in our data.


7.1.5D: The Law as an Instrument of Oppression
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Oppression is the exercise of authority or power in a burdensome, cruel, or unjust manner.
Learning Objectives
- Describe the sociological approach to oppression
Key Points
- Anarchists and other libertarian socialists argue that police and law themselves are oppression. The term oppression, in such instances, refers to the subordination of a given group or social category by unjust use of force or authority in order to achieve the effects of oppression.
- In psychology, racism, sexism and other prejudices are often studied as individual beliefs which, although not necessarily oppressive in themselves, can lead to oppression if they are codified in law or become parts of a culture.
- In sociology, prejudices are often studied as being institutionalized systems of oppression in some societies.
- In sociology and psychology, internalized oppression is the manner in which an oppressed group comes to use against itself the methods of the oppressor.
Key Terms
- anarchist : One who believes in or advocates the absence of hierarchy and authority in most forms (compare anarchism), especially one who works toward the realization of such.
- prejudice : An adverse judgment or opinion formed beforehand or without knowledge of the facts.
Oppression is the exercise of authority or power in a burdensome, cruel, or unjust manner. It can also be defined as an act or instance of oppressing, the state of being oppressed, and the feeling of being heavily burdened, mentally or physically, by troubles, adverse conditions, and anxiety. Injustice refers to the absence of justice. The term may be applied either in reference to a particular event or act, or to a larger status quo. The term generally refers to misuse, abuse, neglect, or malfeasance that is uncorrected or otherwise sanctioned by a legal system. Misuse and abuse with regard to a particular case or context may represent a systemic failure to serve the cause of justice.
Anarchists and other libertarian socialists argue that police and laws themselves are oppression. The term oppression, in such instances, refers to the subordination of a given group or social category by unjust use of force, authority, or societal norms in order to achieve the effects mentioned above. When institutionalized, formally or informally, it may achieve the dimension of systematic oppression. Oppression is customarily experienced as a consequence of, and expressed in, the form of a prevailing, if unconscious, assumption that the given target is in some way inferior. Oppression is rarely limited solely to formal government action: An individual may be the particular focus of oppression or persecution, and in such circumstances, have no group membership in which to share, and thus maybe mitigate the burden of ostracism.
In psychology, racism, sexism and other prejudices are often studied as individual beliefs which, although not necessarily oppressive in themselves, can lead to oppression if they are codified in law or become parts of a culture. By comparison, in sociology, these prejudices are often studied as being institutionalized systems of oppression in some societies. In sociology, the tools of oppression include a progression of denigration, dehumanization, and demonization which often generate scapegoating, which is used to justify aggression against targeted groups and individuals.
In sociology and psychology, internalized oppression is the manner in which an oppressed group comes to use against itself the methods of the oppressor. For example, sometimes members of marginalized groups hold an oppressive view toward their own group, or start to believe in negative stereotypes.
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21.4.1: Set Notation
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Learning Outcomes
- Read set notation.
- Describe sets using set notation.
A set is just a collection of items and there are different ways of representing a set. We want to be able to both read the various ways and be able to write down the representation ourselves in order to best display the set. We have already seen how to represent a set on a number line, but that can be cumbersome, especially if we want to just use a keyboard. Imagine how difficult it would be to text a friend about a cool set if the only way to do this was with a number line. Fortunately, mathematicians have agreed on notation to describe a set.
Example \(\PageIndex{1}\)
If we just have a few items to list, we enclose them in curly brackets "{" and "}" and separate the items with commas. For example,
\[\{\text{Miguel}, \text{Kristin}, \text{Leo}, \text{Shanice}\} \nonumber\]
means the set the contains these four names.
Example \(\PageIndex{2}\)
If we just have a long collection of numbers that have a clear pattern, we use the "..." notation to mean "start here, keep going, and end there". For example,
\[\{3, 6, 9, 12, ..., 90\}\nonumber \]
This set contains more than just the five numbers that are shown. It is clear that the numbers are separated by three each. After the 12, even though it is not explicitly shown, is a 15 which is part of this set. It also contains 18, 21 and keeps going including all the multiples of 3 until it gets to its largest number 90.
Example \(\PageIndex{3}\)
If we just have a collection of numbers that have a clear pattern, but never ends, we use the "..." without a number at the end. For example,
\[\left\{\frac{1}{2},\:\frac{2}{3},\:\frac{3}{4},\:\frac{4}{5},\:...\right\}\nonumber \]
This set contains an infinite number of fractions, since there is no number followed by the "...".
Example \(\PageIndex{4}\)
Sometimes we have a set that it best described by stating a rule. For example, if you want to describe the set of all people who are over 18 years old but not 30 years old, you announce the conditions by putting them to the left of a vertical line segment. We read the line segment as "such that".
\[\left\{x\:|\:x>18\:and\:x\ne30\right\}\nonumber \]
This can be read as "the set of all numbers \(x\) such that \(x\) is greater than 18 and \(x\) is not equal to 30".
Exercise
Describe using set notation the collection of all positive even whole numbers that are not equal to 20 or 50.


5.3: Exercise- Working with the Artificial Data
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In the Chapter_5 data folder, you will find an Excel spreadsheet named simulated_data.xlsx , which contains everything I used to create the simulated data in Figure 5.1. Open the file in Excel (or import it into Google Sheets). You will see that each line is a time point. The first column shows the latency of the time point (in ms), and the second column is the source waveform. There are also columns for the absolute voltage at each electrode site, with the weighting factor for a given site in the second row.
If you look at the equations in the cells that compute the absolute voltage values, you’ll see that the absolute voltage at a given electrode site was computed by multiplying the weighting factor for that site by the source waveform. You’ll also see that the single-ended voltage was created by subtracting the absolute voltage at the ground electrode from the absolute voltage at each other site, and then adding the 60 Hz noise. And you’ll see that the referenced voltage was computed by subtracting the single-ended signal at the Lm electrode from the single-ended signal at a given active site. At the right side of the spreadsheet, you’ll see plots of the absolute voltages, the single-ended voltages, and the referenced voltages. Take a careful look at the equations in the spreadsheet and make sure that you understand how the absolute voltage is related to the source waveform and the weights, how the single-ended voltage is related to the absolute voltage, and how the referenced voltage is related to the single-ended voltage.
The spreadsheet also contains sheets with copies of the data formatted for exporting as a text file (using the Tab delimited Text format), which can easily be imported into ERPLAB (using EEGLAB > ERPLAB > Export & Import ERP > Import ERP from text (universal) ). This allows you to create simulated data and see how the various ERPLAB processes work.
In the spreadsheet, change the weight above the Absolute_Ground label from -.04 to -.10 . You’ll see that this greatly increases the magnitude of the absolute voltage in the ground channel. And because the ground is subtracted from the other signals to create the single-ended voltage, this also changes all the single-ended voltage waveforms. But did it change the referenced waveforms?
No, it did not! The nature of the referencing procedure means that any signals or noise at the ground electrode are subtracted away from the referenced voltages. This means that you can place the ground electrode anywhere on the head, and the location does not matter. You might then wonder why we use a ground electrode at all. The answer is simple: The amplifier will freak out if there isn’t a ground electrode. You need to have a ground, and it needs to be appropriately attached to the head (or anywhere on the body). But the location does not matter.
Now try changing the weight for the Lm channel from -.011 to -.20 . This simulates changing the location of the reference electrode, moving it farther away from the zero line. What did this do to the referenced waveforms? Not only are some of them now larger, the polarity of the Fz waveform has now changed from negative to positive. This demonstrates how the signal at the reference electrode can have a large impact on the referenced waveforms from the so-called active electrodes.
Play around with the spreadsheet some more. For example, try changing the weights for the other electrodes or the magnitude of the 60 Hz noise.


15.0: Introduction
A county health department provides community health screenings in a pharmacy parking lot. A nurse asks 25-year-old Deanna if her 148/88 mm Hg blood pressure reading is usual for her. Deanna doesn’t know, but she says that her grandparents and parents all have high blood pressure. The nurse questions Deanna about any recent headaches, weight gain, swelling in her hands or feet, or visual changes. Deanna denies experiencing any of these. The nurse describes the genetic and behavioral risk factors for high blood pressure as well as potential complications with high blood pressure, such as strokes. The nurse also explains to Deanna that high blood pressure can be reduced with certain healthy behaviors, such as limiting sodium intake and engaging in regular physical activity, and the two discuss an infographic illustrating these and other healthy behaviors. The nurse also provides Deanna with a dated card that indicates Deanna’s blood pressure measurement and encourages Deanna to make an appointment with her primary health care provider. Deanna says she’ll do so, as she doesn’t want to have a chronic illness at such a young age.
If not for the community health screening , Deanna may not have discovered that she has a health problem that she needs to address. Unidentified, and therefore untreated, health conditions can lead to bigger issues. According to the U.S. Department of Health and Human Services (USDHHS) (2022), about 11 million Americans do not know their blood pressure is elevated and, therefore, are not receiving any treatment to control it. This chapter will discuss ways that health promotion and disease prevention can positively impact individual and population health outcomes. It will describe the role of the nurse in performing health promotion and disease prevention.


9.21: Discussion- Logical Fallacy Advertisement
STEP 1 : First, pick a product or service to advertise. It could be a fictitious object, or something real—products such as food items, cleaning or office supplies, furniture, tickets to an event or venue, or services such as tutoring help, lawn care, painting, cleaning, etc.
Create an advertisement for your product using a minimum of 3 fallacies you learned about in the module. You won’t explicitly say what fallacies you are incorporating in your ad, but you should make it obvious enough so that others can easily tell that you are exaggerating or manipulating claims.  You may use whatever format or program you’d like (PhotoShop, Microsoft PowerPoint, Prezi, Canva, YouTube, etc.). Your submission should include links to all of the resources you used to develop your advertisement.
STEP 2 : Post your advertisement to the discussion forum.
STEP 3 : Respond in two separate posts to two classmates (in at least 75 words). Try to guess which fallacies they included and why you think that. Elaborate on their post by thinking of other examples of those same logical fallacies or other ways they could include even more logical fallacies in a similar advertisement.
STEP 4 : At the end of the week, return to your original post and let everyone know which logical fallacies you included in your advertisement.
Contributors and Attributions
- Provided by : Lumen Learning. License : CC BY: Attribution


8.5: Translation-Protein Synthesis
- Define and understand the mechanism of biological translation.
Translation in biology is the process of protein synthesis using the information encoded in mRNA.
Translation process
The process starts when the ribosome binds to mRNA in the cytoplasm, as illustrated in Figure \(\PageIndex{1}\). The ribosome moves along the mRNA from 5' to 3" direction until it reaches the stats codon AUG. The AUG is also the codon for methionine. So, the methionine-tRNA (methionine-loaded aminoacyl-tRNA) with the complementary anticodon UAC arrives and aligns opposite the codon AUG. The aminoacyl-tRNA with a complementary anticodon to the codon next to AUG on the mRNA arrives and aligns opposite the codon.
Ribosome catalyzes a peptide bond formation between \(\ce{-NH2}\) group of the amino acid on the second tRNA with the carboxylate group of the first by nucleophilic acyl substitution mechanism. The first tRNA becomes empty, and the second tRNA becomes peptidyl-tRNA. The ribosome moves on to the next codon -a process called translocation . The empty-tRNA leaves and a new aminoacyl-tRNA with a complementary anticodon to the third codon align on the mRNA and repeat the above process, as illustrated in Figure \(\PageIndex{2}\). The cycle repeats, and the peptide keeps elongating until the ribosome reaches a stop codon. The translation process stops at this point, and the newly formed peptide is released. The first methionine is usually removed from the peptide. The empty tRNA are re-loaded with the amino acid later on, as illustrated in Fig. 8.4.2. The hydrogen bonding and the other intramolecular interaction, like salt bridges, disulfide bonds, etc., make peptides acquire the secondary, tertiary, and sometimes quaternary structure. This is how functional proteins with the appropriate forms are synthesized.
Figure \(\PageIndex{3}\) presents a simulation that summarizes the process of transcription and translation and the trinucleotide codons in DNA, mRNA, anticodons on tRNA, and the corresponding amino acids in the polypeptide that are shown in the simulation. Watch the DNA and RNA video summary by clicking this link or the video below.
Antibiotics that interrupt protein synthesis in bacteria but not in humans are clinically useful. Some examples and their actions are shown in Table 1.
| Antibiotic | Effect on protein synthesis in bacteria |
|---|---|
| Tetracycline | Prevents the aminoacyl-tRNA from binding to the ribosome |
| Erythromycin | Prevent the translocation of the ribosome along the mRNA |
| Streptomycin | Inhibits the initiation of protein synthesis |
| Chloramphenicol | Prevents the new peptide bond from being formed |


12.7: Henry's Law
- Describe the effect of pressure on the solubility of gases in water and apply this concept in health-related conditions.
Solubility of a Gas in Water
Earlier you were introduced to the effect of intermolecular attractive forces on solution formation. The chemical structures of the solute and solvent dictate the types of forces possible and, consequently, are important factors in determining solubility.
For example, under similar conditions, the water solubility of oxygen is approximately three times greater than that of helium.
The solubility of oxygen (nonpolar) in the liquid hydrocarbon hexane, C 6 H 14 (nonpolar), is approximately 20 times greater than it is in water (polar).
Solubility of a Gas and Temperature
Temperature is another factor that affects gas solubility. Gas solubility decreases as temperature increases. This is one of the major impacts resulting from the thermal pollution of natural bodies of water. When the temperature of a river, lake, or stream is raised abnormally high, usually due to the discharge of hot water from some industrial process, the solubility of oxygen in the water is decreased. Decreased levels of dissolved oxygen may have serious consequences for the health of the water’s ecosystems and, in severe cases, can result in large-scale fish kills.
Henry's Law (Solubility of a Gas and Pressure)
The solubility of a gaseous solute is also affected by pressure. Gas solubility increases as the pressure of the gas increases. Carbonated beverages provide a nice illustration of this relationship. The carbonation process involves exposing the beverage to a relatively high pressure of carbon dioxide gas and then sealing the beverage container, thus saturating the beverage with CO 2 at high pressure. When the beverage container is opened, a familiar hiss is heard as the carbon dioxide gas pressure is released, and some of the dissolved carbon dioxide is typically seen leaving solution in the form of small bubbles (Figure \(\PageIndex{1}\)). With time, the dissolved carbon dioxide concentration will decrease to its equilibrium value and the beverage will become “flat.”
Decompression Sickness (“The Bends”)
Decompression sickness (DCS), or “the bends,” is an effect of the increased pressure of the air inhaled by scuba divers when swimming underwater at considerable depths. In addition to the pressure exerted by the atmosphere, divers are subjected to additional pressure due to the water above them, experiencing approximately 2 atm at 10 m of depth. Therefore, the air inhaled by a diver while submerged contains gases at the corresponding higher pressure, and the concentrations of the gases dissolved in the diver’s blood are proportionally higher per Henry’s law.
As the diver ascends to the surface of the water, the atmospheric pressure decreases and the dissolved gases becomes less soluble. If the ascent is too rapid, the gases escaping from the diver’s blood may form bubbles that eventually bursts capillaries and cause a variety of symptoms ranging from rashes and joint pain to paralysis and death. To avoid DCS, divers must ascend from depths at relatively slow speeds (10 or 20 m/min) or otherwise make several decompression stops, pausing for several minutes at given depths during the ascent. This allows excess dissolved gases in the blood to be exhaled. If stops or slow speeds are not an option, then the diver should exhale rapidly during the ascent.
Divers may also use a special mixture of helium and oxygen. Nitrogen in air is replaced by helium. He gas is not as soluble in the blood serum, so the effects of DCS is minimized. Under similar conditions, the water solubility of nitrogen is greater than that of helium.
When these preventive measures are unsuccessful, divers with DCS are often provided hyperbaric oxygen therapy in pressurized vessels called decompression (or recompression) chambers.
Hyperbaric Chamber
Hyperbaric (high pressure) oxygen treatment in another application of Henry's Law. This is an air chamber that is at two to three times the atmospheric pressure. The solubility of oxygen increases with an increase in pressure. A patient placed in a hyperbaric chamber has a higher concentration of oxygen dissolved in blood. The higher concentration of oxygen is toxic to many strains of bacteria. Therefore the hyperbaric chambers are used to treat burn patients, gangrene, tetenus, in surgeries, and to treat some cancers.
The hyperbaric chambers are also used to treat carbon monoxide (CO) poisoning because the higher concentration of oxygen in the chamber can displace the CO bound with hemoglobin faster than atmospheric oxygen does.
Another use of hyperbaric chambers is to treat scuba divers suffering from the bends. If a diver ascends too quickly, the nitrogen dissolved in blood makes bubbles in the vessels that block the blood flow, a condition called bends. The divers suffering from the bends are placed in a hyperbaric chamber at high pressure, and then the pressure is slowly decreased to atmospheric pressure. The nitrogen dissolved in blood under higher pressure and slowly diffuses out through the lungs as the pressure is gradually decreased.
Figure \(\PageIndex{2}\): (a) US Navy divers undergo training in a recompression chamber. (b) Divers receive hyperbaric oxygen therapy.
Summary
The solubility of a gas in a liquid is proportional to the pressure of the gas over the liquid. The higher the pressure of a gas above a liquid, the greater its solubility.
Contributions & Attributions
- Wikipedia
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Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).
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Henry Agnew (UC Davis)


8.5: Conditional Formatting
Learning Outcomes
- Create and apply conditional formatting.
As we have learned so far, Excel has a wide variety of easy to use tools for organizing, sorting, and otherwise marking information. Think back to when we applied styles to a cell to indicate good information or information that needs to be verified. Excel also has the ability to automatically apply such markings through conditional formatting.
With conditional formatting, you provide Excel with a rule, such as “less than 10,” and the program will scan through your data and highlight all the cells that meet that rule. There are several rules already available, but you can also create and apply your own rules and visual clues.
- Select the cells, rows, or columns you wish to have conditional formatting.
- From the Styles group, click on the Conditional Formatting button
- Select the style of formatting you would like. Here we have Highlight Cell Rules.
- Select the specific type of rule you would like to use and then apply your target value. Here we have selected Less Than.
- The formatting will appear automatically so you can see what it will look like. Note that Excel will automatically provide a value, but you can manually change it.
- Hit OK if you wish to apply the formatting. Otherwise, when you leave the formatting menu, it will disappear.
One important tool to keep in mind is the ability to enter your own rules. This can include applying formatting to specific date ranges, to specific text (like names), or even cells that are blank. In this case, you also set the format, so instead of highlighting cells you can choose to strikethrough text or change the font, change the size, or bold the text.
Contributors and Attributions
- Formatting. Authored by : Shelli Carter. Provided by : Lumen Learning. License : CC BY: Attribution


2.7: CHAPTER 7- SEXUALITY
Discussion Topics for Sexual Education Support
Key Points for Caregivers
Sexual health is often overlooked in individuals with intellectual disabilities. They are very vulnerable to sexual abuse and may not understand what consenting to sexual activity means. They may not know how to express their sexuality.  These kinds of topics must also be part of the everyday conversations that take place between intellectually disabled individuals and the people in their lives. When individuals with intellectual disabilities are parents, they need a network of people around them and opportunities to learn about children’s development and showing appropriate affection to children.
Understanding Boundaries and Providing Resources
Key Points for Caregivers
Establishing boundaries in relationships can create safe psychological spaces. When individuals with an intellectual disability express their sexuality in inappropriate ways, their behaviour needs to be addressed just as it would be with non-disabled individuals. Open demonstrations of sexuality may be linked to mental illnesses and must be reported and discussed with other staff and health professionals. In some instances, the behaviour must be reported to police. Caregivers can model respectful behaviour that shows the difference between private and public actions. This is an important way for individuals with intellectual disability to learn about setting boundaries. Resources are available, such as picture cards and the Circle of Relationships activity.
Differentiating Between Challenging Behaviours and Sex Offending Behaviours
Key Points for Caregivers
Inappropriate expressions of sexuality are very serious. Whether they are open or violent, they can traumatize victims and leave offenders facing lifelong consequences. Different professional groups do not all agree on the best treatment approaches. Some view sexually violent behaviours that hurt others and interfere with quality of life for others as challenging . Others view these behaviours as sexual offending . Treatment approaches that view the behaviour as challenging will try to find out why individuals act this way, what the behaviour means, and how it meets an otherwise unmet need. When caregivers are involved, they will admit that the behaviour is unacceptable and hurts others as well as themselves. They will also recognize that the behaviour is occurring for a reason and they will guide individuals toward meeting their needs with more acceptable behaviours.
Treatment approaches that view the behaviour as sexual offending will not focus on what the behaviour means. Here, when caregivers are involved, they will still admit that the behaviour is unacceptable and hurts others as well as themselves. However, they will pay more attention to strategies that prevent the behaviour from re-occurring and to protecting all the people in the community that the individual interacts with.
Although views about treatment approaches are different and each individual is unique, inappropriate expressions of sexuality must all be reported, discussed, and addressed as a team.


13.8: References
Acevedo, H., Rahman, M., & Moreno, M. (2023, April 9). Sonography 3rd trimester and placenta assessment, protocols, and interpretation . StatPearls [Internet]. www.ncbi.nlm.nih.gov/books/NBK572062/
Adekanmi, A. J., Roberts, A., Morhason-Bello, I. O., & Adeyinka, A. O. (2022, February 28). Utilization of uterine and umbilical artery Doppler in the second and third trimesters to predict adverse pregnancy outcomes: A Nigerian experience. Women’s Health Reports (New Rochelle, N.Y.), 3 (1), 256–266. www.ncbi.nlm.nih.gov/pmc/articles/PMC8896215
American College of Obstetricians and Gynecologists. (2017a, June). Ultrasound exams . https://www.acog.org/womens-health/f...trasound-exams
American College of Obstetricians and Gynecologists. (2017b). Committee Opinion No. 723: Guidelines for diagnostic imaging during pregnancy and lactation . https://www.acog.org/clinical/clinic...-and-lactation
American College of Obstetricians and Gynecologists. (2019a, May). Special tests for monitoring fetal well-being . https://www.acog.org/womens-health/f...tal-well-being
American College of Obstetricians and Gynecologists. (2019b). Prenatal genetic testing chart . https://www.acog.org/womens-health/i...-testing-chart
American College of Obstetricians and Gynecologists. (2020a, December). Carrier screening . https://www.acog.org/womens-health/f...rier-screening
American College of Obstetricians and Gynecologists. (2020b). Non-invasive prenatal testing . https://www.acog.org/advocacy/policy...enatal-testing
American College of Obstetricians and Gynecologists. (2020c). Prenatal genetic screening tests . https://www.acog.org/womens-health/f...creening-tests
American College of Obstetricians and Gynecologists. (2021a, March). Amniocentesis . https://www.acog.org/womens-health/faqs/amniocentesis
American College of Obstetricians and Gynecologists. (2021b, May). Bleeding during pregnancy . https://www.acog.org/womens-health/f...ring-pregnancy
American College of Obstetricians and Gynecologists. (2021c, June). ACOG Practice Bulletin No. 229: Antepartum fetal surveillance. Obstetrics & Gynecology, 137 (6), e116 e127. https://pubmed.ncbi.nlm.nih.gov/34011889/
American College of Obstetricians and Gynecologists. (2021d, June). Committee Opinion No. 828: Indications for outpatient antenatal fetal surveillance . https://www.acog.org/clinical/clinic...l-surveillance
American College of Obstetricians and Gynecologists. (2021e, June). Viral hepatitis in pregnancy . https://www.acog.org/womens-health/f...c-in-pregnancy
American College of Obstetricians and Gynecologists. (2022a, June). The Rh factor: How it can affect your pregnancy . https://www.acog.org/womens-health/f...your-pregnancy
American College of Obstetricians and Gynecologists. (2022b, July). Obstetric Care Consensus No. 11: Pregnancy at age 35 years or older . https://www.acog.org/clinical/clinic...years-or-older
American College of Obstetricians and Gynecologists. (2023). Current ACOG Guidance: NIPT summary of recommendations . https://www.acog.org/advocacy/policy...-acog-guidance
Burd, I., Zieve, D., & Ogilvie, I. (2016, May 16). Percutaneous umbilical cord blood sampling, series—Normal anatomy . MedlinePlus. https://medlineplus.gov/ency/present...s/100196_1.htm
Centers for Disease Control and Prevention. (2022, February 16). Viral hepatitis: Perinatal transmission . Centers for Disease Control and Prevention. www.cdc.gov/hepatitis/hbv/perinatalxmtn.htm
Chen, Y., Wu, B., Chen, Y., Ning, W., & Zhang, H. (2022). A risk model for predicting fetuses with trisomy 21 using alpha-fetoprotein variants L2 combined with maternal serum biomarkers in early pregnancy. Reproductive Sciences, 29 (4), 1287–1295. https://doi.org/10.1007/s43032-021-00762-5
Correa-de-Araujo, R., & Yoon, S. S. S. (2021, February). Clinical outcomes in high-risk pregnancies due to advanced maternal age. Journal of Women’s Health, 30 (2), 160–167. www.ncbi.nlm.nih.gov/pmc/articles/PMC8020515/
East, C. E., Smyth, R. M. D., Leader, L. R., Henshall, N. E., Colditz, P. B., Lau, R., & Tan, K. H. (2013, January 31). Vibroacoustic stimulation for fetal assessment in labour in the presence of a nonreassuring fetal heart rate trace. Cochrane Database of Systematic Reviews . www.ncbi.nlm.nih.gov/pmc/articles/PMC7212002/
Fuentes, A. (Ed.). (2018, August). Prenatal test: Multiple marker test [for parents]. Nemours. KidsHealth. https://kidshealth.org/en/parents/pr...le-marker.html
Gatta, G., Grezia, G., Cuccurullo, V., Sardu, C., Iovino, F., Comune, R., Ruggeiro, A., Chirico, M., Forgia, D., Fanizzi, A., Massafra, R., Belfiore, M., Falco, G., Reginelli, A., Brunese, L., Grassi, R., Cappabianca, S., & Viola, L. (2022, January). MRI in pregnancy and precision medicine: A review from literature. Journal of Personalized Medicine, 12 (1), 9. www.ncbi.nlm.nih.gov/pmc/articles/PMC8778056/
Gordon, S., Umandap, C., & Langaker, M. (2023, January 16). Prenatal genetic screening . StatPearls [Internet]. www.ncbi.nlm.nih.gov/books/NBK557702/
Jabaz, D., & Abed, M. (2023, May 22). Sonography 2nd trimester assessment, protocols, and interpretation . StatPearls [Internet]. www.ncbi.nlm.nih.gov/books/NBK570574/
Jones, T. M., & Montero, F. J. (2022, December 11). Chorionic villus sampling . StatPearls [Internet]. www.ncbi.nlm.nih.gov/books/NBK563301/
Kaiser Permanente. (n.d.). Counting kicks . https://healthy.kaiserpermanente.org...ount-the-kicks
Lord, M. Marino, S., & Kole, M. (2023, March). Amniotic fluid index . StatPearls [Internet]. www.ncbi.nlm.nih.gov/books/NBK441881/
Maulik, D. (2022, May). Doppler ultrasound of the umbilical artery for fetal surveillance in singleton pregnancies . UpToDate. https://www.uptodate.com/contents/do...on-pregnancies
Mount Sinai Health System. (2022). Nuchal translucency test . https://www.mountsinai.org/health-li...nslucency-test
Murphy, A. (2023, March). Umbilical arterial Doppler assessment . Radiopaedia. radiopaedia.org/articles/umb...ssment?lang=us
Murugan, V. A., Murphy, B. O., Dupuis, C., Goldstein, A., & Kim, Y. H. (2020). Role of ultrasound in the evaluation of first-trimester pregnancies in the acute setting. Ultrasonography (Seoul, Korea), 39 (2), 178–189. https://doi.org/10.14366/usg.19043
Nguyen, T. P. (Ed.). (2022, July). Prenatal tests: Second trimester [for parents]. Nemours. KidsHealth. https://kidshealth.org/en/parents/te...trimester.html
Palka, C., Guanciali-Franchi, P., Morizio, E., Afonsi, M., Papponetti, M., Sabbatinelli, G., Palka, G., Calabrese, G., & Benn, P. (2019, May 18). Non-invasive prenatal screening: A 20-year experience in Italy. European Journal of Obstetrics & Gynecology and Reproductive Biology: X, 3 , 100050. www.sciencedirect.com/scienc...90161319300857
Ramirez-Montiel, M. L., Casillas-Barrera, M., Morales-Morales, M. P., Ortiz, M. I., Lopez de Lara-Diaz de Leon, O., Carrasco-Blancas, E. R., & Morales, L. R. (2017). Complications associated with amniocentesis in the third trimester of pregnancy. Journal of Clinical Gynecology and Obstetrics, 6 (2), 34–36. https://www.jcgo.org/index.php/JCGO/article/view/452
Raymond, Y. C., Fernando, S., Menezes, M., Meagher, S., Mol, B. W., McLennan, A., Scott, F., Mizia, K., Carey, K., Fleming, G., & Rolnik, D. L. (2022). Cell-free DNA screening for rare autosomal trisomies and segmental chromosome imbalances. Prenatal Diagnosis, 42 (11), 1349–1357. doi.org/10.1002/pd.6233
Ruano, R., & Vega, B. (2019). Fetal surgery: How recent technological advancements are extending its applications. Expert Review of Medical Devices, 16 (8), 643–645. doi.org/10.1080/17434440.2019.1641404
Sapoval, J., Singh, V., Carter, R. E. (2023, January 16). Ultrasound biophysical profile . StatPearls [Internet]. www.ncbi.nlm.nih.gov/books/NBK539866/
Shiefa, S., Amargandhi, M., Bhupendra, J., Moulali, S., & Kristine, T. (2013, January). First trimester maternal serum screening using biochemical markers papp-A and free β-hcg for Down syndrome, Patau syndrome and Edward syndrome. Indian Journal of Clinical Biochemistry , 28 (1), 3–12. www.ncbi.nlm.nih.gov/pmc/articles/PMC3547446/
Srisuro A. (2022, April). The top 6 pregnancy questions I hear from first-time moms . ACOG. https://www.acog.org/womens-health/e...irst-time-moms
U.S. National Library of Medicine. (2021a, August). Nonstress test . https://medlineplus.gov/lab-tests/nonstress-test/
U.S. National Library of Medicine. (2021b). Prenatal cell-free DNA screening . MedlinePlus. https://medlineplus.gov/lab-tests/pr...dna-screening/
U.S. National Library of Medicine. (2022a, April 19). Monitoring your baby before labor . MedlinePlus. https://medlineplus.gov/ency/patient...ons/000485.htm
U.S. National Library of Medicine. (2022b, May 16). Amniocentesis (amniotic fluid test) . MedlinePlus. https://medlineplus.gov/lab-tests/am...ic-fluid-test/
U.S. National Library of Medicine. (2022c, July 22). Alpha-fetoprotein (AFP) test . https://medlineplus.gov/lab-tests/al...tein-afp-test/
U.S. National Library of Medicine. (2022d, November 8). Down syndrome tests . MedlinePlus. https://medlineplus.gov/lab-tests/do...yndrome-tests/
William, W. A., Nelson, G., & Grogan, S. P. (2023, July 31). Sonography 1st trimester assessment, protocols, and interpretation . StatPearls [Internet]. www.ncbi.nlm.nih.gov/books/NBK573070/
Yoham, A., & Casadesus, D. (2023, May 22). Rho(D) Immune Globulin . StatPearls [Internet]. www.ncbi.nlm.nih.gov/books/NBK557884/


9: Social Emotional Development in Early Childhood
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Learning Objectives
After reading this chapter, you should be able to:
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Describe how preschoolers view themselves.
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Summarize Erikson’s stage of initiative versus guilt.
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Discuss the progression of social emotional development during early childhood.
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Explain how children develop their understanding of gender.
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Compare and contrast different styles of parenting.
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Define characteristics of high quality child care.
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Discuss the role of siblings and peers.
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Describe the types of play.
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Summarize the influence on social and emotional competence.
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Identify the effects of stress on three- to five-year olds.
In early childhood, children’s understanding of themselves and their role in the world expands greatly.
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9.1: Social and Emotional Milestones
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Here is a table of social and emotional milestones that children typically experience during early childhood.
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9.2: Interactionism and Views of Self
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Early childhood is a time of forming an initial sense of self. A self-concept or idea of who we are, what we are capable of doing, and how we think and feel is a social process that involves taking into consideration how others view us. So, in order to develop a sense of self, you must have interaction with others. Interactionist theorists, Cooley and Mead offer two interesting explanations of how a sense of self develops.
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9.3: Erikson- Initiative vs. Guilt
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Psychologist Erik Erikson argues that children in early childhood go through a stage of “initiative vs. guilt”. If the child is placed in an environment where he/she can explore, make decisions, and initiate activities, they have achieved initiative. On the other hand, if the child is put in an environment where initiation is repressed through criticism and control, he/she will develop a sense of guilt.
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9.4: Gender Identity, Gender Constancy, and Gender Roles
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Another important dimension of the self is the sense of self as male or female. Preschool-aged children become increasingly interested in finding out the differences between boys and girls both physically and in terms of what activities are acceptable for each. While 2-year-olds can identify some differences and learn whether they are boys or girls, preschoolers become more interested in what it means to be male or female.
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9.5: Family Life
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Relationships between parents and children continue to play a significant role in children’s development during early childhood. We will explore two models of parenting styles. Keep in mind that most parents do not follow any model completely. Real people tend to fall somewhere in between these styles. And sometimes parenting styles change from one child to the next or in times when the parent has more or less time and energy for parenting.
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9.6: Peers
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Relationships within the family (parent-child and siblings) are not the only significant relationships in a child’s life. Peer relationships are also important. Social interaction with another child who is similar in age, skills, and knowledge provokes the development of many social skills that are valuable for the rest of life (Bukowski, Buhrmester, & Underwood, 2011).
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9.7: Play
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Freud saw play as a means for children to release pent-up emotions and to deal with emotionally distressing situations in a more secure environment. Vygotsky and Piaget saw play as a way of children developing their intellectual abilities (Dyer & Moneta, 2006). Piaget created stages of play that correspond with his stages of cognitive development.
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9.8: Social Understanding
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As we have seen, children’s experience of relationships at home and the peer group contributes to an expanding repertoire of social and emotional skills and also to broadened social understanding. In these relationships, children develop expectations for specific people (leading, for example, to secure or insecure attachments to parents), understanding of how to interact with adults and peers, and developing self-concept based on how others respond to them.
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9.9: Personality
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Parents often scrutinize their child’s preferences, characteristics, and responses for clues of a developing personality. They are quite right to do so because temperament is a foundation for personality growth. But temperament (defined as early-emerging differences in reactivity and self-regulation) is not the whole story.
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9.10: Social and Emotional Competence
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Social and personality development is built from the social, biological, and representational influences discussed above. These influences result in important developmental outcomes that matter to children, parents, and society: a young adult’s capacity to engage in socially constructive actions (helping, caring, sharing with others), to curb hostile or aggressive impulses, to live according to meaningful moral values.
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9.11: Childhood Stress and Development
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Children experience different types of stressors. Normal, everyday stress can provide an opportunity for young children to build coping skills and poses little risk to development. Even more long-lasting stressful events such as changing schools or losing a loved one can be managed fairly well. But children who experience toxic stress or who live in extremely stressful situations of abuse over long periods of time can suffer long-lasting effects.
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9.S: Summary
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Thumbnail: www.pexels.com/photo/tilt-sh...short-1027931/


18.12: Occurrence, Preparation, and Properties of the Noble Gases
- Describe the properties, preparation, and uses of the noble gases
The elements in group 18 are the noble gases (helium, neon, argon, krypton, xenon, and radon). They earned the name “noble” because they were assumed to be nonreactive since they have filled valence shells. In 1962, Dr. Neil Bartlett at the University of British Columbia proved this assumption to be false.
These elements are present in the atmosphere in small amounts. Some natural gas contains 1–2% helium by mass. Helium is isolated from natural gas by liquefying the condensable components, leaving only helium as a gas. The United States possesses most of the world’s commercial supply of this element in its helium-bearing gas fields. Argon, neon, krypton, and xenon come from the fractional distillation of liquid air. Radon comes from other radioactive elements. More recently, it was observed that this radioactive gas is present in very small amounts in soils and minerals. Its accumulation in well-insulated, tightly sealed buildings, however, constitutes a health hazard, primarily lung cancer.
The boiling points and melting points of the noble gases are extremely low relative to those of other substances of comparable atomic or molecular masses. This is because only weak London dispersion forces are present, and these forces can hold the atoms together only when molecular motion is very slight, as it is at very low temperatures. Helium is the only substance known that does not solidify on cooling at normal pressure. It remains liquid close to absolute zero (0.001 K) at ordinary pressures, but it solidifies under elevated pressure.
Helium is used for filling balloons and lighter-than-air craft because it does not burn, making it safer to use than hydrogen. Helium at high pressures is not a narcotic like nitrogen. Thus, mixtures of oxygen and helium are important for divers working under high pressures. Using a helium-oxygen mixture avoids the disoriented mental state known as nitrogen narcosis, the so-called rapture of the deep. Helium is important as an inert atmosphere for the melting and welding of easily oxidizable metals and for many chemical processes that are sensitive to air.
Liquid helium (boiling point, 4.2 K) is an important coolant to reach the low temperatures necessary for cryogenic research, and it is essential for achieving the low temperatures necessary to produce superconduction in traditional superconducting materials used in powerful magnets and other devices. This cooling ability is necessary for the magnets used for magnetic resonance imaging, a common medical diagnostic procedure. The other common coolant is liquid nitrogen (boiling point, 77 K), which is significantly cheaper.
Neon is a component of neon lamps and signs. Passing an electric spark through a tube containing neon at low pressure generates the familiar red glow of neon. It is possible to change the color of the light by mixing argon or mercury vapor with the neon or by utilizing glass tubes of a special color.
Argon was useful in the manufacture of gas-filled electric light bulbs, where its lower heat conductivity and chemical inertness made it preferable to nitrogen for inhibiting the vaporization of the tungsten filament and prolonging the life of the bulb. Fluorescent tubes commonly contain a mixture of argon and mercury vapor. Argon is the third most abundant gas in dry air.
Krypton-xenon flash tubes are used to take high-speed photographs. An electric discharge through such a tube gives a very intense light that lasts only \(\dfrac{1}{50,000}\) of a second. Krypton forms a difluoride, KrF 2 , which is thermally unstable at room temperature.
Stable compounds of xenon form when xenon reacts with fluorine. Xenon difluoride, XeF 2 , forms after heating an excess of xenon gas with fluorine gas and then cooling. The material forms colorless crystals, which are stable at room temperature in a dry atmosphere. Xenon tetrafluoride, XeF 4 , (Figure \(\PageIndex{1}\)) and xenon hexafluoride, XeF 6 , are prepared in an analogous manner, with a stoichiometric amount of fluorine and an excess of fluorine, respectively. Compounds with oxygen are prepared by replacing fluorine atoms in the xenon fluorides with oxygen.
When XeF 6 reacts with water, a solution of XeO 3 results and the xenon remains in the 6+-oxidation state:
\[\ce{XeF6}(s)+\ce{3H2O}(l)⟶\ce{XeO3}(aq)+\ce{6HF}(aq) \nonumber \]
Dry, solid xenon trioxide, XeO 3 , is extremely explosive—it will spontaneously detonate. Both XeF 6 and XeO 3 disproportionate in basic solution, producing xenon, oxygen, and salts of the perxenate ion, \(\ce{XeO6^4-}\), in which xenon reaches its maximum oxidation sate of 8+.
Radon apparently forms RnF 2 —evidence of this compound comes from radiochemical tracer techniques.
Unstable compounds of argon form at low temperatures, but stable compounds of helium and neon are not known.
Summary
The most significant property of the noble gases (group 18) is their inactivity. They occur in low concentrations in the atmosphere. They find uses as inert atmospheres, neon signs, and as coolants. The three heaviest noble gases react with fluorine to form fluorides. The xenon fluorides are the best characterized as the starting materials for a few other noble gas compounds.
Glossary
- halide
- compound containing an anion of a group 17 element in the 1− oxidation state (fluoride, F − ; chloride, Cl − ; bromide, Br − ; and iodide, I − )
- interhalogen
- compound formed from two or more different halogens


3.4.7: Slater's Rules for Effective Nuclear Charge
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Skills to Develop
- Describe how to calculate the shielding constant using Slater's rules
- Identify a periodic trend for effective nuclear charge
- Calculate effective nuclear charge
Effective nuclear charge is really important, because it determines the size and energy of orbitals, which determine most properties of atoms. So it's useful to be able to predict effective nuclear charge! Slater's rules give a simple approximation of effective nuclear charge that works pretty well.
Based on the last section, we can expect that effective nuclear charge will depend on the number of electrons that might get between, so it depends on the electron we are looking at. For any electron, to find the effective nuclear charge it feels, we need to know how many other electrons might get in the way, and how much time it spends near the nucleus. Based on these, we will calculate a shielding constant , S. Then,
\[Z_{eff}=Z-S\]
where Z is the actual nuclear charge (which is the same as the atomic number) and Z eff is the effective nuclear charge.
To calculate S, we will write out all the electrons in atom until we get to the group of the electron we want, like this:
(1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)(5s,5p) etc.
- Each other electron (not counting the one we have picked) in the same group () as the chosen electron, contributes 0.35 to S. (This means electrons in the same group shield each other 35%.
- If the chosen electron is d or f, every electron in groups to the left contributes 1.00 to S. (This means that d and f electrons are shielded 100% by electrons with lower n or same n and lower ℓ.)
- If the chosen electron is s or p, all the electrons in the next lower shell (n - 1) contribute 0.85 to S. (This means that s and p electrons are shielded 85% by the electrons one shell lower.) And all the electrons in even lower shells contribute 1.00 to S. (All electrons in shells n - 2 or lower shield 100%.)
The outcome of this is that Z eff changes suddenly when going from one period to another. As you go from Li to Be, Z eff (for the new electron) increases, because you add one proton (Z + 1) and it is only shielded 35% (S + 0.35). When you get to B, you added one proton, and it still shields 35%. So Z eff increases until you go from Ne to Na. Now, suddenly, the (1s) electrons shield 100% instead of 85%, and the (2s,2p) shield 85% instead of 35%! So Z eff goes down suddenly. From Na to Ar, Z eff increases slowly again. From Ar to K, it drops again.
For an example, let's calculate Z eff for a d electron in Zn, atomic number 30. Notice that although 4s is filled, we don't include it because it comes to the right of the d electrons we are looking at.
(1s)(2s,2p)(3s,3p)(3d)
\[S=18(1)+9(0.35)=21.15\]
\[Z_{eff}=31-26=5\]
You can see that just like changing periods (going to a new shell), going from the d-block to the p-block also gives a drop in Z eff (partly because you actually are going to a new shell, as well as subshell).


6.6: Determinants. Jacobians. Bijective Linear Operators
We assume the reader to be familiar with elements of linear algebra. Thus we only briefly recall some definitions and well-known rules.
Given a linear operator \(\phi : E^{n} \rightarrow E^{n}\left(\text { or } \phi : C^{n} \rightarrow C^{n}\right),\) with matrix
\[
[\phi]=\left(v_{i k}\right), \quad i, k=1, \ldots, n,
\]
we define the determinant of \([\phi]\) by
\[
\begin{aligned} \operatorname{det}[\phi]=\operatorname{det}\left(v_{i k}\right) &=\left|\begin{array}{cccc}{v_{11}} & {v_{12}} & {\dots} & {v_{1 n}} \\ {v_{21}} & {v_{22}} & {\dots} & {v_{2 n}} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ {v_{n 1}} & {v_{n 2}} & {\dots} & {v_{n n}}\end{array}\right| \\[12pt] &=\sum(-1)^{\lambda} v_{1 k_{1}} v_{2 k_{2}} \ldots v_{n k_{n}} \end{aligned}
\]
where the sum is over all ordered \(n\)-tuples \(\left(k_{1}, \ldots, k_{n}\right)\) of distinct integers \(k_{j}\left(1 \leq k_{j} \leq n\right),\) and
\[
\lambda=\left\{\begin{array}{ll}{0} & {\text { if } \prod_{j<m}\left(k_{m}-k_{j}\right)>0 \text { and }} \\ {1} & {\text { if } \prod_{j<m}\left(k_{m}-k_{j}\right)<0}\end{array}\right.
\]
Recall (Problem 12 in §2) that a set \(B=\left\{\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{n}\right\}\) in a vector space \(E\) is a basis iff
(i) \(B\) spans \(E,\) i.e., each \(\vec{v} \in E\) has the form
\[
\vec{v}=\sum_{i=1}^{n} a_{i} \vec{v}_{i}
\]
for some scalars \(a_{i},\) and
(ii) this representation is unique.
The latter is true iff the \(\vec{v}_{i}\) are independent, i.e.,
\[
\sum_{i=1}^{n} a_{i} \vec{v}_{i}=\overrightarrow{0} \Longleftrightarrow a_{i}=0, i=1, \ldots, n.
\]
If \(E\) has a basis of \(n\) vectors, we call \(E\) n-dimensional (e.g., \(E^{n}\) and \(C^{n} )\).
Determinants and bases satisfy the following rules.
(a) Multiplication rule. If \(\phi, g : E^{n} \rightarrow E^{n}\left(\text { or } C^{n} \rightarrow C^{n}\right)\) are linear, then
\[
\operatorname{det}[g] \cdot \operatorname{det}[\phi]=\operatorname{det}([g][\phi])=\operatorname{det}[g \circ \phi]
\]
(see §2, Theorem 3 and Note 4).
(b) If \(\phi(\vec{x})=\vec{x}\) (identity map), then \([\phi]=\left(v_{i k}\right)\), where
\[
v_{i k}=\left\{\begin{array}{ll}{0} & {\text { if } i \neq k \text { and }} \\ {1} & {\text { if } i=k}\end{array}\right.
\]
hence det \([\phi]=1 .(\text { Why } ?)\) See also the Problems.
(c) An \(n\) -dimensional space \(E\) is spanned by a set of \(n\) vectors iff they are independent. If so, each basis consists of exactly \(n\) vectors.
For any function \(f : E^{n} \rightarrow E^{n}\) (or \(f : C^{n} \rightarrow C^{n} ),\) we define the \(f\)-induced Jacobian map \(J_{f} : E^{n} \rightarrow E^{1}\left(J_{f} : C^{n} \rightarrow C\right)\) by setting
\[
J_{f}(\vec{x})=\operatorname{det}\left(v_{i k}\right),
\]
where \(v_{i k}=D_{k} f_{i}(\vec{x}), \vec{x} \in E^{n}\left(C^{n}\right),\) and \(f=\left(f_{1}, \ldots, f_{n}\right)\).
The determinant
\[
J_{f}(\vec{p})=\operatorname{det}\left(D_{k} f_{i}(\vec{p})\right)
\]
is called the Jacobian of \(f\) at \(\vec{p}\).
By our conventions, it is always defined, as are the functions \(D_{k} f_{i}\).
Explicitly, \(J_{f}(\vec{p})\) is the determinant of the right-side matrix in formula \((14)\) in §3. Briefly,
\[
J_{f}=\operatorname{det}\left(D_{k} f_{i}\right).
\]
By Definition 2 and Note 2 in §5,
\[
J_{f}(\vec{p})=\operatorname{det}\left[d^{1} f(\vec{p} ; \cdot)\right].
\]
If \(f\) is differentiable at \(\vec{p}\),
\[
J_{f}(\vec{p})=\operatorname{det}\left[f^{\prime}(\vec{p})\right].
\]
Note 1. More generally, given any functions \(v_{i k} : E^{\prime} \rightarrow E^{1}(C),\) we can define a map \(f : E^{\prime} \rightarrow E^{1}(C)\) by
\[
f(\vec{x})=\operatorname{det}\left(v_{i k}(\vec{x})\right);
\]
briefly \(f=\operatorname{det}\left(v_{i k}\right), i, k=1, \ldots, n\).
We then call \(f\) a functional determinant.
If \(E^{\prime}=E^{n}\left(C^{n}\right)\) then \(f\) is a function of \(n\) variables, since \(\vec{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right)\). If all \(v_{i k}\) are continuous or differentiable at some \(\vec{p} \in E^{\prime},\) so is \(f ;\) for by \((1), f\) is a finite sum of functions of the form
\[
(-1)^{\lambda} v_{i k_{1}} v_{i k_{2}} \dots v_{i k_{n}},
\]
and each of these is continuous or differentiable if the \(v_{i k_{i}}\) are (see Problems 7 and 8 in §3).
Note 2. Hence the Jacobian map \(J_{f}\) is continuous or differentiable at \(\vec{p}\) if all the partially derived functions \(D_{k} f_{i}(i, k \leq n)\) are.
If, in addition, \(J_{f}(\vec{p}) \neq 0,\) then \(J_{f} \neq 0\) on some globe about \(\vec{p}.\) (Apply Problem 7 in Chapter 4, §2, to \(\left|J_{f}\right|.)\)
In classical notation, one writes
\[
\frac{\partial\left(f_{1}, \ldots, f_{n}\right)}{\partial\left(x_{1}, \ldots, x_{n}\right)} \text { or } \frac{\partial\left(y_{1}, \ldots, y_{n}\right)}{\partial\left(x_{1}, \ldots, x_{n}\right)}
\]
for \(J_{f}(\vec{x}) .\) Here \(\left(y_{1}, \ldots, y_{n}\right)=f\left(x_{1}, \ldots, x_{n}\right)\).
The remarks made in §4 apply to this "variable" notation too. The chain rule easily yields the following corollary.
If \(f : E^{n} \rightarrow E^{n}\) and \(g : E^{n} \rightarrow E^{n}\) (or \(f, g : C^{n} \rightarrow C^{n})\) are differentiable at \(\vec{p}\) and \(\vec{q}=f(\vec{p}),\) respectively, and if
\[h=g \circ f,\]
then
\[J_{h}(\vec{p})=J_{g}(\vec{q}) \cdot J_{f}(\vec{p})=\operatorname{det}\left(z_{i k}\right),\]
where
\[z_{i k}=D_{k} h_{i}(\vec{p}), \quad i, k=1, \ldots, n;\]
or, setting
\[\begin{aligned}\left(u_{1}, \ldots, u_{n}\right) &=g\left(y_{1}, \ldots, y_{n}\right) \text { and } \\\left(y_{1}, \ldots, y_{n}\right) &=f\left(x_{1}, \ldots, x_{n}\right) \text { ("variables")}, \end{aligned}\]
we have
\[\frac{\partial\left(u_{1}, \ldots, u_{n}\right)}{\partial\left(x_{1}, \ldots, x_{n}\right)}=\frac{\partial\left(u_{1}, \ldots, u_{n}\right)}{\partial\left(y_{1}, \ldots, y_{n}\right)} \cdot \frac{\partial\left(y_{1}, \ldots, y_{n}\right)}{\partial\left(x_{1}, \ldots, x_{n}\right)}=\operatorname{det}\left(z_{i k}\right),\]
where
\[z_{i k}=\frac{\partial u_{i}}{\partial x_{k}}, \quad i, k=1, \ldots, n.\]
- Proof
-
By Note 2 in §4,
\[\left[h^{\prime}(\vec{p})\right]=\left[g^{\prime}(\vec{q})\right] \cdot\left[f^{\prime}(\vec{p})\right].\]
Thus by rule (a) above,
\[\operatorname{det}\left[h^{\prime}(\vec{p})\right]=\operatorname{det}\left[g^{\prime}(\vec{q})\right] \cdot \operatorname{det}\left[f^{\prime}(\vec{p})\right],\]
i.e.,
\[J_{h}(\vec{p})=J_{g}(\vec{q}) \cdot J_{f}(\vec{p}).\]
Also, if \(\left[h^{\prime}(\vec{p})\right]=\left(z_{i k}\right),\) Definition 2 yields \(z_{i k}=D_{k} h_{i}(\vec{p})\).
This proves (i), hence (ii) also. \(\quad \square\)
In practice, Jacobians mostly occur when a change of variables is made. For instance, in \(E^{2},\) we may pass from Cartesian coordinates \((x, y)\) to another system \((u, v)\) such that
\[x=f_{1}(u, v) \text { and } y=f_{2}(u, v).\]
We then set \(f=\left(f_{1}, f_{2}\right)\) and obtain \(f : E^{2} \rightarrow E^{2}\),
\[J_{f}=\operatorname{det}\left(D_{k} f_{i}\right), \quad k, i=1,2.\]
Let \(x=f_{1}(r, \theta)=r \cos \theta\) and \(y=f_{2}(r, \theta)=r \sin \theta\).
Then using the "variable" notation, we obtain \(J_{f}(r, \theta)\) as
\[\begin{aligned} \frac{\partial(x, y)}{\partial(r, \theta)}=\left|\begin{array}{ll}{\frac{\partial x}{\partial r}} & {\frac{\partial x}{\partial \theta}} \\ {\frac{\partial y}{\partial r}} & {\frac{\partial y}{\partial \theta}}\end{array}\right| &=\left|\begin{array}{cc}{\cos \theta} & {-r \sin \theta} \\ {\sin \theta} & {r \cos \theta}\end{array}\right| \\ &=r \cos ^{2} \theta+r \sin ^{2} \theta=r. \end{aligned}\]
Thus here \(J_{f}(r, \theta)=r\) for all \(r, \theta \in E^{1} ; J_{f}\) is independent of \(\theta\).
We now concentrate on one-to-one (invertible) functions.
For a linear map \(\phi : E^{n} \rightarrow E^{n}\left(\text {or} \phi : C^{n} \rightarrow C^{n}\right),\) the following are equivalent:
(i) \(\phi\) is one-to-one;
(ii) the column vectors \(\vec{v}_{1}, \ldots, \vec{v}_{n}\) of the matrix \([\phi]\) are independent;
(iii) \(\phi\) is onto \(E^{n}\left(C^{n}\right)\);
(iv) \(\operatorname{det}[\phi] \neq 0\).
- Proof
-
Assume (i) and let
\[\sum_{k=1}^{n} c_{k} \vec{v}_{k}=\overrightarrow{0}.\]
To deduce (ii), we must show that all \(c_{k}\) vanish.
Now, by Note 3 in §2, \(\vec{v}_{k}=\phi\left(\vec{e}_{k}\right);\) so by linearity,
\[\sum_{k=1}^{n} c_{k} \vec{v}_{k}=\overrightarrow{0}\]
implies
\[\phi\left(\sum_{k=1}^{n} c_{k} \vec{e}_{k}\right)=\overrightarrow{0}.\]
As \(\phi\) is one-to-one, it can vanish at \(\overrightarrow{0}\) only. Thus
\[\sum_{k=1}^{n} c_{k} \vec{e}_{k}=\overrightarrow{0}.\]
Hence by Theorem 2 in Chapter 3, §§1-3, \(c_{k}=0, k=1, \ldots, n,\) and (ii) follows.
Next, assume (ii); so, by rule (c) above, \(\left\{\vec{v}_{1}, \ldots, \vec{v}_{n}\right\}\) is a basis.
Thus each \(\vec{y} \in E^{n}\left(C^{n}\right)\) has the form
\[\vec{y}=\sum_{k=1}^{n} a_{k} \vec{v}_{k}=\sum_{k=1}^{n} a_{k} \phi\left(\vec{e}_{k}\right)=\phi\left(\sum_{k=1}^{n} a_{k} \vec{e}_{k}\right)=\phi(\vec{x}),\]
where
\[\vec{x}=\sum_{k=1}^{n} a_{k} \vec{e}_{k} \text { (uniquely).}\]
Hence (ii) implies both (iii) and (i). (Why?)
Now assume (iii). Then each \(\vec{y} \in E^{n}\left(C^{n}\right)\) has the form \(\vec{y}=\phi(\vec{x}),\) where
\[\vec{x}=\sum_{k=1}^{n} x_{k} \vec{e}_{k},\]
by Theorem 2 in Chapter 3, §§1-3. Hence again
\[\vec{y}=\sum_{k=1}^{n} x_{k} \phi\left(\vec{e}_{k}\right)=\sum_{k=1}^{n} x_{k} \vec{v}_{k};\]
so the \(\vec{v}_{k}\) span all of \(E^{n}\left(C^{n}\right).\) By rule (c) above, this implies (ii), hence (i), too. Thus (i), (ii), and (iii) are equivalent.
Also, by rules (a) and (b), we have
\[\operatorname{det}[\phi] \cdot \operatorname{det}\left[\phi^{-1}\right]=\operatorname{det}\left[\phi \circ \phi^{-1}\right]=1\]
if \(\phi\) is one-to-one (for \(\phi \circ \phi^{-1}\) is the identity map). Hence \(\operatorname{det}[\phi] \neq 0\) if (i) holds.
For the converse, suppose \(\phi\) is not one-to-one. Then by (ii), the \(\vec{v}_{k}\) are not independent. Thus one of them is a linear combination of the others, say,
\[\vec{v}_{1}=\sum_{k=2}^{n} a_{k} \vec{v}_{k}.\]
But by linear algebra (Problem 13(iii)), \(\operatorname{det}[\phi]\) does not change if \(\vec{v}_{1}\) is replaced by
\[\vec{v}_{1}-\sum_{k=2}^{n} a_{k} \vec{v}_{k}=\overrightarrow{0}.\]
Thus \(\operatorname{det}[\phi]=0\) (one column turning to \(\overrightarrow{0}).\) This completes the proof. \(\quad \square\)
Note 3. Maps that are both onto and one-to-one are called bijective. Such is \(\phi\) in Theorem 1. This means that the equation
\[\phi(\vec{x})=\vec{y}\]
has a unique solution
\[\vec{x}=\phi^{-1}(\vec{y})\]
for each \(\vec{y}.\) Componentwise, by Theorem 1, the equations
\[\sum_{k=1}^{n} x_{k} v_{i k}=y_{i}, \quad i=1, \ldots, n,\]
have a unique solution for the \(x_{k}\) iff \(\operatorname{det}\left(v_{i k}\right) \neq 0\).
If \(\phi \in L\left(E^{\prime}, E\right)\) is bijective, with \(E^{\prime}\) and \(E\) complete, then \(\phi^{-1} \in L\left(E, E^{\prime}\right).\)
- Proof for \(E=E^{n}\left(C^{n}\right)\)
-
The notation \(\phi \in L\left(E^{\prime}, E\right)\) means that \(\phi : E^{\prime} \rightarrow E\) is linear and continuous.
As \(\phi\) is bijective, \(\phi^{-1} : E \rightarrow E^{\prime}\) is linear (Problem 12).
If \(E=E^{n}\left(C^{n}\right),\) it is continuous, too (Theorem 2 in §2).
Thus \(\phi^{-1} \in L\left(E, E^{\prime}\right). \quad \square\)
Note. The case \(E=E^{n}\left(C^{n}\right)\) suffices for an undergraduate course. (The beginner is advised to omit the "starred" §8.) Corollary 2 and Theorem 2 below, however, are valid in the general case. So is Theorem 1 in §7.
Let \(E, E^{\prime}\) and \(\phi\) be as in Corollary 2. Set
\[\left\|\phi^{-1}\right\|=\frac{1}{\varepsilon}.\]
Then any map \(\theta \in L\left(E^{\prime}, E\right)\) with \(\|\theta-\phi\|<\varepsilon\) is one-to-one, and \(\theta^{-1}\) is uniformly continuous.
- Proof
-
Proof. By Corollary 2, \(\phi^{-1} \in L\left(E, E^{\prime}\right),\) so \(\left\|\phi^{-1}\right\|\) is defined and \(>0\) (for \(\phi^{-1}\) is not the zero map, being one-to-one).
Thus we may set
\[\varepsilon=\frac{1}{\left\|\phi^{-1}\right\|}, \quad\left\|\phi^{-1}\right\|=\frac{1}{\varepsilon}.\]
Clearly \(\vec{x}=\phi^{-1}(\vec{y})\) if \(\vec{y}=\phi(\vec{x}).\) Also,
\[\left|\phi^{-1}(\vec{y})\right| \leq \frac{1}{\varepsilon}|\vec{y}|\]
by Note 5 in §2, Hence
\[|\vec{y}| \geq \varepsilon\left|\phi^{-1}(\vec{y})\right|,\]
i.e.,
\[|\phi(\vec{x})| \geq \varepsilon|\vec{x}|\]
for all \(\vec{x} \in E^{\prime}\) and \(\vec{y} \in E\).
Now suppose \(\phi \in L\left(E^{\prime}, E\right)\) and \(\|\theta-\phi\|=\sigma<\varepsilon\).
Obviously, \(\theta=\phi-(\phi-\theta),\) and by Note 5 in §2,
\[|(\phi-\theta)(\vec{x})| \leq\|\phi-\theta\||\vec{x}|=\sigma|\vec{x}|.\]
Thus for every \(\vec{x} \in E^{\prime}\),
\[\begin{aligned}|\theta(\vec{x})| & \geq|\phi(\vec{x})|-|(\phi-\theta)(\vec{x})| \\ & \geq|\phi(\vec{x})|-\sigma|\vec{x}| \\ & \geq(\varepsilon-\sigma)|\vec{x}| \end{aligned}\]
by (2). Therefore, given \(\vec{p} \neq \vec{r}\) in \(E^{\prime}\) and setting \(\vec{x}=\vec{p}-\vec{r} \neq \overrightarrow{0},\) we obtain
\[|\theta(\vec{p})-\theta(\vec{r})|=|\theta(\vec{p}-\vec{r})|=|\theta(\vec{x})| \geq(\varepsilon-\sigma)|\vec{x}|>0\]
(since \(\sigma<\varepsilon )\).
We see that \(\vec{p} \neq \vec{r}\) implies \(\theta(\vec{p}) \neq \theta(\vec{r});\) so \(\theta\) is one-to-one, indeed.
Also, setting \(\theta(\vec{x})=\vec{z}\) and \(\vec{x}=\theta^{-1}(\vec{z})\) in (3), we get
\[|\vec{z}| \geq(\varepsilon-\sigma)\left|\theta^{-1}(\vec{z})\right|;\]
that is,
\[\left|\theta^{-1}(\vec{z})\right| \leq(\varepsilon-\sigma)^{-1}|\vec{z}|\]
for all \(\vec{z}\) in the range of \(\theta\) (domain of \(\theta^{-1})\).
Thus \(\theta^{-1}\) is linearly bounded (by Theorem 1 in §2), hence uniformly continuous, as claimed.\(\quad \square\)
If \(E^{\prime}=E=E^{n}\left(C^{n}\right)\) in Theorem 2 above, then for given \(\phi\) and \(\delta>0,\) there always is \(\delta^{\prime}>0\) such that
\[\|\theta-\phi\|<\delta^{\prime} \text { implies }\left\|\theta^{-1}-\phi^{-1}\right\|<\delta.\]
In other words, the transformation \(\phi \rightarrow \phi^{-1}\) is continuous on \(L(E), E=\) \(E^{n}\left(C^{n}\right).\)
- Proof
-
First, since \(E^{\prime}=E=E^{n}\left(C^{n}\right), \theta\) is bijective by Theorem 1(iii), so \(\theta^{-1} \in L(E)\).
As before, set \(\|\theta-\phi\|=\sigma<\varepsilon\).
By Note 5 in §2, formula (5) above implies that
\[\left\|\theta^{-1}\right\| \leq \frac{1}{\varepsilon-\sigma}.\]
Also,
\[\phi^{-1} \circ(\theta-\phi) \circ \theta^{-1}=\phi^{-1}-\theta^{-1}\]
(see Problem 11).
Hence by Corollary 4 in §2, recalling that \(\left\|\phi^{-1}\right\|=1 / \varepsilon,\) we get
\[\left\|\theta^{-1}-\phi^{-1}\right\| \leq\left\|\phi^{-1}\right\| \cdot\|\theta-\phi\| \cdot\left\|\theta^{-1}\right\| \leq \frac{\sigma}{\varepsilon(\varepsilon-\sigma)} \rightarrow 0 \text { as } \sigma \rightarrow 0. \quad \square\]


20.6: Activities
A standard Alberta license plate has three letters followed by three or four digits.
- How many different vehicles can the province license with this scheme?
- Do you think the province was right to expand license plates by adding another digit, or do you think it should have added another letter instead? (Or, as a third possibility, is it irrelevant in practical terms?)
- Hint.
-
The figure \(26^3 = 17576\) may help you decide.
 How many different sequences of rolls are possible?
- Describe how Task a relates to the problem of determining \(\vert \Sigma^{\ast}_{10} \vert\) for a suitable alphabet \(\Sigma\text{.}\)
Let \(\Sigma = \{a, b, c, \ldots, y, z\}\text{.}\) How many words in \(\Sigma^{\ast}_5\) end in the letter \(z\text{?}\) How many do not?
You and your five housemates pick names out of a hat each week to determine who is going to clean the toilet. Over a three-week period, how many different sequences of toilet bowl cleaners could be determined in this fashion
- if names are placed back in the hat after each draw?
- if names are not placed back in the hat after each draw?
How many natural numbers between \(1\) and \(1,000,000\) (inclusive) contain the digit \(5\text{?}\)
- Hint.
-
You might instead count how many numbers don't contain the digit \(5\text{.}\)
How many natural numbers between \(100\) and \(999\) (inclusive) have no repeated digits? Of these, how many are odd?
- Hint.
-
There's no rule that when you “construct” an arbitrary object of this type that you have to choose the first digit first.
Use the Pigeonhole Principle to prove that in every set of three integers there exists a pair whose difference is even.
- Hint.
-
What kinds of numbers add up to an even sum?
You have a list of the names of twenty students. Ten of the students are domestic students and the other ten are out-of-province students. How many students must you select from the list to be certain to form a group that contains at least one domestic student and at least one out-of-province student?
Let \(n\) be a fixed natural number. Determine the smallest number \(M\) for which the following statement is true: every subset of
\begin{equation*} \mathbb{N}_{<2n+1} = \{0,1,2,3,\ldots ,2n\} \end{equation*}
of size \(M\) contains at least one odd number.
 There are \(T\) toys on the floor and \(n\) empty toy storage boxes. You randomly throw toys into boxes, and when you're done the box with the most toys contains \(N\) toys.
- What is the smallest that \(N\) could be when \(T = 2 n + 1\text{?}\)
- What is the smallest that \(N\) could be when \(T = k n + 1\text{?}\)
- Now suppose that the number of toys \(T\) satisfies
\begin{equation*} T \lt \dfrac{n (n - 1)}{2} \text{.} \end{equation*}
Prove that when you are done cleaning up, there will be (at least) one pair of boxes that contain the same number of toys.
- Hint.
-
Argue the contrapositive by assuming that every box ends up a different number of toys. What is the fewest number of toys you could have started with?


3.6: Reaction Mechanisms
- Distinguish net reactions from elementary reactions (steps)
- Identify the molecularity of elementary reactions
- Write a balanced chemical equation for a process given its reaction mechanism
- Derive the rate law consistent with a given reaction mechanism
A balanced equation for a chemical reaction indicates what is reacting and what is produced, but it reveals nothing about how the reaction actually takes place. The reaction mechanism (or reaction path) is the process, or pathway, by which a reaction occurs. A chemical reaction often occurs in steps, although it may not always be obvious to an observer. The decomposition of ozone, for example, appears to follow a mechanism with two steps:
\ce{O}+\ce{O3}(g)⟶\ce{2O2}(g) \label{12.7.1}\]
We call each step in a reaction mechanism an elementary reaction . Elementary reactions occur exactly as they are written and cannot be broken down into simpler steps. Elementary reactions add up to the overall reaction, which, for the decomposition, is:
Notice that the oxygen atom produced in the first step of this mechanism is consumed in the second step and therefore does not appear as a product in the overall reaction. Species that are produced in one step and consumed in a subsequent step are called intermediates .
While the overall reaction equation for the decomposition of ozone indicates that two molecules of ozone react to give three molecules of oxygen, the mechanism of the reaction does not involve the collision and reaction of two ozone molecules. Rather, it involves a molecule of ozone decomposing to an oxygen molecule and an intermediate oxygen atom; the oxygen atom then reacts with a second ozone molecule to give two oxygen molecules. These two elementary reactions occur exactly as they are shown in the reaction mechanism.
Unimolecular Elementary Reactions
The molecularity of an elementary reaction is the number of reactant species (atoms, molecules, or ions).  For example, the reaction:
illustrates a unimolecular elementary reaction that occurs as one part of a two-step reaction mechanism. However, some unimolecular reactions may have only a single reaction in the reaction mechanism. (In other words, an elementary reaction can also be an overall reaction in some cases.) For example, the gas-phase decomposition of cyclobutane, C 4 H 8 , to ethylene, C 2 H 4 , occurs via a unimolecular, single-step mechanism:
For these unimolecular reactions to occur, all that is required is the separation of parts of single reactant molecules into products.
Chemical bonds do not simply fall apart during chemical reactions. Energy is required to break chemical bonds. The activation energy for the decomposition of C 4 H 8 , for example, is 261 kJ per mole. This means that it requires 261 kilojoules to distort one mole of these molecules into activated complexes that decompose into products:
In a sample of C 4 H 8 , a few of the rapidly moving C 4 H 8 molecules collide with other rapidly moving molecules and pick up additional energy. When the C 4 H 8 molecules gain enough energy, they can transform into an activated complex, and the formation of ethylene molecules can occur. In effect, a particularly energetic collision knocks a C 4 H 8 molecule into the geometry of the activated complex. However, only a small fraction of gas molecules travel at sufficiently high speeds with large enough kinetic energies to accomplish this. Hence, at any given moment, only a few molecules pick up enough energy from collisions to react.
The rate of decomposition of C 4 H 8 is directly proportional to its concentration. Doubling the concentration of C 4 H 8 in a sample gives twice as many molecules per liter. Although the fraction of molecules with enough energy to react remains the same, the total number of such molecules is twice as great. Consequently, there is twice as much C 4 H 8 per liter, and the reaction rate is twice as fast:
A similar relationship applies to any unimolecular elementary reaction; the reaction rate is directly proportional to the concentration of the reactant, and the reaction exhibits first-order behavior. The proportionality constant is the rate constant for the particular unimolecular reaction.
Bimolecular Elementary Reactions
The collision and combination of two molecules or atoms to form an activated complex in an elementary reaction is called a bimolecular reaction . There are two types of bimolecular elementary reactions:
\[A+B⟶\ce{products} \label{12.7.6}\]
and
For the first type, in which the two reactant molecules are different, the rate law is first-order in A and first order in B :
For the second type, in which two identical molecules collide and react, the rate law is second order in A :
Some chemical reactions have mechanisms that consist of a single bimolecular elementary reaction. One example is the reaction of nitrogen dioxide with carbon monoxide:
(see Figure \(\PageIndex{1}\))
Bimolecular elementary reactions may also be involved as steps in a multistep reaction mechanism. The reaction of atomic oxygen with ozone is one example:
\[\ce{O}(g)+\ce{O3}(g)⟶\ce{2O2}(g) \label{12.7.12}\]
Termolecular Elementary Reactions
An elementary termolecular reaction involves the simultaneous collision of three atoms, molecules, or ions. Termolecular elementary reactions are uncommon because the probability of three particles colliding simultaneously is less than one one-thousandth of the probability of two particles colliding. There are, however, a few established termolecular elementary reactions. The reaction of nitric oxide with oxygen appears to involve termolecular steps:
\[\ce{2NO + O2 ⟶ 2NO2}\\
\ce{rate}=k[\ce{NO}]^2[\ce{O2}] \label{12.7.13}\]
Likewise, the reaction of nitric oxide with chlorine appears to involve termolecular steps:
\[\ce{2NO + Cl2 ⟶ 2NOCl}\\
\ce{rate}=k[\ce{NO}]^2[\ce{Cl2}] \label{12.7.14}\]
Relating Reaction Mechanisms to Rate Laws
It's often the case that one step in a multistep reaction mechanism is significantly slower than the others. Because a reaction cannot proceed faster than its slowest step, this step will limit the rate at which the overall reaction occurs. The slowest step is therefore called the rate-limiting step, or rate-determining step , of the reaction (Figure \(\PageIndex{2}\)).
Rate laws may be derived directly from the chemical equations for elementary reactions. This is only true for elementary reactions, and not the case for overall chemical reactions. The balanced equations most often encountered represent the overall change for some chemical system, and very often this is the result of some multistep reaction mechanism. In every case, we must determine the overall rate law from experimental data and deduce the mechanism from the rate law (and sometimes from other data). The reaction of NO 2 and CO provides an illustrative example:
For temperatures above 225 °C, the rate law has been found to be:
The reaction is first order with respect to NO 2 and first-order with respect to CO. This is consistent with a single-step bimolecular mechanism and it is possible that this is the mechanism for this reaction at high temperatures.
At temperatures below 225 °C , the reaction is described by a rate law that is second order with respect to NO 2 :
This is consistent with a mechanism that involves the following two elementary reactions, the first of which is slower and is therefore the rate-determining step:
\[\ce{NO2}(g)+\ce{NO2}(g)⟶\ce{NO3}(g)+\ce{NO}(g)\:\ce{(slow)}\\
\ce{NO3}(g)+\ce{CO}(g)⟶\ce{NO2}(g)+\ce{CO2}(g)\:\ce{(fast)}\]
The rate-determining step gives a rate law showing second-order dependence on the NO 2 concentration, and the sum of the two equations gives the net overall reaction.
In general, when the rate-determining (slower) step is the first step in a mechanism, the rate law for the overall reaction is the same as the rate law for this step. However, when the rate-determining step is preceded by a step involving an equilibrium reaction, the rate law for the overall reaction may be more difficult to derive.
An elementary reaction is at equilibrium when it proceeds in both the forward and reverse directions at equal rates. Consider the dimerization of NO to N 2 O 2 , with k 1 used to represent the rate constant of the forward reaction and k -1 used to represent the rate constant of the reverse reaction:
\[\ce{NO + NO ⇌ N2O2}\]
\[\ce{rate_{forward}=rate_{reverse}}\]
If N 2 O 2 was an intermediate in a mechanism, this expression could be rearranged to represent the concentration of N 2 O 2 in the overall rate law expression using algebraic manipulation:
Nitrogen monoxide, NO, reacts with hydrogen, H 2 , according to the following equation:
\[\ce{2NO}+\ce{2H2}⟶\ce{N2}+\ce{2H2O}\nonumber\]
What would the rate law be if the mechanism for this reaction were:
\[\ce{2NO}+\ce{H2}⟶\ce{N2}+\ce{H2O2}\:\ce{(slow)}\\
\ce{H2O2}+\ce{H2}⟶\ce{2H2O}\:\ce{(fast)}\nonumber\]
Solution
In the given mechanism, the first step is the rate-determining step. The rate law for this step is first-order with respect to H 2 and second-order with respect to NO.
\[\ce{rate}=k[\ce{NO}]^2[\ce{H2}]\nonumber\]
Because intermediates are present in very low amounts in reactions, and because we do not control these amounts, reaction intermedietes are not included as part of the overall rate law expression. Yet if they are involved in the rate-determining step, they can be included in an individual elementary reaction of a mechanism. Example \(\PageIndex{2}\) will illustrate how to derive overall rate laws from mechanisms involving equilibrium steps preceding the rate-determining step.
Mechanism Nitryl chloride (NO 2 Cl) decomposes to nitrogen dioxide (NO 2 ) and chlorine gas (Cl 2 ) according to the following mechanism:
- \(\ce{2NO2Cl}(g)⇌\ce{ClO2}(g)+\ce{N2O}(g)+\ce{ClO}(g)\) (fast, k 1 represents the rate constant for the forward reaction and k −1 the rate constant for the reverse reaction)
- \(\ce{N2O}(g)+\ce{ClO2}(g)⇌\ce{NO2}(g)+\ce{NOCl}(g)\) (fast, k 2 for the forward reaction, k −2 for the reverse reaction)
- \(\ce{NOCl + ClO ⟶ NO2 + Cl2}\) (slow, k 3 the rate constant for the forward reaction)
Determine the overall reaction, write the rate law expression for each elementary reaction, identify any intermediates, and determine the overall rate law expression.
Solution
For the overall reaction, simply sum the three steps, cancel intermediates, and combine like formulas:
Next, write the rate law expression for each elementary reaction. Remember that for elementary reactions that are part of a mechanism, the rate law expression can be derived directly from the stoichiometry:
k_1\ce{[NO2Cl]2}&=k_{−1}\ce{[ClO2][N2O][ClO]}\\
k_2\ce{[N2O][ClO2]}&=k_{−2}\ce{[NO2][NOCl]}\\
\ce{Rate}&=k_3\ce{[NOCl][ClO]}
\end{align*}\]
The third step, which is the slow step, is the rate-determining step. Therefore, the overall rate law expression could be written as Rate = k 3 [NOCl][ClO]. However, both NOCl and ClO are intermediates. Algebraic expressions must be used to represent [NOCl] and [ClO] such that no intermediates remain in the overall rate law expression.
- Using elementary reaction 1, \(\ce{[ClO]}=\dfrac{k_1\ce{[NO2Cl]^2}}{k_{−1}\ce{[ClO2][N2O]}}\).
- Using elementary reaction 2, \(\ce{[NOCl]}=\dfrac{k_2\ce{[N2O][ClO2]}}{k_{−2}\ce{[NO2]}}\).
Now substitute these algebraic expressions into the overall rate law expression and simplify:
\ce{rate}&=k_3\left(\dfrac{k_2\ce{[N2O][ClO2]}}{k_{−2}\ce{[NO2]}}\right)\left(\dfrac{k_1\ce{[NO2Cl]^2}}{k_{−1}\ce{[ClO2][N2O]}}\right)\\
\ce{rate}&=\dfrac{k_3k_2k_1\ce{[NO2Cl]^2}}{k_{−2}k_{−1}\ce{[NO2]}}
\end{align*}\]
Notice that this rate law shows an inverse dependence on the concentration of one of the product species, consistent with the presence of an equilibrium step in the reaction mechanism.
Atomic chlorine in the atmosphere reacts with ozone in the following pair of elementary reactions:
\(\ce{Cl}+\ce{O3}(g)⟶\ce{ClO}(g)+\ce{O2}(g)\hspace{20px}(\textrm{rate constant }k_1)\)
The second step is the rate-determining step. Determine the overall reaction, write the rate law expression for each elementary reaction, identify any intermediates, and determine the overall rate law expression.
- Answer
-
- overall reaction: \(\ce{O3}(g)+\ce{O}⟶\ce{2O2}(g)\)
- rate 1 = k 1 [O 3 ][Cl]; rate 2 = k 2 [ ClO ][O]
- intermediate: ClO(g)
- overall rate = k 2 k 1 [O 3 ][Cl][O]
Summary
The sequence of individual steps, or elementary reactions, by which reactants are converted into products during the course of a reaction is called the reaction mechanism. The overall rate of a reaction is determined by the rate of the slowest step, called the rate-determining step. Unimolecular elementary reactions have first-order rate laws, while bimolecular elementary reactions have second-order rate laws. By comparing the rate laws derived from a reaction mechanism to that determined experimentally, the mechanism may be deemed either incorrect or plausible.
Footnotes
- This question is taken from the Chemistry Advanced Placement Examination and is used with the permission of the Educational Testing Service.
Glossary
- bimolecular reaction
- elementary reaction involving the collision and combination of two reactant species
- elementary reaction
- reaction that takes place precisely as depicted in its chemical equation
- intermediate
- molecule or ion produced in one step of a reaction mechanism and consumed in another
- molecularity
- number of reactant species (atoms, molecules or ions) involved in an elementary reaction
- rate-determining step
- (also, rate-limiting step) slowest elementary reaction in a reaction mechanism; determines the rate of the overall reaction
- reaction mechanism
- stepwise sequence of elementary reactions by which a chemical change takes place
- termolecular reaction
- elementary reaction involving the simultaneous collision and combination of three reactant species
- unimolecular reaction
- elementary reaction involving the rearrangement of a single reactant species to produce one or more molecules of product
Contributors and Attributions
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).