7.3: The Central Limit Theorem for Sums
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Suppose \(X\) is a random variable with a distribution that may be known or unknown (it can be any distribution) and suppose:
- \(\mu_{x}\) = the mean of \(X\)
- \(\sigma_{x}\) = the standard deviation of \(X\)
If you draw random samples of size \(n\), then as \(n\) increases, the random variable \(\sum X\) consisting of sums tends to be normally distributed and
\[\sum X \sim N((n)(\mu_{x}), (\sqrt{n})(\sigma_{x})).\]
The central limit theorem for sums says that if you keep drawing larger and larger samples and taking their sums, the sums form their own normal distribution (the sampling distribution), which approaches a normal distribution as the sample size increases. The normal distribution has a mean equal to the original mean multiplied by the sample size and a standard deviation equal to the original standard deviation multiplied by the square root of the sample size.
The random variable \(\sum X\) has the following z -score associated with it:
- \(\sum x\) is one sum.
-
\(z = \frac{\sum x - (n)(\mu_{x})}{(\sqrt{n})(\sigma_{x})}\)
- \((n)(\mu_{x})\)= the mean of \(\sum X\)
- \((\sqrt{n})(\sigma_{x})\)= standard deviation of \(\sum X\)
Calculator
To find probabilities for sums on the calculator, follow these steps.
2
nd
DISTR
2:
normalcdf
normalcdf
(lower value of the area, upper value of the area, (\(n\))(mean), (\(\sqrt{n}\))(standard deviation))
where:
- mean is the mean of the original distribution
- standard deviation is the standard deviation of the original distribution
- sample size \(= n\)
Example \(\PageIndex{1}\)
An unknown distribution has a mean of 90 and a standard deviation of 15. A sample of size 80 is drawn randomly from the population.
- Find the probability that the sum of the 80 values (or the total of the 80 values) is more than 7,500.
- Find the sum that is 1.5 standard deviations above the mean of the sums.
Answer
Let \(X =\) one value from the original unknown population. The probability question asks you to find a probability for the sum (or total of) 80 values.
\(\sum X =\) the sum or total of 80 values. Since \(\mu_{x} = 90\), \(\sigma_{x} = 15\), and \(n = 80\), \(\sum X \sim N((80)(90),(\sqrt{80})(15))\)
- mean of the sums \(= (n)(\mu_{x}) = (80)(90) = 7,200\)
- standard deviation of the sums \(= (\sqrt{n})(\sigma_{x}) = (\sqrt{80})(15) = (80)(15)\)
- sum of 80 values \(= \sum X = 7,500\)
a. Find \(P(\sum X > 7,500)\)
\(P(\sum X > 7,500) = 0.0127\)
normalcdf
(lower value, upper value, mean of sums,
stdev
of sums)
The parameter list is abbreviated \(\left(lower, upper, (n)(\mu_{x}, (\sqrt{n}(\sigma_{x})\right)\)
normalcdf
\(\left(7500,1E99,(80)(90),(\sqrt{80})(15)\right) = 0.0127\)
REMINDER
1E99 = 10 99 .
Press the
EE
key for E.
b. Find \(\sum x\) where \(z = 1.5\).
\(\sum x = (n)(\nu_{x}) + (z)(\sqrt{n})(\sigma_{x}) = (80)(90) + (1.5)(\sqrt{80})(15) = 7,401.2\)
Exercise \(\PageIndex{1}\)
An unknown distribution has a mean of 45 and a standard deviation of eight. A sample size of 50 is drawn randomly from the population. Find the probability that the sum of the 50 values is more than 2,400.
Answer
0.0040
Calculator
To find percentiles for sums on the calculator, follow these steps.
2 nd DIStR
3:invNorm
\(k = \text{invNorm} (\text{area to the left of} k, (n)(\text{mean}), (\sqrt{n})(\text{standard deviation}))\)
where:
- \(k\) is the \(k\) th percentile
- mean is the mean of the original distribution
- standard deviation is the standard deviation of the original distribution
- sample size \(= n\)
Example \(\PageIndex{2}\)
In a recent study reported Oct. 29, 2012 on the Flurry Blog, the mean age of tablet users is 34 years. Suppose the standard deviation is 15 years. The sample of size is 50.
- What are the mean and standard deviation for the sum of the ages of tablet users? What is the distribution?
- Find the probability that the sum of the ages is between 1,500 and 1,800 years.
- Find the 80 th percentile for the sum of the 50 ages.
Answer
-
\(\mu_{x} - n\mu_{x} = 1,700\) and \(\sigma_{\sum X} = \sqrt{n}\sigma_{X} = (\sqrt{50})(15) = 106.01\)
The distribution is normal for sums by the central limit theorem. - \(P(1500 < \sum X < 1800) = (1,500, 1,800, (50)(34), (\sqrt{50})(15)) = 0.7974\)
-
Let \(k\) = the 80
th
percentile.
\(k = (0.80,(50)(34),(\sqrt{50})(15)) = 1,789.3\)
Exercise \(\PageIndex{2}\)
In a recent study reported Oct. 29, 2012 on the Flurry Blog, the mean age of tablet users is 35 years. Suppose the standard deviation is ten years. The sample size is 39.
- What are the mean and standard deviation for the sum of the ages of tablet users? What is the distribution?
- Find the probability that the sum of the ages is between 1,400 and 1,500 years.
- Find the 90 th percentile for the sum of the 39 ages.
Answer
-
\(\mu_{\sum X} = n\mu_{X} = 1,365\) and \(\sigma_{\sum X} = \sqrt{n}\sigma_{x} = 62.4\)
The distribution is normal for sums by the central limit theorem. - \(P(1400 < \sum_{X} < 1500) = \text{normalcdf} (1400,1500,(39)(35),(\sqrt{39})(10)) = 0.2723\)
-
Let \(k\) = the 90
th
percentile.
\(k = \text{invNorm} (0.90,(39)(35),(\sqrt{39}) (10)) = 1445.0\)
Example \(\PageIndex{3}\)
The mean number of minutes for app engagement by a tablet user is 8.2 minutes. Suppose the standard deviation is one minute. Take a sample of size 70.
- What are the mean and standard deviation for the sums?
- Find the 95 th percentile for the sum of the sample. Interpret this value in a complete sentence.
- Find the probability that the sum of the sample is at least ten hours.
Answer
- \(\mu_{\sum X} = n\mu_{X}= 70(8.2) = 574\) minutes and \(\sigma_{\sum X} (\sqrt{n})(\sigma_{x}) = (\sqrt{70})(1) = 8.37\) minutes
-
Let \(k\) = the 95
th
percentile.
\(k = \text{invNorm} (0.95,(70)(8.2),(\sqrt{70})(1)) = 587.76\) minutes
Ninety five percent of the app engagement times are at most 587.76 minutes. -
ten hours = 600 minutes
\(P(\sum X \geq 600) = \text{normalcdf}(600,E99,(70)(8.2),(\sqrt{70})(1)) = 0.0009\)
Exercise \(\PageIndex{3}\)
The mean number of minutes for app engagement by a table use is 8.2 minutes. Suppose the standard deviation is one minute. Take a sample size of 70.
- What is the probability that the sum of the sample is between seven hours and ten hours? What does this mean in context of the problem?
- Find the 84 th and 16 th percentiles for the sum of the sample. Interpret these values in context.
Answer
-
7 hours = 420 minutes
10 hours = 600 minutes
\(\text{normalcdf} P(420 \leq \sum X \leq 600) = \text{normalcdf}(420,600,(70)(8.2),\sqrt{70}(1)) = 0.9991\)
This means that for this sample sums there is a 99.9% chance that the sums of usage minutes will be between 420 minutes and 600 minutes. -
\(\text{invNorm}(0.84,(70)(8.2)\),\(\sqrt{70}(1)) = 582.32\)
\(\text{invNorm}(0.16,(70)(8.2),(\sqrt{70}(1)) = 565.68\)
Since 84% of the app engagement times are at most 582.32 minutes and 16% of the app engagement times are at most 565.68 minutes, we may state that 68% of the app engagement times are between 565.68 minutes and 582.32 minutes.
References
- Farago, Peter. “The Truth About Cats and Dogs: Smartphone vs Tablet Usage Differences.” The Flurry Blog, 2013. Posted October 29, 2012. Available online at blog.flurry.com (accessed May 17, 2013).
Review
The central limit theorem tells us that for a population with any distribution, the distribution of the sums for the sample means approaches a normal distribution as the sample size increases. In other words, if the sample size is large enough, the distribution of the sums can be approximated by a normal distribution even if the original population is not normally distributed. Additionally, if the original population has a mean of \(\mu_{x}\) and a standard deviation of \(\sigma_{x}\), the mean of the sums is \(n\)\(\mu_{x}\) and the standard deviation is (\(\sqrt{n}\))(\(\sigma_{x}\)) where \(n\) is the sample size.
Formula Review
- The Central Limit Theorem for Sums: \(\sum X ~ N[(n)(\mu_{x}, (\sqrt{n})(\sigma_{x}))]\)
- Mean for Sums \((\sum X): (n)(\mu_{x})\)
- The Central Limit Theorem for Sums \(z\)-score and standard deviation for sums: \(z \text{ for the sample mean} = \frac{\sum x - (n)(\mu_{x})}{(\sqrt{n})(\sigma_{x})}\)
- Standard deviation for Sums \((\sum X): (\sqrt{n})(\sigma_{x})\)


5.1: Sugar and Salt
Sodium chloride also known as table salt, is an ionic compound with the chemical formula \(\ce{NaCl}\), representing a 1:1 ratio of sodium and chloride ions. It is commonly used as a condiment and food preservative. Salt can be created by adding two very reactive elements together: sodium Na (s) metal and chlorine Cl 2 (g) gas.
\[ \ce{2Na (s) + Cl2(g) \rightarrow 2NaCl (s)} \label{eq1}\]
The element sodium (Figure \(\PageIndex{1a}\)) is a very reactive metal; given the opportunity, it will react with the sweat on your hands and form sodium hydroxide, which is a very corrosive substance. The element chlorine (Figure \(\PageIndex{1b}\)) is a pale yellow, corrosive gas that should not be inhaled due to its poisonous nature. Bring these two hazardous substances together, however, and they react to make the ionic compound sodium chloride (Figure \(\PageIndex{1c}\)), known simply as salt.
Salt is necessary for life. \(\ce{Na^{+}}\) ions are one of the main ions in the human body and are necessary to regulate the fluid balance in the body. \(\ce{Cl^{−}}\) ions are necessary for proper nerve function and respiration. Both of these ions are supplied by salt. The taste of salt is one of the fundamental tastes; salt is probably the most ancient flavoring known, and one of the few rocks we eat. Clearly when the elemental sodium and chlorine combine (Equation \ref{eq1}), the resulting salt product has radically different properties (both physical and chemical). This reaction is spectacular to observed (Video \(\PageIndex{1}\)).
Video \(\PageIndex{1}\): Making Table Salt using Sodium Metal and Chlorine gas
Another compound is sugar, which is the generic name for sweet, soluble carbohydrates, many of which are used in food. Sugar has the chemical formulate \(\ce{C12H22O11}\) and is constructed from different elements than salt: carbon , hydrogen and oxygen . While sugar qualitatively resembles table salt (often confused in the kitchen), they two have distinctly different physical and chemical properties. There are various types of sugar derived from different sources. While sugar is made with carbon, hydrogen and oxygen, it is considerably hard to syntheses from the constituent elements than table salt (Equation \ref{eq1}). However, the thermal decomposition is considerably easiler and can be represented as as a dehydration of sucrose to pure carbon and water vapor in Equation \ref{eq2} and demonstrated in Video \(\PageIndex{2}\).
\[\ce{C12H22O11 (s) + heat → 12C (s) + 11H2O (g)} \label{eq2}\]
Video \(\PageIndex{2}\): A science experiment in the kitchen shows what happens to sugar molecules when they are heated. The experiment did not disappoint!
As with salt, sugar has radically different properties (both physical and chemical) than its constituent elements. That is a central feature of chemical reactions as this chapter will discuss.
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17.1: Introduction
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Small molecules like O2 or CO2 can cross cellular membranes unassisted; neither the hydrophilic surfaces nor the hydrophobic interior of the phospholipid bilayer are barriers to their transit. On the other hand, most molecules (even water!) need the help of membrane transport proteins to get in or out of cells and organelles.
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17.2: Membrane Transport
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Only a few small, relatively uncharged molecules can cross a membrane unassisted (i.e., by passive diffusion). Hydrophilic molecules that must enter or leave cells do so with help, i.e., by facilitated transport. Passive and facilitated transport release the free energy inherent in concentration gradients as molecules diffuse across a membrane.
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17.3: Ligand and Voltage Gated Channels in Neurotransmission
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When neurotransmitters bind to their receptors, ion channels in responding neuron or muscle cells open. The resulting influx of Na+ ions disrupts the resting potential of the target cell. The effect is only transient if the membrane potential remains negative. However, if enough Na+ ions enter the cell, the membrane becomes depolarized. If the cell experiences hyperpolarization, a localized reversal of normal membrane polarity (say from –70 mV to +65mV or more) will generate an action potential.
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17.4: Endocytosis and Exocytosis
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Endocytosis is a mechanism for internalizing large extracellular molecules (e.g., proteins), insoluble particles, or even microorganisms. The three main types of exocytosis are phagocytosis, pinocytosis and receptor-mediated endocytosis. Pinocytosis is non-specific. Phagocytosis targets large structures (e.g., bacteria, food particles…) and is not particularly specific. As its name suggests, receptor-mediated endocytosis is specific for substances recognized by a cell-surface receptor.
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17.5: Directing the Traffic of Proteins in Cells
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Each polypeptide protein translated by ribosomes from a sequence of bases in an mRNA has a specific functional location, either in the cytoplasm, on cellular membranes, inside organelles or in extracellular fluids. In this section we consider the movement and sorting of proteins in the endomembrane system as well as the transport of proteins into and out of organelles.
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17.6: How Cells are Held Together and How they Communicate
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Proteins and glycoproteins on cell surfaces play a major role in how cells interact with their surroundings and with other cells. Some of the proteins in the glycocalyx of adjacent cells interact to form cell-cell junctions, while others interact with extracellular proteins and carbohydrates to form the extracellular matrix (ECM). Still others are part of receptor systems that bind hormones and other signaling molecules at the cell surface, conveying information into the cell by signal moves.
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17.7: 17.7 Signal Transduction
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When hydrophobic chemical effector molecules such as steroid hormones reach a target cell they can cross the hydrophobic membrane and bind to an intracellular receptor to initiate a response. When large effector molecules (e.g., protein hormones) or highly polar hormones (e.g., adrenalin) reach a target cell, they can’t cross the cell membrane. Instead, they bind to transmembrane protein receptors on cell surfaces.
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17.8: Key Words and Terms
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6.9: Describing a Reaction - Energy Diagrams and Transition States
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Objectives
After completing this section, you should be able to
- sketch the reaction energy diagram for a single-step reaction, given some indication of whether the reaction is fast or slow, exothermic or endothermic.
- interpret the reaction energy diagram for a single-step process (e.g., use the diagram to decide whether the reaction is exothermic or endothermic).
- suggest possible transition-state structures for simple one-step processes.
- assess the likelihood of a reaction occurring at room temperature, given the value of the activation energy ΔG ‡ .
Key Terms
Make certain that you can define, and use in context, the key terms below.
- activation energy, ΔG ‡
- reaction energy diagram
- transition state
Study Notes
You may have been taught to use the term “activated complex” rather than “transition state,” as the two are often used interchangeably. Similarly, the activation energy of a reaction is often represented by the symbol E act or E a .
You may recall from general chemistry that it is often convenient to describe chemical reactions with energy diagrams. In an energy diagram, the vertical axis represents the overall energy of the reactants, while the horizontal axis is the ‘ reaction coordinate ’, tracing from left to right the progress of the reaction from starting compounds to final products. The energy diagram for a typical one-step reaction might look like this:
Despite its apparent simplicity, this energy diagram conveys some very important ideas about the thermodynamics and kinetics of the reaction. Recall that when we talk about the thermodynamics of a reaction, we are concerned with the difference in energy between reactants and products, and whether a reaction is ‘downhill’ (exergonic, energy releasing) or ‘uphill (endergonic, energy absorbing). When we talk about kinetics , on the other hand, we are concerned with the rate of the reaction, regardless of whether it is uphill or downhill thermodynamically.
First, let’s review what this energy diagram tells us about the thermodynamics of the reaction illustrated by the energy diagram above. The energy level of the products is lower than that of the reactants. This tells us that the change in standard Gibbs Free Energy for the reaction ( Δ G˚ rnx ) is negative. In other words, the reaction is exergonic, or ‘downhill’. Recall that the Δ G˚ rnx term encapsulates both Δ H˚ rnx , the change in enthalpy (heat) and Δ S˚ rnx , the change in entropy (disorder):
\[ΔG˚ = ΔH˚- TΔS˚ \nonumber \]
where T is the absolute temperature in Kelvin. For chemical processes where the entropy change is small (~0), the enthalpy change is essentially the same as the change in Gibbs Free Energy. Energy diagrams for these processes will often plot the enthalpy (H) instead of Free Energy for simplicity.
The standard Gibbs Free Energy change for a reaction can be related to the reaction's equilibrium constant (\(K_{eq}\_) by a simple equation:
\[ΔG˚ = -RT \ln K_{eq} \nonumber \]
where:
- K eq = [product] / [reactant] at equilibrium
- R = 8.314 J ×K -1 ×mol -1 or 1.987 cal× K -1 ×mol -1
- T = temperature in Kelvin (K)
If you do the math, you see that a negative value for Δ G˚ rnx (an exergonic reaction) corresponds - as it should by intuition - to K eq being greater than 1, an equilibrium constant which favors product formation.
In a hypothetical endergonic (energy-absorbing) reaction the products would have a higher energy than reactants and thus Δ G˚ rnx would be positive and K eq would be less than 1, favoring reactants.
Now, let's move to kinetics. Look again at the energy diagram for exergonic reaction: although it is ‘downhill’ overall, it isn’t a straight downhill run.
First, an ‘energy barrier’ must be overcome to get to the product side. The height of this energy barrier, you may recall, is called the ‘ activation energy’ ( Δ G ‡ ). The activation energy is what determines the kinetics of a reaction: the higher the energy hill, the slower the reaction. At the very top of the energy barrier, the reaction is at its transition state (TS), which is the point at which the bonds are in the process of breaking and forming. The transition state is an ‘ activated complex’ : a transient and dynamic state that, unlike more stable species, does not have any definable lifetime. It may help to imagine a transition state as being analogous to the exact moment that a baseball is struck by a bat. Transition states are drawn with dotted lines representing bonds that are in the process of breaking or forming, and the drawing is often enclosed by brackets. Here is a picture of a likely transition state for a substitution reaction between hydroxide and chloromethane:
Note that this species is absent from the chemical equation (that is it is neither a reactant nor product)
\[\ce{CH_3Cl + HO^{-} \rightarrow CH_3OH + Cl^{-}} \nonumber \]
This reaction involves a collision between two molecules: for this reason, we say that it has second order kinetics . The rate expression for this type of reaction is:
\[\text{rate} = k[\text{reactant 1}][\text{reactant 2}] \nonumber\]
... which tells us that the rate of the reaction depends on the rate constant \(k\) as well as on the concentration of both reactants. The rate constant can be determined experimentally by measuring the rate of the reaction with different starting reactant concentrations. The rate constant depends on the activation energy, of course, but also on temperature: a higher temperature means a higher k and a faster reaction, all else being equal. This should make intuitive sense: when there is more heat energy in the system, more of the reactant molecules are able to get over the energy barrier.
Here is one more interesting and useful expression. Consider a simple reaction where the reactants are \(\ce{A}\) and \(\ce{B}\), and the product is \(\ce{AB}\) (this is referred to as a condensation reaction , because two molecules are coming together, or condensing). If we know the rate constant \(k\) for the forward reaction and the rate constant \(k_{reverse}\) for the reverse reaction (where AB splits apart into \(\ce{A}\) and \(\ce{B}\)), we can simply take the quotient to find our equilibrium constant \(K_{eq}\):
\[A + B \rightleftharpoons AB \nonumber\]
with
\[K _{ eq }=\frac{[ AB ]}{[ A ][ B ]}=\frac{k_{\text {forward }}}{k_{\text {reverse }}} \nonumber\]
This too should make some intuitive sense; if the forward rate constant is higher than the reverse rate constant, equilibrium should lie towards products.


5.7: Projections
Computer screens and printers are two-dimensional display devices. We must somehow convert three-dimensional images to two dimensions in order to display them. This task is done by another kind of matrix operator called a projection .
To build a projection, we first choose a projection plane in the three- dimensional space of the object we wish to view. All points of the object are then projected onto the plane. There are many different kinds of projections, corresponding to various geometric rules for mapping points in space onto a plane. We begin with the parallel projection illustrated in Figure 1, wherein the dotted lines between the points and their projections in the plane are all parallel to one another. These dotted lines are called projectors .
Range
The projection plane is called the range of the projection. We will assume for now that the projection plane passes through the origin. If it does not, we may later compose the necessary translations with our projection. Three points, not all in a line, are required to determine a plane. We will take the origin as one of the points and suppose that the other two points are the vectors \(r_1\) and \(r_2\). From these vectors, we form the matrix
\[\mathrm{R}=\left[\mathrm{r}_{1} \mathrm{r}_{2}\right] \nonumber \]
which determines the range of the projection.
Null-Space
To complete the specification of the parallel projection, we must specify the direction of the projectors. The line through the origin in this direction is called the null-space of the projection. Note that any line in the three-dimensional object that is parallel to the null-space will disappear to a point when projected. The null-space may be specified by a vector \(n\) pointing in the direction of the projectors.
Orthogonal Projection
If the null-space is perpendicular to the range, we say that the projection is an orthogonal projection . The matrix operator for orthogonal projection is a function of the range. If we assume that \(R\) is specified in three-dimensional coordinates (not homogeneous!), we have the following definition for the orthogonal projection onto the range of \(R\):
\[\mathrm{P}(\mathrm{R})=\mathrm{R}\left(\mathrm{R}^{T} \mathrm{R}\right)^{-1} \mathrm{R}^{T} \in \mathscr{R}^{3 \times 3} \nonumber \]
In homogeneous coordinates, we embed this 3×3 projection matrix in the general 4×4 transformation:
\[\mathrm{P}_{h}(\mathrm{R})=\left[\begin{array}{ll}
\mathrm{P}(\mathrm{R}) & 0 \\
0^{T} & 1
\end{array}\right] \nonumber \]
Suppose we choose the \(x−y\) plane as our projection plane. Vectors \(r_1\) and \(r_2\) can be any two points in the plane (besides the origin), so let's take
\[\mathrm{r}_{1}=\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right], \mathrm{r}_{2}=\left[\begin{array}{l}
1 \\
1 \\
0
\end{array}\right] \nonumber \]
Then the range matrix is
\[\mathrm{R}=\left[\begin{array}{ll}
1 & 1 \\
0 & 1 \\
0 & 0
\end{array}\right] \nonumber \]
The orthogonal projection looking down on the \(x−y\) plane from the z-axis is
\[\begin{align}
\mathrm{P}(\mathrm{R}) &=\left[\begin{array}{ll}
1 & 1 \\
0 & 1 \\
0 & 0
\end{array}\right]\left(\left[\begin{array}{lll}
1 & 0 & 0 \\
1 & 1 & 0
\end{array}\right]\left[\begin{array}{ll}
1 & 1 \\
0 & 1 \\
0 & 0
\end{array}\right]\right)^{-1}\left[\begin{array}{lll}
1 & 0 & 0 \\
1 & 1 & 0
\end{array}\right] \nonumber \\
&=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{array}\right] .
\end{align} \nonumber \]
In homogeneous coordinates,
\[\mathrm{P}_{h}(\mathrm{R})=\left[\begin{array}{llll}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] \nonumber \]
Let the vectors \(\mathrm{r}_{1}=\left[\begin{array}{l}
2 \\
0 \\
0
\end{array}\right]\) and \(\mathrm{r}_{2}=\left[\begin{array}{l}
0 \\
3 \\
0
\end{array}\right]\) specify the range of an orthogonal projection. Find P(R) and P\(_h\)(R). Compare with
Example 1
and explain any similarities.
In the chapter on linear algebra you learned that the projection of \(w\) onto \(x\) is given by
\[\mathrm{z}=\frac{(\mathrm{x}, \mathrm{w}) \mathrm{x}}{(\mathrm{x}, \mathrm{x})} \nonumber \]
This is an orthogonal projection of \(w\) onto a line, but it is closely related to the projection into the plane just described. To see the similarity, let's work on the expression for \(z\) a little:
\[\begin{align}
\mathbf{z} &=\frac{\mathrm{x}(\mathrm{x}, \mathrm{w})}{(\mathrm{x}, \mathrm{x})} \nonumber \\
&=\mathrm{x}(\mathrm{x}, \mathrm{x})^{-1}(\mathrm{x}, \mathrm{w}) \nonumber \\
&=\mathrm{x}\left(\mathrm{x}^{T} \mathrm{x}\right)^{-1}\left(\mathrm{x}^{T} \mathrm{w}\right) \nonumber\\
&=\left[\mathrm{x}\left(\mathrm{x}^{T} \mathrm{x}\right)^{-1} \mathrm{x}^{T}\right] \mathrm{w} \nonumber\\
&=\mathrm{P}(\mathrm{x}) \mathrm{w} .
\end{align} \nonumber \]
So we see that it can be written in the same way as the projection onto the plane. The only difference is that the range is now one-dimensional and specified by \(x\) in place of \(R\).
Oblique Projection
More generally, we may have a null-space \(n\) that is not perpendicular to the range R. The projection shown in Figure 1 is an oblique projection. Once again we start with nonhomogeneous coordinates for \(n\) and \(R\) and write the 3×3 oblique projection as
\[\mathrm{E}(\mathrm{R}, \mathrm{n})=\mathrm{R}\left\{\mathrm{R}^{T}[I--P(n)] \mathrm{R}\right\}^{-1} \mathrm{R}^{T}[I--P(n)] \nonumber \]
where
\[\mathrm{P}(\mathrm{n})=\mathrm{n}\left(\mathrm{n}^{T} \mathrm{n}\right)^{-1} \mathrm{n}^{T}=\frac{\mathrm{n} \mathrm{n}^{T}}{\|\mathrm{n}\|^{2}} \nonumber \]
As with the orthogonal projection, we can return to homogeneous coordinates by
\[\mathrm{E}_{h}(\mathrm{R}, \mathrm{n})=\left[\begin{array}{lll}
\mathrm{E}(\mathrm{R} & \mathrm{n}) & 0 \\
\mathrm{O}^{T} & & 1
\end{array}\right] \nonumber \]
Prove and interpret the following properties of paralle1 projections (both orthogonal and oblique):
- P\(^2\) = P; E\(^2\) = E.
- \(PR\)=R; \(ER\)=R.
- \(En\)=0; \(Pn\)=0. (First show that R\(^T\)n=0 when \(n\) is orthogonal to \(R\)).
Display from Projections
Even after we have used a projection, the image points are in three-dimensional homogeneous coordinates. How then do we display them on a two-dimensional display? If we had chosen the \(x−y\) plane as the range of our projection, we could let the output device represent the \(x−y\) plane and ignore the z-coordinate of each point. The z-coordinates of the projected image would all be 0 anyway since the projected points would lie in the \(x−y\) plane. The fourth coordinate of 1 may also be ignored for display. But even if the projection plane is not the \(x−y\) plane, we may perform a rotation to align the two planes and again let the output device represent the \(x−y\) plane.
Perspective Projections
To obtain a perspective projection, we first choose a projection plane as we did for parallel projections. Instead of choosing a null-space parallel to all projectors, we now choose a viewpoint. The projectors all pass through the viewpoint in a perspective projection, as shown in Figure 2 . For a viewpoint at \(z=d\) on the z-axis and a projection plane coinciding with the \(x−y\) plane, the three-dimensional homogeneous operator for perspective projection is
\[\mathbf{M}(d)=\left[\begin{array}{llll}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & \frac{-1}{d} & 1
\end{array}\right] \nonumber \]
The first thing you should notice about the perspective projection M(\(d\)) is that it violates the structure given in Equation 2 from "Vector Graphics: Three-Dimensional Homogeneous Coordinates" by having the 4,3 position equal to −\(\frac{1}{d}\) rather than to 0. This means that the fourth coordinates will not remain 1 in the new point matrix
\[\mathrm{G}_{\text {new }}=\mathbf{M}(d) \mathbf{G} \nonumber \]
This situation is interpreted to mean that the new point \((x,y,z,w)\) must be renormalized to \((\frac{x}{w}, \frac{y}{w} \frac{z}{w}1)\) before the operation is considered complete. Renormalization adds another computational step to the image transformation process. This is undesirable, but we are stuck with it if we wish to include perspective projections in our repertoire of transformations. Note that renormalization is a point-by-point process since \(w\) may be different for each point.
How does renormalization affect composition of operators? We might expect that, each time we perform an operation requiring renormalization, we will be forced to stop and do the renormalization before going on with other operations. In this respect we are fortunate: we may put off the renormalization until all transformations are complete and renormalize once just before displaying the image. Thus we are able to compose perspective projections at will with other transformations.
What is the perspective projection matrix for a viewpoint at infinity, M(\(\infty\))? Interpret the result.
Generalization of Projections
“Projection” is a technical term in linear algebra. A square matrix of any size is considered a projection if it obeys the property of Exercise 2(a) , P\(^2\)=P. All of our geometric projection matrices have this property. We have considered only three-dimensional spaces with a two-dimensional range and a one-dimensional null-space. In general, the dimensionality of the space may be split in any way between range and null-space. If we reinterpret homogeneous coordinates as four-dimensional space, we have projections with three-dimensional range and one-dimensional null-space, but the perspective projection is actually not quite a projection in the technical sense even though M\(^2(d)\)=M(\(d\)) because perspective projection also includes renormalization.


29.2: Voice Leading Borrowed Chords
29.2 Voice Leading Borrowed Chords
Borrowed chords , or chords making use of mode mixture, are handled similarly to their diatonic versions in terms of voice leading, e.g., ii ø iiø7 is handled similarly to iiii7. In any of the borrowed chords with ♭6^ , resolve this tendency tone down by step whenever possible.
Principle 29.2.1. Voice Leading Borrowed Chords.
Resolve ♭6^ down by step when it is a chord member of a borrowed chord.
Another situation where you may encounter a borrowed chord is the VV–♭VIVIcadence, which should be handled the same as the deceptive cadence in minor (see Principle 26.7.4 )—resolve the 3rd of the VV chord up by step and move the other two voices in contrary motion to the bass.
There are no specific rules for other borrowed chords such as ♭IIIIII, ♭VIIVII, ii, and vv; simply voice lead them as smoothly as possible while avoiding objectionable parallels.


6.4: ATP: Adenosine Triphosphate
Skills to Develop
- Explain the role of ATP as the cellular energy currency
- Describe how energy is released through hydrolysis of ATP
Even exergonic, energy-releasing reactions require a small amount of activation energy in order to proceed. However, consider endergonic reactions, which require much more energy input, because their products have more free energy than their reactants. Within the cell, where does energy to power such reactions come from? The answer lies with an energy-supplying molecule called adenosine triphosphate , or ATP . ATP is a small, relatively simple molecule (Figure \(\PageIndex{1}\)), but within some of its bonds, it contains the potential for a quick burst of energy that can be harnessed to perform cellular work. This molecule can be thought of as the primary energy currency of cells in much the same way that money is the currency that people exchange for things they need. ATP is used to power the majority of energy-requiring cellular reactions.
As its name suggests, adenosine triphosphate is comprised of adenosine bound to three phosphate groups (Figure \(\PageIndex{1}\)). Adenosine is a nucleoside consisting of the nitrogenous base adenine and a five-carbon sugar, ribose. The three phosphate groups, in order of closest to furthest from the ribose sugar, are labeled alpha, beta, and gamma. Together, these chemical groups constitute an energy powerhouse. However, not all bonds within this molecule exist in a particularly high-energy state. Both bonds that link the phosphates are equally high-energy bonds ( phosphoanhydride bonds ) that, when broken, release sufficient energy to power a variety of cellular reactions and processes. These high-energy bonds are the bonds between the second and third (or beta and gamma) phosphate groups and between the first and second phosphate groups. The reason that these bonds are considered “high-energy” is because the products of such bond breaking—adenosine diphosphate (ADP) and one inorganic phosphate group (P i )—have considerably lower free energy than the reactants: ATP and a water molecule.  In other words, ATP is hydrolyzed into ADP in the following reaction:
Like most chemical reactions, the hydrolysis of ATP to ADP is reversible. The reverse reaction regenerates ATP from ADP + P i . Indeed, cells rely on the regeneration of ATP just as people rely on the regeneration of spent money through some sort of income. Since ATP hydrolysis releases energy, ATP regeneration must require an input of free energy. The formation of ATP is expressed in this equation:
Two prominent questions remain with regard to the use of ATP as an energy source. Exactly how much free energy is released with the hydrolysis of ATP, and how is that free energy used to do cellular work? The calculated ∆G for the hydrolysis of one mole of ATP into ADP and P i is −7.3 kcal/mole (−30.5 kJ/mol). Since this calculation is true under standard conditions, it would be expected that a different value exists under cellular conditions. In fact, the ∆G for the hydrolysis of one mole of ATP in a living cell is almost double the value at standard conditions: 14 kcal/mol (−57 kJ/mol).
ATP is a highly unstable molecule. Unless quickly used to perform work, ATP spontaneously dissociates into ADP + P i , and the free energy released during this process is lost as heat. The second question posed above, that is, how the energy released by ATP hydrolysis is used to perform work inside the cell, depends on a strategy called energy coupling. Cells couple the exergonic reaction of ATP hydrolysis with endergonic reactions, allowing them to proceed. One example of energy coupling using ATP involves a transmembrane ion pump that is extremely important for cellular function. This sodium-potassium pump (Na + /K + pump) drives sodium out of the cell and potassium into the cell (Figure \(\PageIndex{2}\)). A large percentage of a cell’s ATP is spent powering this pump, because cellular processes bring a great deal of sodium into the cell and potassium out of the cell. The pump works constantly to stabilize cellular concentrations of sodium and potassium. In order for the pump to turn one cycle (exporting three Na + ions and importing two K + ions), one molecule of ATP must be hydrolyzed. When ATP is hydrolyzed, its gamma phosphate doesn’t simply float away, but is actually transferred onto the pump protein. This process of a phosphate group binding to a molecule is called phosphorylation. As with most cases of ATP hydrolysis, a phosphate from ATP is transferred onto another molecule. In a phosphorylated state, the Na + /K + pump has more free energy and is triggered to undergo a conformational change. This change allows it to release Na + to the outside of the cell. It then binds extracellular K + , which, through another conformational change, causes the phosphate to detach from the pump. This release of phosphate triggers the K + to be released to the inside of the cell. Essentially, the energy released from the hydrolysis of ATP is coupled with the energy required to power the pump and transport Na + and K + ions. ATP performs cellular work using this basic form of energy coupling through phosphorylation.
Art Connection
The hydrolysis of one ATP molecule releases 7.3 kcal/mol of energy (∆G = −7.3 kcal/mol of energy). If it takes 2.1 kcal/mol of energy to move one Na + across the membrane (∆G = +2.1 kcal/mol of energy), how many sodium ions could be moved by the hydrolysis of one ATP molecule?
Often during cellular metabolic reactions, such as the synthesis and breakdown of nutrients, certain molecules must be altered slightly in their conformation to become substrates for the next step in the reaction series. One example is during the very first steps of cellular respiration, when a molecule of the sugar glucose is broken down in the process of glycolysis. In the first step of this process, ATP is required for the phosphorylation of glucose, creating a high-energy but unstable intermediate. This phosphorylation reaction powers a conformational change that allows the phosphorylated glucose molecule to be converted to the phosphorylated sugar fructose. Fructose is a necessary intermediate for glycolysis to move forward. Here, the exergonic reaction of ATP hydrolysis is coupled with the endergonic reaction of converting glucose into a phosphorylated intermediate in the pathway. Once again, the energy released by breaking a phosphate bond within ATP was used for the phosphorylation of another molecule, creating an unstable intermediate and powering an important conformational change.
Link to Learning
See an interactive animation of the ATP-producing glycolysis process at this site .
Summary
ATP is the primary energy-supplying molecule for living cells. ATP is made up of a nucleotide, a five-carbon sugar, and three phosphate groups. The bonds that connect the phosphates (phosphoanhydride bonds) have high-energy content. The energy released from the hydrolysis of ATP into ADP + P i is used to perform cellular work. Cells use ATP to perform work by coupling the exergonic reaction of ATP hydrolysis with endergonic reactions. ATP donates its phosphate group to another molecule via a process known as phosphorylation. The phosphorylated molecule is at a higher-energy state and is less stable than its unphosphorylated form, and this added energy from the addition of the phosphate allows the molecule to undergo its endergonic reaction.
Art Connections
Figure \(\PageIndex{2}\): The hydrolysis of one ATP molecule releases 7.3 kcal/mol of energy (∆G = −7.3 kcal/mol of energy). If it takes 2.1 kcal/mol of energy to move one Na + across the membrane (∆G = +2.1 kcal/mol of energy), how many sodium ions could be moved by the hydrolysis of one ATP molecule?
- Answer
-
Three sodium ions could be moved by the hydrolysis of one ATP molecule. The ∆G of the coupled reaction must be negative. Movement of three sodium ions across the membrane will take 6.3 kcal of energy (2.1 kcal × 3 Na + ions = 6.3 kcal). Hydrolysis of ATP provides 7.3 kcal of energy, more than enough to power this reaction. Movement of four sodium ions across the membrane, however, would require 8.4 kcal of energy, more than one ATP molecule can provide.
Glossary
- ATP
- adenosine triphosphate, the cell’s energy currency
- phosphoanhydride bond
- bond that connects phosphates in an ATP molecule


1.8: Proteins
Proteins are the third main group of organic compounds in the cell - in fact if you dried out a cell you would find that about 2/3 of the dry dust you were left with would consist of protein. Like carbohydrates and fats, proteins contain C, H and O but they all also contain nitrogen . Many also contain sulphur and phosphorus.
In the cell, proteins are an important part of the plasma membrane of the cell, but their most essential role is as enzymes . These are molecules that act as biological catalysts and are necessary for biochemical reactions to proceed. Protein is also found as keratin in the skin, feathers and hair, in muscles, as well as in antibodies and some hormones.
Proteins are built up of long chains of smaller molecules called amino acids . There are 20 common types of amino acid and different numbers of these arranged in different orders create the multitude of individual proteins that exist in an animal’s body.
Long chains of amino acids often link with other amino acid chains and the resulting protein molecule may twist, spiral and fold up to make a complex 3-dimensional shape. As an example, see the diagram of the protein lysozyme below. Believe it or not, this is a small and relatively simple protein.
It is this shape that determines how proteins behave in cells, particularly when they are acting as enzymes. If for any reason this shape is altered, the protein stops working. This is what happens if proteins are heated or put in a solution that is too acidic or alkaline. Think what happens to the “white’” of an egg when it is cooked. The “white” contains the protein albumin, which is changed or “ denatured ” permanently by cooking. The catastrophic effect that heat has on enzymes is one of the reasons animals die if exposed to high temperatures.
In the animal’s diet, proteins are found in meat (muscle), dairy products, seeds, nuts and legumes like soya. When the enzymes in the gut digest proteins they break them down into the separate amino acids, which are small enough to be absorbed into the blood.


10.7: Hypothesis Testing of a Single Mean and Single Proportion (Worksheet)
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Name: ______________________________
Section: _____________________________
Student ID#:__________________________
Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.
Student Learning Outcomes
- The student will select the appropriate distributions to use in each case.
- The student will conduct hypothesis tests and interpret the results.
Television Survey
In a recent survey, it was stated that Americans watch television on average four hours per day. Assume that \(\sigma = 2\). Using your class as the sample, conduct a hypothesis test to determine if the average for students at your school is lower.
- \(H_{0}\): _____________
- \(H_{a}\): _____________
- In words, define the random variable. __________ = ______________________
- The distribution to use for the test is _______________________.
- Determine the test statistic using your data.
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Draw a graph and label it appropriately.Shade the actual level of significance.
- Graph: Figure 9.7.1.
- Determine the \(p\text{-value}\).
- Do you or do you not reject the null hypothesis? Why?
- Write a clear conclusion using a complete sentence.
Language Survey
About 42.3% of Californians and 19.6% of all Americans over age five speak a language other than English at home. Using your class as the sample, conduct a hypothesis test to determine if the percent of the students at your school who speak a language other than English at home is different from 42.3%.
- \(H_{0}\): ___________
- \(H_{a}\): ___________
- In words, define the random variable. __________ = _______________
- The distribution to use for the test is ________________
- Determine the test statistic using your data.
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Draw a graph and label it appropriately. Shade the actual level of significance.
- Graph: Figure 9.7.2.
- Determine the \(p\text{-value}\).
- Do you or do you not reject the null hypothesis? Why?
- Write a clear conclusion using a complete sentence.
Jeans Survey
Suppose that young adults own an average of three pairs of jeans. Survey eight people from your class to determine if the average is higher than three. Assume the population is normal.
- \(H_{0}\): _____________
- \(H_{a}\): _____________
- In words, define the random variable. __________ = ______________________
- The distribution to use for the test is _______________________.
- Determine the test statistic using your data.
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Draw a graph and label it appropriately. Shade the actual level of significance.
- Graph: Figure 9.7.3.
- Determine the \(p\text{-value}\).
- Do you or do you not reject the null hypothesis? Why?
- Write a clear conclusion using a complete sentence.


6.3: Catalysis
- To understand how catalysts increase the reaction rate and the selectivity of chemical reactions.
Catalysts are substances that increase the reaction rate of a chemical reaction without being consumed in the process. A catalyst, therefore, does not appear in the overall stoichiometry of the reaction it catalyzes, but it must appear in at least one of the elementary reactions in the mechanism for the catalyzed reaction. The catalyzed pathway has a lower E a , but the net change in energy that results from the reaction (the difference between the energy of the reactants and the energy of the products) is not affected by the presence of a catalyst ( Figur e \(\PageIndex{1}\)). Nevertheless, because of its lower E a , the reaction rate of a catalyzed reaction is faster than the reaction rate of the uncatalyzed reaction at the same temperature. Because a catalyst decreases the height of the energy barrier, its presence increases the reaction rates of both the forward and the reverse reactions by the same amount. In this section, we will examine the three major classes of catalysts: heterogeneous catalysts, homogeneous catalysts, and enzymes.
A catalyst affects E a , not Δ E .
Heterogeneous Catalysis
In heterogeneous catalysis , the catalyst is in a different phase from the reactants. At least one of the reactants interacts with the solid surface in a physical process called adsorption in such a way that a chemical bond in the reactant becomes weak and then breaks. Poisons are substances that bind irreversibly to catalysts, preventing reactants from adsorbing and thus reducing or destroying the catalyst’s efficiency.
An example of heterogeneous catalysis is the interaction of hydrogen gas with the surface of a metal, such as Ni, Pd, or Pt. As shown in part (a) in Figure \(\PageIndex{2}\) , the hydrogen–hydrogen bonds break and produce individual adsorbed hydrogen atoms on the surface of the metal. Because the adsorbed atoms can move around on the surface, two hydrogen atoms can collide and form a molecule of hydrogen gas that can then leave the surface in the reverse process, called desorption. Adsorbed H atoms on a metal surface are substantially more reactive than a hydrogen molecule. Because the relatively strong H–H bond (dissociation energy = 432 kJ/mol) has already been broken, the energy barrier for most reactions of H 2 is substantially lower on the catalyst surface.
Figure \(\PageIndex{2}\) shows a process called hydrogenation , in which hydrogen atoms are added to the double bond of an alkene, such as ethylene, to give a product that contains C–C single bonds, in this case ethane. Hydrogenation is used in the food industry to convert vegetable oils, which consist of long chains of alkenes, to more commercially valuable solid derivatives that contain alkyl chains. Hydrogenation of some of the double bonds in polyunsaturated vegetable oils, for example, produces margarine, a product with a melting point, texture, and other physical properties similar to those of butter.
Several important examples of industrial heterogeneous catalytic reactions are in Table \(\PageIndex{1}\). Although the mechanisms of these reactions are considerably more complex than the simple hydrogenation reaction described here, they all involve adsorption of the reactants onto a solid catalytic surface, chemical reaction of the adsorbed species (sometimes via a number of intermediate species), and finally desorption of the products from the surface.
| Commercial Process | Catalyst | Initial Reaction | Final Commercial Product |
|---|---|---|---|
| contact process | V 2 O 5 or Pt | 2SO 2 + O 2 → 2SO 3 | H 2 SO 4 |
| Haber process | Fe, K 2 O, Al 2 O 3 | N 2 + 3H 2 → 2NH 3 | NH 3 |
| Ostwald process | Pt and Rh | 4NH 3 + 5O 2 → 4NO + 6H 2 O | HNO 3 |
| water–gas shift reaction | Fe, Cr 2 O 3 , or Cu | CO + H 2 O → CO 2 + H 2 | H 2 for NH 3 , CH 3 OH , and other fuels |
| steam reforming | Ni | CH 4 + H 2 O → CO + 3H 2 | H 2 |
| methanol synthesis | ZnO and Cr 2 O 3 | CO + 2H 2 → CH 3 OH | CH 3 OH |
| Sohio process | bismuth phosphomolybdate | \(\mathrm{CH}_2\textrm{=CHCH}_3+\mathrm{NH_3}+\mathrm{\frac{3}{2}O_2}\rightarrow\mathrm{CH_2}\textrm{=CHCN}+\mathrm{3H_2O}\) | \(\underset{\textrm{acrylonitrile}}{\mathrm{CH_2}\textrm{=CHCN}}\) |
| catalytic hydrogenation | Ni, Pd, or Pt | RCH=CHR′ + H2 → RCH 2 —CH 2 R′ | partially hydrogenated oils for margarine, and so forth |
Homogeneous Catalysis
In homogeneous catalysis , the catalyst is in the same phase as the reactant(s). The number of collisions between reactants and catalyst is at a maximum because the catalyst is uniformly dispersed throughout the reaction mixture. Many homogeneous catalysts in industry are transition metal compounds ( Table \(\PageIndex{2}\)), but recovering these expensive catalysts from solution has been a major challenge. As an added barrier to their widespread commercial use, many homogeneous catalysts can be used only at relatively low temperatures, and even then they tend to decompose slowly in solution. Despite these problems, a number of commercially viable processes have been developed in recent years. High-density polyethylene and polypropylene are produced by homogeneous catalysis.
| Commercial Process | Catalyst | Reactants | Final Product |
|---|---|---|---|
| Union Carbide | [Rh(CO) 2 I 2 ] − | CO + CH 3 OH | CH 3 CO 2 H |
| hydroperoxide process | Mo(VI) complexes | CH 3 CH=CH 2 + R–O–O–H | |
| hydroformylation | Rh/PR 3 complexes | RCH=CH 2 + CO + H 2 | RCH 2 CH 2 CHO |
| adiponitrile process | Ni/PR 3 complexes | 2HCN + CH 2 =CHCH=CH 2 | NCCH 2 CH 2 CH 2 CH 2 CN used to synthesize nylon |
| olefin polymerization | (RC 5 H 5 ) 2 ZrCl 2 | CH 2 =CH 2 | –(CH 2 CH 2 –) n : high-density polyethylene |
Enzymes
Enzymes, catalysts that occur naturally in living organisms, are almost all protein molecules with typical molecular masses of 20,000–100,000 amu. Some are homogeneous catalysts that react in aqueous solution within a cellular compartment of an organism. Others are heterogeneous catalysts embedded within the membranes that separate cells and cellular compartments from their surroundings. The reactant in an enzyme-catalyzed reaction is called a substrate .
Because enzymes can increase reaction rates by enormous factors (up to 10 17 times the uncatalyzed rate) and tend to be very specific, typically producing only a single product in quantitative yield, they are the focus of active research. At the same time, enzymes are usually expensive to obtain, they often cease functioning at temperatures greater than 37 °C, have limited stability in solution, and have such high specificity that they are confined to turning one particular set of reactants into one particular product. This means that separate processes using different enzymes must be developed for chemically similar reactions, which is time-consuming and expensive. Thus far, enzymes have found only limited industrial applications, although they are used as ingredients in laundry detergents, contact lens cleaners, and meat tenderizers. The enzymes in these applications tend to be proteases, which are able to cleave the amide bonds that hold amino acids together in proteins. Meat tenderizers, for example, contain a protease called papain, which is isolated from papaya juice. It cleaves some of the long, fibrous protein molecules that make inexpensive cuts of beef tough, producing a piece of meat that is more tender. Some insects, like the bombadier beetle, carry an enzyme capable of catalyzing the decomposition of hydrogen peroxide to water ( Figure \(\PageIndex{3}\)).
Enzyme inhibitors cause a decrease in the reaction rate of an enzyme-catalyzed reaction by binding to a specific portion of an enzyme and thus slowing or preventing a reaction from occurring. Irreversible inhibitors are therefore the equivalent of poisons in heterogeneous catalysis. One of the oldest and most widely used commercial enzyme inhibitors is aspirin, which selectively inhibits one of the enzymes involved in the synthesis of molecules that trigger inflammation. The design and synthesis of related molecules that are more effective, more selective, and less toxic than aspirin are important objectives of biomedical research.
Summary
Catalysts participate in a chemical reaction and increase its rate. They do not appear in the reaction’s net equation and are not consumed during the reaction. Catalysts allow a reaction to proceed via a pathway that has a lower activation energy than the uncatalyzed reaction. In heterogeneous catalysis, catalysts provide a surface to which reactants bind in a process of adsorption. In homogeneous catalysis, catalysts are in the same phase as the reactants. Enzymes are biological catalysts that produce large increases in reaction rates and tend to be specific for certain reactants and products. The reactant in an enzyme-catalyzed reaction is called a substrate. Enzyme inhibitors cause a decrease in the reaction rate of an enzyme-catalyzed reaction.


<IP_ADDRESS>: Reliability in Unobtrusive Research
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Learning Objectives
- Define stability and describe strategies for overcoming problems of stability.
- Define reproducibility and describe strategies for overcoming problems of reproducibility.
- Define accuracy and describe strategies for overcoming problems of accuracy.
This final section of the chapter investigates a few particularities related to reliability in unobtrusive research projects (Krippendorff, 2009)Krippendorff, K. (2009). Testing the reliability of content analysis data: What is involved and why. In K. Krippendorff & M. A. Bock (Eds.), The content analysis reader (pp. 350–357). Thousand Oaks, CA: Sage. that warrant our attention. These particularities have to do with how and by whom the coding of data occurs. Issues of stability, reproducibility, and accuracy all speak to the unique problems—and opportunities—with establishing reliability in unobtrusive research projects.
Stability refers to the extent to which the results of coding vary across different time periods. If stability is a problem, it will reveal itself when the same person codes the same content at different times and comes up with different results. Coding is said to be stable when the same content has been coded multiple times by the same person with the same result each time.  Ambiguities in the text itself might also contribute to problems of stability. While you cannot alter your original textual data sources, simply being aware of possible ambiguities in the data as you code may help reduce the likelihood of problems with stability. It is also possible that problems with stability may result from a simple coding error, such as inadvertently jotting a 1 instead of a 10 on your code sheet.
Reproducibility , sometimes referred to as intercoder reliability (Lombard, Snyder-Duch, & Campanella Bracken, 2010),Lombard, M., Snyder-Duch, J., & Campanella Bracken, C. (2004). Practical resources for assessing and reporting intercoder reliability in content analysis research projects. Retrieved from astro.temple.edu/~lombard/reliability is the extent to which one’s coding procedures will result in the same results when the same text is coded by different people. Cognitive differences among the individuals coding data may result in problems with reproducibility, as could ambiguous coding instructions. Random coding errors might also cause problems. One way of overcoming problems of reproducibility is to have coders code together. While working as a graduate research assistant, I participated in a content analysis project in which four individuals shared the responsibility for coding data. To reduce the potential for reproducibility problems with our coding, we conducted our coding at the same time in the same room, while sitting around a large, round table. We coded at the same time in the same room so that we could consult one another when we ran into problems or had questions about what we were coding. Resolving those ambiguities together meant that we grew to have a shared understanding of how to code various bits of data.
Finally, accuracy refers to the extent to which one’s coding procedures correspond to some preexisting standard. This presumes that a standard coding strategy has already been established for whatever text you’re analyzing. It may not be the case that official standards have been set, but perusing the prior literature for the collective wisdom on coding on your particular area is time well spent. Scholarship focused on similar data or coding procedures will no doubt help you to clarify and improve your own coding procedures.
KEY TAKEAWAYS
- Stability can become an issue in unobtrusive research project when the results of coding by the same person vary across different time periods.
- Reproducibility has to do with multiple coders’ results being the same for the same text.
- Accuracy refers to the extent to which one’s coding procedures correspond to some preexisting standard.
Exercises
- With a peer, create a code sheet that you can use to record instances of violence on the television program of your choice. Now, on your own, watch two or three episodes of that program, coding for instances of violence as you go along. Your peer should do the same. Finally, compare your code sheet to that of your peer. How similar were you in your coding? Where do your coding results differ, and why? Which issues of reliability may be relevant?


22.5: Activities
From a pool of eleven students (five first-year, six senior), how many ways are there to form:
- A committee of three students?
- A committee consisting of three first-year students and four senior students?
- A committee of six students if two of the senior students refuse to be together on the committee?
- A committee consisting of four first-year students and three senior students if two of the first-year students refuse to be together on the team?
From the alphabet \(\Sigma = \{0,1\}\text{:}\)
- How many words of length \(10\) contain exactly six \(0\)s?
- How many contain at least three \(1\)s?
From the alphabet \(\Sigma = \{0,1,2\}\text{:}\)
- How many words of length \(10\) contain exactly four \(2\)s?
- How many contain at most seven \(0\)s?
Figure \(\PageIndex{1}\) contains a diagram in a pyramid shape. The unfilled circles represent “positions” in the pyramid, and the smaller dots represent “dividers” between positions. Consider “paths” through this pyramid that begin at the peak position and end on the lowest level. The filled circles joined by line segments represent one such path.
- How many such paths are there?
- How many paths are there that change direction exactly once? Exactly twice? At every step?
(For each case described in this task, you should be able to arrive at an answer without explicitly determining all such paths.)
You get to the final exam of one your courses and are faced with twelve questions. In how many ways can you fulfill the requirements exam if the instructions ask you to:
- Answer any ten of the questions?
- Answer any seven of the first eight questions and any three of the last four questions?
- Answer ten of the questions, at least five of which must be from the first eight questions and at least three of which must be from the last four questions?
A course instructor for a class of twenty is feeling particularly lazy and doesn't bother to mark the final exams. Instead, she decides that for each of the letter grades A, B, C, she will randomly assign that grade to exactly six students, and the last two unlucky students will be assigned a grade of D. How many different course outcomes are there?
How many ways are there to split \(m n \) people into \(m\) groups of equal size?
Suppose you have \(2 n\) teddy bears that are identical except for a number stitched into the paw of the right foot. Of these bears, \(n\) have the number \(0\) on their foot, and the remaining \(n\) bears have a unique number from \(1, 2, 3, \ldots, n\text{.}\) How many ways can you choose \(n\) of the bears, with the understanding that any of the bears labelled \(0\) are interchangeable?
- Hint.
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Break into cases based on how many bears labelled \(0\) will be in your collection.
Consider the set \(\{ 1, 2, 3, \ldots, 2n \}\text{.}\) How many subsets of size \(2\) are there such that the two elements therein have an even sum?
Consider the set \(\{ 1, 2, 3, \ldots, n \}\text{.}\) How many subsets of size \(3\) are there such that no two of the three elements therein are consecutive?
- Hint.
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It might be easier to count the subsets of size \(3\) that do contain (at least) two consecutive numbers.


6.4: Goal Setting
Discover Your Values and Goals
Assessing Your Core Values
The journey of achieving success in college begins with a single step: identifying your personal values. Your personal values are your core beliefs and guiding principles. They shape the roles you play in daily life. They color your interests and passions and frame your thoughts and words. In essence, your values are a compass that helps you make decisions and choices.
 They shape how you interact with others. They determine your priorities (whether you’re conscious about it or not), and they shape the choices you make. They are the measures by which you judge yourself and they’re also the measures by which you judge others.
When your actions are consistent with your values, you feel peaceful with the choices you make even if the outcome of those choices is not positive. When some action or decision is not aligned with your values, you feel conflicted and remorseful.
You can assess your core values by checking to see if it meets these three criteria:
- It has been freely chosen from amongst alternatives after consideration and thought.
- It is prized, cherished and considered precious and is pubically affirmed when appropriate.
- It is acted upon consistently, modeled for others, and pursued even when there are consequences for doing so.
Why Find Your Core Values?
This might seem like a rhetorical question, but in fact, it is quite important to understand the power and importance of core values.  The stronger the foundation the better and greater the person you will be able to become.
What are your values, then? Which are most important to you, and which are least important? How do your values fit into your educational goals? How do your educational goals relate to your future career? Are you spending your time in a way that prioritizes your goals? Does your budget reflect your goals?
To help you answer these questions, you can use a “self-assessment” survey. These surveys can help you evaluate your personal identity—your thoughts, actions, attitudes, beliefs, values, and behaviors—in relationship to the task at hand, like going to college and preparing for a career.
Many different self-assessment surveys are available from college career centers and online sites. Some are designed as personality tests, like the Keirsey Temperament Sorter, or as inventories, like the Myers-Briggs Type Indicator (MTBI®), the most widely used personality inventory in history. You may also come across instruments designed as scales, or measures, games, surveys, and more. These descriptors are often interchangeably used, although most often they refer to questionnaires. The distinctions are not as important as whether or not the instrument meets your self-assessment needs.
You can visit a career counselor at your college’s Career Services Office and request the MBTI assessment to further explore your values. You can also use other assessments available through your school.
In Task #2 (see below) you will sample several self-assessment surveys to gain insights into your personal identity, values, educational goals, and career goals. By better understanding the interconnections, you are in a better position to make solid college and career choices.
Stages Of Life
Keep in mind that your personal values and interests can and do change as you get older. This is evidenced in research conducted by a number of contemporary social scientists, like Erik Erikson and Daniel Levinson. Their studies show how our values affect our choices and how our choices can characterize the stage of life we’re in.
For example, college students, ages 18–26, tend to make choices that are tentative (more short-range) and support a desire for autonomy. Later, during ages 27–31, young adults may rethink decisions and lean toward more permanent choices. In ages 32–42, adults tend to have a greater sense of commitment and stability, as shown by their choices. In essence, our personal identity and values change over time, but they continue to affect our choices and can illuminate the stage of life we’re in.
Keeping in mind that there are many phases of life, you can expect to see changes in your values and choices as you get older. You may experience a significant change in perspective while you are in college! To better understand your relationship with your values, you can continually reassess what is important to you. Make a commitment to examining your thinking, actions, and choices, and keep taking self-assessment tests. This will put you in a stronger position to manage changes in your educational goals, your career, living situation, hobbies, friends, and other aspects of your life. Changes are part of normal life transitions.
Setting Goals
Some people are goal-oriented and seem to easily make decisions that lead to achieving their goals, while others seem just to “go with the flow” and accept what life gives them. While the latter may sound pleasantly relaxed, moving through life without goals may not lead anywhere at all. The fact that you’re in college now shows you already have the major goal to complete your college program.
A goal is a result we intend to reach mostly through our own actions. Things we do may move us closer to or farther away from that result. Studying moves us closer to success in a difficult course, while sleeping through the final examination may completely prevent reaching that goal. That’s fairly obvious in an extreme case, yet a lot of college students don’t reach their goal of graduating. The problem may be a lack of commitment to the goal, but often students have conflicting goals.  Also, consider whether your goals support your core values. You are more likely to achieve a goal that is aligned directly with your values.
Consider These Four Student Scenarios
Juan
To help his widowed mother, Juan went to work full time after high school but now, a few years later, he’s dissatisfied with the kinds of jobs he has been able to get and has begun taking classes toward an Associates Degree in Computer Science in the evenings. He’s often tired after work and his mother would like him to spend more time at home, and his girlfriend also wants to spend more time with him. Sometimes he cuts class to visit his mother or spend time with his girlfriend.
Becky
In her senior year of college, Becky has just been elected president of her sorority and is excited about planning a major community service project. She knows she should be spending more time on her senior thesis, but she feels her community project may gain her contacts that can help her find a better job after graduation. Besides, the sorority project is a lot more fun, and she’s enjoying the esteem of her position. Even if she doesn’t do well on her thesis, she’s sure she’ll pass.
Morgan
After an easy time in high school, Morgan is surprised their college classes are so hard. They have enough time to study for their first-year courses, but they also have a lot of friends and fun things to do. Sometimes they're surprised to look up from their computer to see it’s midnight already, and they haven't yet started reading the assigned chapter! Where does the time go? When they're stressed, however, they can’t study well, so they plan to get up early and read the chapter before class, but then they turn back to their computer to see who’s online.
Sachito
Sachito was successful in cutting back her hours at work to give her more time for her college classes, but it’s difficult for her to get much studying done at home. Her husband has been wonderful about taking care of their young daughter, but he can’t do everything, and lately, he’s been hinting more about asking her sister to babysit so that the two of them can go out in the evening the way they used to. Lately, when she’s had to study on a weekend, he leaves with his friends, and Sachito ends up spending the day with her daughter—and not getting much studying done.
What do these very different students have in common? Each student has goals that conflict in one or more ways. Each needs to develop strategies to meet their other goals without threatening their academic success. And all of them have time management issues to work through, three because they feel they don’t have enough time to do everything they want or need to do, and one because even though he has enough time, he needs to learn how to manage it more effectively. For all four of them, motivation and attitude will be important as they develop strategies to achieve their goals.
It all begins with setting goals and thinking about values and priorities!
Benefits of Goal Setting
Setting goals can turn your dreams into reality. You may have a dream to one day graduate from college, buy a new car, own your own home, travel abroad, etc. Any of these dreams can be broken down into a detailed goal and plan of action. For example, maybe you want to buy a home in 20 years. You will need $40,000 as a down payment. That’s a lot of money and may not feel achievable. But, if you break that $40,000 into 20 years, that’s $2,000 a year. That sounds more manageable. And if we break it down even more, you can buy that house if you save about $165 a month, or $42 a week, or $6 a day! Can you save $6 a day, maybe by packing your lunch instead of the drive-thru? Our big dream is now an achievable, realistic goal.
Setting goals has many benefits, including turning your dreams into reality. Goal setting allows you to create a plan to focus on your goal, rather than dreaming about the future. It also reduces anxiety and worry. It is much less anxiety-producing to focus on saving $6 a day than it is to save $40,000. It is also motivating because you will be able to measure your progress and successes. At the end of one year, you will have saved $2,000, which will motivate you to keep saving and maybe even increase your saving goal. You will use your time and resources more wisely, often leading to faster and increased results.
As you think about your own goals, think about more than just being a student. You’re also a person with your own core values, individual needs, and desires, hopes and dreams, plans and schemes. Your long-term goals likely include graduation and a career but may also involve social relationships with others, a romantic relationship, family, hobbies or other activities, where and how you live, and so on. While you are a student you may not be actively pursuing all your goals with the same fervor, but they remain goals and are still important in your life. Think about what goals you would like to achieve academically, vocationally (career), financially, personally, physically, and spiritually.
Types of Goals
There are different types of goals, based on time and topic.
Long-term goals may begin with graduating from college and everything you want to happen thereafter. Often your long-term goals (graduating with a bachelor’s degree) guide your mid-term goals (transferring to a University), and your short-term goals (getting an A on your upcoming exam) become steps for reaching those larger goals. Thinking about your goals in this way helps you realize how even the little things you do every day can keep you moving toward your most important long-term goals. Common long-term goals include things like earning your Bachelor’s degree, owning a home, getting a job in your career area, buying a new car, etc.
Mid-term goals involve plans for this school year or your time here at college or goals you want to achieve within the next six months to two years. Mid-term goals are often stepping stones to your long-term goals, but they can also be independent goals. For example, you may have a goal of transferring to University, which is a midterm goal that brings you closer to your long-term goal of getting your Bachelor’s degree. Or, you may have a goal to pay off your credit card debt within the next 12 months or to save for a car that you plan to buy next year. When making mid-term goals related to your long-term goals, make a list of accomplishments that will lead you to your final goal.
Short-term goals focus on today and the next few days and perhaps weeks. Short-term goals expect accomplishment in a short period of time, such as trying to get a bill paid in the next few days or getting an A on your upcoming exam. The definition of a short-term goal need not relate to any specific length of time. In other words, one may achieve (or fail to achieve) a short-term goal in a day, week, month, year, etc. The time-frame for a short-term goal relates to its context in the overall timeline that it is being applied to. For instance, one could measure a short-term goal for a month-long project in days; whereas one might measure a short-term goal for someone’s lifetime in months or in years. Often, people define short-term goals in relation to their mid-term of long-term goals.
An example of how short-term and mid-term goals relate to long-term goals is wanting to earn your Bachelor’s degree. If you have a goal of earning your Bachelor’s degree in four years, a mid-term goal is getting your Associates Degree and getting accepted to your top choice University in two years. This can be broken down into a series of short-term goals such as your GPA goal for this semester, your goal grade on an upcoming exam, and the amount of time you plan to study this weekend. Every long-term goal can be broken down into smaller steps and eventually lead to the question, “what do I have to do today to achieve my goal?”
You will make goals in different areas of life and at different times in your life. At this point in your life, academic goals may take precedent but there are also other areas to consider.
Academic – You clearly already have an academic goal and are actively working on pursuing it. Academic goals may include things like a target GPA, completing your Associate’s Degree, or transferring to a University. It may also include short-term goals like completing your homework before the weekend.
Career – At this point, your career goals are closely linked to your academic goals, such as getting a degree or certificate in your chosen career field. You may also have careers goals of gaining experience in your field through internships and work experience.
Financial – Your financial goals are often tied to your career goals. You may have a salary goal or you may have the goal of saving for a home, a car or a vacation. You may also have goals to reduce debt and manage your budget.
Health/Physical – Almost all of us have worked on physical goals. Many people have the goal to lose weight, to increase their exercise or to drink more water. Other health goals could include establishing a regular sleep schedule, eating more fruits and vegetables, or seeing your doctor regularly. Health goals can also include mental health such as meditating or working to reduce stress and anxiety.
Social/Relationships – Even though it may feel like it sometimes, your life is more than school and work. You should also establish goals for your social relationships. For example, make a goal to stay in contact with a friend who moved, or to visit your family every week, or to have a date with your significant other once a week.
Spiritual – Many people have religious goals, such as attending church regularly, practicing daily prayer, or joining a church group. Even if you aren’t religious, you may have spiritual goals such as time alone to meditate.
Personal/Hobbies – In addition to work and school, you may have hobbies or personal interests that you want to devote time and energy to. Perhaps you have a goal of rebuilding a motorcycle or learning how to knit or sew.
SMART Goals
Turn your dreams into reality by following the SMART goal-setting process. SMART goals are commonly associated with Peter Drucker’s management by objectives concept. It gives structure and organization to the goal-setting process by establishing defined actions, milestones, objectives and deadlines. Creating SMART goals helps with motivation and focus and keeps you moving forward. Every goal can be made into a SMART goal!
When writing your goals, follow these SMART guidelines. You should literally write them down because the act of finding the best words to describe your goals helps you think more clearly about them.
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Goals should be SPECIFIC (S)
- What exactly do you want to achieve? Avoid vague terms like “good,” and “more.” The more specific you are, the most likely you are to succeed.
- A specific goal has a much greater chance of being accomplished than a general goal.
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To set a specific goal,
answer the six “W” questions
:
Who: Who is involved?
What: What do I want to accomplish?
Where: Identify a location.
When: Establish a time frame.
Which: Identify requirements and constraints.
Why: Specific reasons, purpose or benefits of accomplishing the goal.
“I will get a 3.5 GPA this semester so that I can apply to the Surgical Tech Program.”
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Goals should be MEASURABLE (M)
- Break your goal down into measurable elements so you have concrete evidence of your progress.
- Using numbers, quantities or time is a good way to ensure measurability.
- When you measure your progress, you stay on track, reach your target dates, and experience success!
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To determine if your goal is measurable, ask…
How much?
How many?
How often?
How will I know when it is accomplished?
“I will study 18 hours per week, 3 hours per day for six days a week.”
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Goals should be ATTAINABLE (A)
- A goal should be something to strive for and reach for but something that is achievable and attainable . For example, completing an Associate’s Degree in one year may not be attainable while working full time with a family.
- Ask yourself if you have the time, money, resources and talent to make it happen
- Weigh the effort, time and other costs your goal will take against the benefits and other priorities you have in life.
- You can attain most any goal you set when you plan your steps wisely and establish a time frame that allows you to carry out those steps.
“I will complete 9 credit hours this semester while working part-time.”
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Goals should be REALISTIC (R)
- Your goal should be realistic and relevant . Ask yourself if your goal and timeline is realistic for your life, why is the goal important to you, and what is the objective behind your goal? What makes the goal worthwhile for YOU?
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Be sure the goal is relevant to you.
- Why is this goal important to you? (Make sure your goal aligns with your values.)
- What are the benefits and rewards of accomplishing this goal?
- Why will you be able to stay committed in the long-run?
- Is it something that will still be important to you a month or year from now?
“I will become a Surgical Technician in two years to pursue my interests and values in helping others and provide for my family.”
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Goals should have a TIMEABLE (T)
- Your goal should have a clear deadline. This will help you stay accountable and motivated.
- Keep the timeline realistic but also a little challenging to create a sense of accountability and avoid procrastination.
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With no deadline, there’s no sense of urgency, which leads to procrastination.
- “Someday,” “soon,” and “eventually” are not deadlines.
- Be specific with each deadline for each step along the way.
“I will complete the draft of my research paper one-week before the deadline.”
You can watch this supplemental video on SMART goals if you wish to explore the idea further.
Putting Your Goals Into Action
Be certain you want to reach the goal. We are willing to work hard and sacrifice to reach goals we really care about, ones that support our core values. But, we’re likely to give up when we encounter obstacles if we don’t feel strongly about a goal. If you’re doing something only because your parents or someone else wants you to, then it’s not your own personal goal—and you may have some more thinking to do about your life.
Writing down your goals helps you to organize your thoughts and be clear with your goals, ensuring you meet the SMART goal criteria. When you write your goals, state them positively, stating what you will do rather than what you won’t do. When you focus on doing something, that behavior often increases. On the other hand, when you focus on not doing something, that behavior also often increases. For example, if you have a goal to increase your health, you may focus on increasing your water intake to at least 64 ounces per day. This will lead you to think about and drink more water! But, if you focus on not drinking soda, you are likely to think about soda all day and end up drinking more.
After you have written down your goal, post it in a visible place to remind you every day of what it is you are working toward. When you see your goal, ask yourself, “Did my choices today help move more toward my goal? Are my actions supporting my goals?” Being reminded of your goal can help you stay motivated and focused.
Consider sharing your goal with friends, family or classmates. Sharing your goal with supportive people who care about you will help you stay on track. Share your goal with people you know will be encouraging and cheer you on as you work toward your goal. In return, offer the same support for your friends’ goals and dreams.
How to Stay Motivated
Watch this video that adapts a presentation by Daniel Pink that explains “the surprising truth about what motivates us.” Do you think you need a big reward in order to stay motivated to work on your goal? Or, do you need a strict consequence if you don’t meet your goal? Watch the video to find out.
KEY TAKEAWAYS
- There are three criteria for identifying your core values. Identifying your core values is the first step of the goal-setting process.
- Self-assessments can help you define your core values.
- Your stage of life influences your personal values and interests.
- Goal setting is a process with many rewards and benefits that allows you to get what you want from life.
- There are short-term, mid-term and long-term goals. They are often stepping stones to meet bigger goals.
- You will create goals in several areas of your life including academic, financial, career and personal.
- The SMART goal setting model is a very effective system for identifying and creating goals.
- A Personal Action Plan can help you define your goals using the SMART goal model.
Journal Assignment
After reading the above content and watching the above videos, take some time to consider each of the questions below and document your thoughts:
- What is the difference between a fixed and growth mindset when it comes to setting and achieving goals? Identify an example of each in your own life.
- What are some common obstacles that might prevent someone from achieving a goal? How can a growth mindset help overcome these obstacles?
- Should people focus more on long-term or short-term goals when working to develop a growth mindset? Why?
- Imagine you set a goal but don't achieve it on your first attempt. What adjustments could you make while maintaining a growth mindset?
- How do you think goal setting can impact your motivation, resilience, and self-confidence over time?
Licenses and Attributions
" Chapter 3: Discover Your Values and Goals " from Learning Framework: Effective Strategies for College Success 2.0 . Copyright © 2025 by Texas Higher Education Coordinating Board is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.
CC LICENSED CONTENT, ORIGINAL
- Discover Your Values and Goals. Authored by : Heather Syrett. Provided by : Austin Community College. License : CC BY-NC-SA-4.0
CC LICENSED CONTENT, SPECIFIC ATTRIBUTION
- Personal Identity in EDUC 1300. Authored by : Linda Bruce. Provided by : Lumen Learning. Located at : https://courses.lumenlearning.com/sanjacinto-learningframework/chapter/personal-identity/ . License : CC BY 4.0
- Finding Your Core Values. Provided by : Thousand Insights. Located at : https://thousandinsights.wordpress.com/articles/on-leadership/finding-your-core-values/ . License : CC BY-SA 4.0
- Goal. Provided by : Wikipedia. Located at : https://en.wikipedia.org/wiki/Goal . License : CC BY-SA 3.0
- SMART_Criteria. Provided by : Wikipedia. Located at : https://en.wikipedia.org/wiki/SMART_criteria . License : CC BY-SA 3.0
ALL RIGHTS RESERVED CONTENT
- Life Stage Progression Assessment. Provided by : Career Test for the Soul. Located at : http://www.career-test.biz/stages_assessment.htm . License : All Rights Reserved
- SMART Goals - Quick Overview. Provided by : DecisionSkills. Located at : https://www.youtube.com/watch?v=1-SvuFIQjK8 . License : All Rights Reserved . License Terms : Standard YouTube License
- Drive: The Surprising Truth About What Motivates Us. Authored by : Daniel Pink. Provided by : RSA Animate. Located at : https://www.youtube.com/watch?v=u6XAPnuFjJc . License : All Rights Reserved . License Terms : Standard YouTube License


4.6A: Epithelial Membranes
The mucous membranes are linings of mostly endodermal origin, covered in epithelium, which are involved in absorption and secretion.
- Describe the function of mucous membranes
Key Points
- The mucous membranes are linings of ectodermal origin. It consists of an epithelium layer and an underlying lamina propria of loose connective tissue.
- The mucus membranes are involved in absorption and secretion.
- Most mucosal membranes contain stratified squamous or simple columnar epithelial tissue types.
- Submucosal exocrine glands secrete mucus to facilitate the movement of particles along the body’s various tubes, such as the throat and the intestines.
Key Terms
- mucous membrane : Linings of cavities that are exposed to the external environment and to internal organs.
The mucous membranes are linings of ectodermal origin. It consists of an epithelium layer and an underlying lamina propria of loose connective tissue. These mucus membranes are involved in absorption and secretion. They line cavities that are exposed to the external environment and internal organs. These membranes exist in the hollow organs of the digestive, respiratory, and urogenital tracts.
The term “mucous membrane” refers to where they are found in the body; not every mucous membrane secretes mucus. Secreted mucous traps the pathogens in the body, preventing any further progression of microbes.
Most mucous membranes contain stratified squamous or simple columnar epithelial tissue. The epithelial tissue sheet lies directly over the layer of loose connective tissue called lamina propria. In some mucosa, the lamina propria rests on a deeper, third layer of smooth muscle.
The submucosa is the tissue that connects the mucosa to the muscle outside the tube. Submucosal glands consist of exocrine glands that secrete mucus. These glands excrete mucus to facilitate the movement of particles along the body’s various tubes, such as the throat and intestines. The submucosal glands are a companion to unicellular goblet cells, which also produce mucus, and are found lining the same tubes.
General organization of the gastrointestinial tract : Illustration of mucosa in relation to other lining components.