4.2.E: More Problems on Limits and Continuity
Complete the proof of Corollary 1.
\(\left[\text { Hint: Consider }\left\{f\left(x_{m}\right)\right\} \text { and }\left\{f\left(x_{m}^{\prime}\right)\right\}, \text { with }\right.\)
\[
x_{m} \rightarrow p \text { and } x_{m}^{\prime} \rightarrow p .
\]
By Chapter 3, §14, Corollary \(4, p\) is also the limit of
\[
x_{1}, x_{1}^{\prime}, x_{2}, x_{2}^{\prime}, \ldots ,
\]
so, by assumption,
\[
f\left(x_{1}\right), f\left(x_{1}^{\prime}\right), \ldots \text { converges (to } q, \text { say } ) .
\]
Hence \(\left\{f\left(x_{m}\right)\right\}\) and \(\left\{f\left(x_{m}^{\prime}\right)\right\}\) must have the same limit \(q .(\mathrm{Why} ?) ]\)
Complete the converse proof of Theorem 2 (cf. proof of Theorem 1 in
Chapter 3, §17).
Define \(f, g : E^{1} \rightarrow E^{1}\) by setting
(i) \(f(x)=2 ; g(y)=3\) if \(y \neq 2,\) and \(g(2)=0 ;\) or
(ii) \(f(x)=2\) if \(x\) is rational and \(f(x)=2 x\) otherwise; \(g\) as in \((\mathrm{i})\).
In both cases, show that
\[
\lim _{x \rightarrow 1} f(x)=2 \text { and } \lim _{y \rightarrow 2} g(y)=3 \text { but not } \lim _{x \rightarrow 1} g(f(x))=3 .
\]
Prove Theorem 3 from "\(\varepsilon, \delta\) " definitions. Also prove (both ways) that if \(f\) is relatively continuous on \(B,\) and \(g\) on \(f[B],\) then \(g \circ f\) is relatively continuous on \(B\).
Complete the missing details in Examples (A) and (B).
[Hint for (B): Verify that
\[
\left(1-\frac{1}{n+1}\right)^{-n-1}=\left(\frac{n}{n+1}\right)^{-n-1}=\left(\frac{n+1}{n}\right)^{n+1}=\left(1+\frac{1}{n}\right)\left(1+\frac{1}{n}\right)^{n} \rightarrow e . ]
\]
\(\Rightarrow 6 .\) Given \(f, g, h : A \rightarrow E^{*}, A \subseteq(S, \rho),\) with
\[
f(x) \leq h(x) \leq g(x)
\]
for \(x \in G_{\neg p}(\delta) \cap A\) for some \(\delta>0 .\) Prove that if
\[
\lim _{x \rightarrow p} f(x)=\lim _{x \rightarrow p} g(x)=q ,
\]
also then
\[
\lim _{x \rightarrow p} h(x)=q.
\]
Use Theorem 1.
[Hint: Take any
\[
\left\{x_{m}\right\} \subseteq A-\{p\} \text { with } x_{m} \rightarrow p .
\]
Then \(f\left(x_{m}\right) \rightarrow q, g\left(x_{m}\right) \rightarrow q,\) and
\[
\left(\forall x_{m} \in A \cap G_{\neg p}(\delta)\right) \quad f\left(x_{m}\right) \leq h\left(x_{m}\right) \leq g\left(x_{m}\right) .
\]
Now apply Corollary 3 of Chapter \(3, §15 . ]\)
\(\Rightarrow 7 .\) Given \(f, g : A \rightarrow E^{*}, A \subseteq(S, \rho),\) with \(f(x) \rightarrow q\) and \(g(x) \rightarrow r\) as \(x \rightarrow p\) \((p \in S),\) prove the following:
(i) If \(q>r,\) then
\[
(\exists \delta>0)\left(\forall x \in A \cap G_{\neg p}(\delta)\right) \quad f(x)>g(x) .
\]
(ii) (Passage to the limit in inequalities.) If
\[
(\forall \delta>0)\left(\exists x \in A \cap G_{\neg p}(\delta)\right) \quad f(x) \leq g(x) ,
\]
then \(q \leq r .\) (Observe that here \(A\) clusters at \(p\) necessarily, so the limits are unique.)
[Hint: Proceed as in Problem \(6 ;\) use Corollary 1 of Chapter \(3, §15 .]\)
Do Problems 6 and 7 using only Definition 2 of §1.
\([\text { Hint: Here prove } 7(\text { ii }) \text { first. }]\)
Do Examples \((a)-(d)\) of §1 using Theorem 1.
[Hint: For \((\mathrm{c}),\) use also Example (a) in Chapter \(3, §16 . ]\)
Addition and multiplication in \(E^{1}\) may be treated as functions
\[
f, g : E^{2} \rightarrow E^{1}
\]
with
\[
f(x, y)=x+y \text { and } g(x, y)=x y .
\]
Show that \(f\) and \(g\) are continuous on \(E^{2}\) (see footnote 2 in Chapter 3 §15). Similarly, show that the standard metric
\[
\rho(x, y)=|x-y|
\]
is a continuous mapping from \(E^{2}\) to \(E^{1}\).
\([\text { Hint: Use Theorems } 1,2, \text { and, } 4 \text { of Chapter } 3, §15 \text { and the sequential criterion. }]\)
Using Corollary 2 and formula \((9),\) find \(\lim _{x \rightarrow 0}(1 \pm m x)^{1 / x}\) for a fixed \(m \in N .\)
\(\Rightarrow 12 .\) Let \(a>0\) in \(E^{1} .\) Prove that \(\lim _{x \rightarrow 0} a^{x}=1\).
\(\left[\text { Hint: Let } n=f(x) \text { be the integral part of } \frac{1}{x}(x \neq 0) . \text { Verify that }\right.\)
\[
a^{-1 /(n+1)} \leq a^{x} \leq a^{1 / n} \text { if } a \geq 1 ,
\]
with inequalities reversed if \(0<a<1 .\) Then proceed as in Example \((\mathrm{A}),\) noting that
\[
\lim _{n \rightarrow \infty} a^{1 / n}=1=\lim _{n \rightarrow \infty} a^{-1 /(n+1)}
\]
by Problem 20 of Chapter \(3, §15 .(\text { Explain! })]\)
\(\Rightarrow 13 .\) Given \(f, g : A \rightarrow E^{*}, A \subseteq(S, \rho),\) with
\[
f \leq g \quad \text { for } x \text { in } G_{\neg p}(\delta) \cap A .
\]
Prove that
(a) if \(\lim _{x \rightarrow p} f(x)=+\infty,\) then also \(\lim _{x \rightarrow p} g(x)=+\infty\);
(b) if \(\lim _{x \rightarrow p} g(x)=-\infty,\) then also \(\lim _{x \rightarrow p} f(x)=-\infty\).
Do it it two ways:
(i) Use definitions only, such as \(\left(2^{\prime}\right)\) in \(§1\).
(ii) Use Problem 10 of Chapter 2, §13 and the sequential criterion.
\(\Rightarrow 14 .\) Prove that
(i) if \(a>1\) in \(E^{1},\) then
\[
\lim _{x \rightarrow+\infty} \frac{a^{x}}{x}=+\infty \text { and } \lim _{x \rightarrow+\infty} \frac{a^{-x}}{x}=0 ;
\]
(ii) if \(0<a<1,\) then
\[
\lim _{x \rightarrow+\infty} \frac{a^{x}}{x}=0 \text { and } \lim _{x \rightarrow+\infty} \frac{a^{-x}}{x}=+\infty ;
\]
(iii) if \(a>1\) and \(0 \leq q \in E^{1},\) then
\[
\lim _{x \rightarrow+\infty} \frac{a^{x}}{x^{q}}=+\infty \text { and } \lim _{x \rightarrow+\infty} \frac{a^{-x}}{x^{q}}=0 ;
\]
\((\text { iv })\) if \(0<a<1\) and \(0 \leq q \in E^{1},\) then
\[
\lim _{x \rightarrow+\infty} \frac{a^{x}}{x^{q}}=0 \text { and } \lim _{x \rightarrow+\infty} \frac{a^{-x}}{x^{q}}=+\infty .
\]
[Hint: (i) From Problems 17 and 10 of Chapter \(3, §15,\) obtain
\[
\lim \frac{a^{n}}{n}=+\infty .
\]
Then proceed as in Examples \((\mathrm{A})-(\mathrm{C}) ;\) (iii) reduces to (i) by the method used in
Problem 18 of Chapter \(3, §15 . ]\)
\(\Rightarrow * 15 .\) For a map \(f :(S, \rho) \rightarrow\left(T, \rho^{\prime}\right),\) show that the following statements are equivalent:
(i) \(f\) is continuous on \(S\).
(ii) \((\forall A \subseteq S) f[\overline{A}] \subseteq \overline{f[A]}\).
(iii) \((\forall B \subseteq T) f^{-1}[\overline{B}] \supseteq \overline{f^{-1}[B]}\).
(iv) \(f^{-1}[B]\) is closed in \((S, \rho)\) whenever \(B\) is closed in \(\left(T, \rho^{\prime}\right)\).
(v) \(f^{-1}[B]\) is open in \((S, \rho)\) whenever \(B\) is open in \(\left(T, \rho^{\prime}\right)\).
[Hints: \( (i) \Longrightarrow(\mathrm{ii})\): Use Theorem 3 of Chapter } 3, §16 and the sequential criterion to show that
\[
p \in \overline{A} \Longrightarrow f(p) \in \overline{f[A]} .
\]
(ii) \(\Longrightarrow(\text { iii }) :\) Let \(A=f^{-1}[B] .\) Then \(f[A] \subseteq B,\) so by \((\text { ii })\),
\[
f[\overline{A}] \subseteq \overline{f[A]} \subseteq \overline{B} .
\]
Hence
\[
\overline{f^{-1}[B]}=\overline{A} \subseteq f^{-1}[f[\overline{A}]] \subseteq f^{-1}[\overline{B}] . \quad \text { (Why?) }
\]
(iii) \(\Longrightarrow(\mathrm{iv}) :\) If \(B\) is closed, \(B=\overline{B}\) (Chapter 3, §16, Theorem 4(ii)), so by (iii),
\[
f^{-1}[B]=f^{-1}[\overline{B}] \supseteq \overline{f^{-1}[B]} ; \text { deduce (iv) } .
\]
\((\mathrm{iv}) \Longrightarrow(\mathrm{v}) :\) Pass to complements in \((\mathrm{iv})\).
\((\mathrm{v}) \Longrightarrow(\mathrm{i}) :\) Assume \((\mathrm{v}) .\) Take any \(p \in S\) and use Definition 1 in \(§1 . ]\)
Let \(f : E^{1} \rightarrow E^{1}\) be continuous. Define \(g : E^{1} \rightarrow E^{2}\) by
\[
g(x)=(x, f(x)) .
\]
Prove that
(a) \(g\) and \(g^{-1}\) are one to one and continuous;
(b) the range of \(g,\) i.e., the set
\[
D_{g}^{\prime}=\left\{(x, f(x)) | x \in E^{1}\right\} ,
\]
is closed in \(E^{2}\).
[Hint: Use Theorem 2 of Chapter 3, §15, Theorem 4 of Chapter 3, §16, and the sequential criterion.]


2.4: Percentages and Proportions
Percents
In the 2004 vice-presidential debates, John Edwards claimed that US forces have suffered "90% of the coalition casualties" in Iraq. Dick Cheney disputed this, saying that in fact Iraqi security forces and coalition allies "have taken almost 50 percent" of the casualties. Who is correct? How can we make sense of these numbers?
Percent literally means “per 100,” or “parts per hundred.” When we write 40%, this is equivalent to the fraction 40100 or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since
\[\dfrac{80}{200} = \dfrac{10}{25} = \dfrac{40}{100} \nonumber\]
Example \(\PageIndex{1}\)
243 people out of 400 state that they like dogs. What percent is this?
Solution
\(\dfrac{243}{400}=0.6075=\dfrac{60.75}{100}\) This is 60.75%.
Notice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.
Example \(\PageIndex{2}\)
Write each as a percent:
- \(\dfrac{1}{4}\)
- 0.02
- 2.35
Solution
- \(\dfrac{1}{4} = 0.25 = 25\% \)
- 0.02 = 2%
- 2.35 = 235%
Percents
If we have a part that is some percent of a whole , then
\(\text {Percent} = \dfrac{\text {part}}{\text {whole}}\), or equivalently, \(\text {part}=\text {percent}\cdot \text {whole} \)
To do the calculations, we write the percent as a decimal.
Example \(\PageIndex{3}\)
The sales tax in a town is 9.4%. How much tax will you pay on a $140 purchase?
Solution
Here, $140 is the whole, and we want to find 9.4% of $140. We start by writing the percent as a decimal by moving the decimal point two places to the left (which is equivalent to dividing by 100). We can then compute:
tax = 0.094 (140) = $13.16 in tax.
Example \(\PageIndex{4}\)
In the news, you hear “tuition is expected to increase by 7% next year.” If tuition this year was $1200 per quarter, what will it be next year?
Solution
The tuition next year will be the current tuition plus an additional 7%, so it will be 107% of this year’s tuition:
$1200(1.07) = $1284.
Alternatively, we could have first calculated 7% of $1200: $1200(0.07) = $84.
Notice this is not the expected tuition for next year (we could only wish). Instead, this is the expected increase , so to calculate the expected tuition, we’ll need to add this change to the previous year’s tuition:
$1200 + $84 = $1284.
Try it Now 1
A TV originally priced at $799 is on sale for 30% off. There is then a 9.2% sales tax. Find the price after including the discount and sales tax.
Example \(\PageIndex{5}\)
The value of a car dropped from $7400 to $6800 over the last year. What percent decrease is this?
Solution
To compute the percent change, we first need to find the dollar value change: $6800-$7400 = -$600. Often we will take the absolute value of this amount, which is called the absolute change :
\(\mid -600 \mid = 600\)
Since we are computing the decrease relative to the starting value, we compute this percent out of $7400:
\(\dfrac{600}{7400} = 0.081 = 8.1\%\) decrease. This is called a relative change .
Absolute and Relative Change
Given two quantities,
\[\begin{align} \text{Absolute change} &= |\text{ending quantity} - \text{starting quantity}| \\[4pt] \text{Relative change} &= \dfrac{\text {absolute change}}{\text {staring quantity}} \end{align}\]
Absolute change has the same units as the original quantity and relative change gives a percent change.
The starting quantity is called the base of the percent change.
The base of a percent is very important. For example, while Nixon was president, it was argued that marijuana was a “gateway” drug, claiming that 80% of marijuana smokers went on to use harder drugs like cocaine. The problem is, this isn’t true. The true claim is that 80% of harder drug users first smoked marijuana. The difference is one of base: 80% of marijuana smokers using hard drugs, vs. 80% of hard drug users having smoked marijuana. These numbers are not equivalent. As it turns out, only one in 2,400 marijuana users actually go on to use harder drugs. 2
Example \(\PageIndex{6}\)
There are about 75 QFC supermarkets in the U.S. Albertsons has about 215 stores. Compare the size of the two companies.
Solution
When we make comparisons, we must ask first whether an absolute or relative comparison. The absolute difference is 215 – 75 = 140. From this, we could say “Albertsons has 140 more stores than QFC.” However, if you wrote this in an article or paper, that number does not mean much. The relative difference may be more meaningful. There are two different relative changes we could calculate, depending on which store we use as the base:
Using QFC as the base, \(\dfrac{140}{75} = 1.867\).
This tells us Albertsons is 186.7% larger than QFC.
Using Albertsons as the base, \(\dfrac{140}{215} = 0.651\).
This tells us QFC is 65.1% smaller than Albertsons.
Notice both of these are showing percent differences . We could also calculate the size of Albertsons relative to QFC: \(\dfrac{215}{75} = 2.867\), which tells us Albertsons is 2.867 times the size of QFC. Likewise, we could calculate the size of QFC relative to Albertsons: \(\dfrac{75}{215} = 0.349\), which tells us that QFC is 34.9% of the size of Albertsons.
Example \(\PageIndex{7}\)
Suppose a stock drops in value by 60% one week, then increases in value the next week by 75%. Is the value higher or lower than where it started?
Solution
To answer this question, suppose the value started at $100. After one week, the value dropped by 60%:
$100 - $100(0.60) = $100 - $60 = $40.
In the next week, notice that base of the percent has changed to the new value, $40. Computing the 75% increase:
$40 + $40(0.75) = $40 + $30 = $70.
In the end, the stock is still $30 lower, or \(\dfrac{$30}{$100} = 30\%\) lower, valued than it started.
Try it Now 2
The U.S. federal debt at the end of 2001 was $5.77 trillion, and grew to $6.20 trillion by the end of 2002. At the end of 2005 it was $7.91 trillion, and grew to $8.45 trillion by the end of 2006 3 . Calculate the absolute and relative increase for 2001-2002 and 2005-2006. Which year saw a larger increase in federal debt?
Example \(\PageIndex{8}\)
A Seattle Times article on high school graduation rates reported “The number of schools graduating 60 percent or fewer students in four years – sometimes referred to as “dropout factories” – decreased by 17 during that time period. The number of kids attending schools with such low graduation rates was cut in half.”
- Is the “decrease by 17” number a useful comparison?
- Considering the last sentence, can we conclude that the number of “dropout factories” was originally 34?
Solution
- This number is hard to evaluate, since we have no basis for judging whether this is a larger or small change. If the number of “dropout factories” dropped from 20 to 3, that’d be a very significant change, but if the number dropped from 217 to 200, that’d be less of an improvement.
- The last sentence provides relative change which helps put the first sentence in perspective. We can estimate that the number of “dropout factories” was probably previously around 34. However, it’s possible that students simply moved schools rather than the school improving, so that estimate might not be fully accurate.
Example \(\PageIndex{9}\)
In the 2004 vice-presidential debates, Edwards' claimed that US forces have suffered "90% of the coalition casualties" in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies "have taken almost 50 percent" of the casualties. Who is correct?
Solution
Without more information, it is hard for us to judge who is correct, but we can easily conclude that these two percents are talking about different things, so one does not necessarily contradict the other. Edwards’ claim was a percent with coalition forces as the base of the percent, while Cheney’s claim was a percent with both coalition and Iraqi security forces as the base of the percent. It turns out both statistics are in fact fairly accurate.
Try it Now 3
In the 2012 presidential elections, one candidate argued that “the president’s plan will cut $716 billion from Medicare, leading to fewer services for seniors,” while the other candidate rebuts that “our plan does not cut current spending and actually expands benefits for seniors, while implementing cost saving measures.” Are these claims in conflict, in agreement, or not comparable because they’re talking about different things?
We’ll wrap up our review of percents with a couple cautions. First, when talking about a change of quantities that are already measured in percents, we have to be careful in how we describe the change.
Example \(\PageIndex{10}\)
A politician’s support increases from 40% of voters to 50% of voters. Describe the change.
Solution
We could describe this using an absolute change: \(\mid 50\% - 40\% \mid = 10\%\). Notice that since the original quantities were percents, this change also has the units of percent. In this case, it is best to describe this as an increase of 10 percentage points .
In contrast, we could compute the percent change: \(\dfrac {10\%}{40\%} = 0.25 = 25\%\) increase. This is the relative change, and we’d say the politician’s support has increased by 25%.
Lastly, a caution against averaging percents.
Example \(\PageIndex{11}\)
A basketball player scores on 40% of two-point field goal attempts and on 30% of three-point field goal attempts. Find the player’s overall field goal percentage.
Solution
It is very tempting to average these values, and claim the overall average is 35%, but this is likely not correct, since most players make many more two-point attempts than three-point attempts. We don’t actually have enough information to answer the question. Suppose the player attempted 200 two-point field goals and 100 three-point field goals. Then they made 200(0.40) = 80 two-point shots and 100(0.30) = 30 three-point shots. Overall, they made 110 shots out of 300, for a \(\dfrac {110}{300} = 0.367 = 36.7\%\) overall field goal percentage.
Proportions and Rates
If you wanted to power the city of Seattle using wind power, how many windmills would you need to install? Questions like these can be answered using rates and proportions.
Rates
A rate is the ratio (fraction) of two quantities.
A unit rate is a rate with a denominator of one.
Example \(\PageIndex{12}\)
Your car can drive 300 miles on a tank of 15 gallons. Express this as a rate.
Solution
Expressed as a rate, \(\dfrac {300 \text { miles}}{15 \text { gallons}}\) We can divide to find a unit rate: \(\dfrac {20 \text { miles}}{1 \text { gallon}}\), which we could also write as \(20\dfrac {\text {miles}}{\text {gallon}}\), or just 20 miles per gallon
Proportion Equation
A proportion equation is an equation showing the equivalence of two rates or ratios.
A key to solving a proportion equation is “cross multiplication.” When two rations are equal to each other, two products obtained by cross multiplication are equal.
Example \(\PageIndex{13}\)
Solve the proportion \(\dfrac {5}{3}= \dfrac {x}{6}\) for the unknown value \(x\).
Solution
This proportion is asking us to find a fraction with denominator 6 that is equivalent to the fraction \(\dfrac {5}{3}\). We can solve this by cross multiplication, giving
\[\begin{align*}
3x &= 5\cdot 6\\
3x &= 30\\
x &= 10
\end{align*} \nonumber \]
Example \(\PageIndex{14}\)
A map scale indicates that ½ inch on the map corresponds with 3 real miles. How many miles apart are two cities that are \(2\dfrac{1}{4}\) inches apart on the map?
Solution
We can set up a proportion by setting equal two \(\dfrac {\text {map inches}}{\text {real miles}}\) rates, and introducing a variable, \(x\), to represent the unknown quantity – the mile distance between the cities.
\[\dfrac{\frac {1}{2} \text { Map inch}}{3 \text { miles}} = \dfrac{2\frac{1}{4} \text { map inch}}{x \text { miles}} \nonumber \]
By cross multiplication, we obtain: (rewriting the mixed number \(2\dfrac{1}{4}\) as \(\dfrac{9}{4}\))
\[\begin{align*}
\dfrac{1}{2}x &=3\cdot \left ( \dfrac{3}{9} \right )\\
\dfrac{1}{2}x &= \dfrac{27}{4},\; \text {Multiply both sides by 2}\\
x &= \dfrac{27}{4}\cdot 2\\
x &= \dfrac{27}{2}\\
x &= 13\dfrac{1}{2}\, \text {miles}
\end{align*} \nonumber \]
Reference
- References (3)
- http://tvtropes.org/pmwiki/pmwiki.php/Main/LiesDamnedLiesAndStatistics
- www.whitehouse.gov/sites/defa...s/hist07z1.xls
Contributors and Attributions
-
Saburo Matsumoto
CC-BY-4.0


1.1: Discussion- The Role of Human Resources
You are being hired as an HR leader for a small IT communication business that doesn’t currently have an HR function. The business owner wants you to hire people and keep them government compliant, but you know you can do more. This IT communication business wants to bring their communication products to two new international markets this year, wants to come up with innovation around portable devices, and hopes to increase their staff by 50% to support all the new activity.
Discussion Prompt
The small business owner doesn’t understand everything you, as HR professional, can do for him. Put together a list of all the functions you can perform to make the organization more successful if staffed with an HR department and how they will benefit the business. Then, review the suggestions of two of your peers, and make comment on their suggestions.
Grading
Share your opinions below and respond to two of your classmates’ thoughts.
| Criteria | Not Evident | Developing | Exemplary | Points |
|---|---|---|---|---|
| Submit your initial response |
0 pts
No post made |
5 pts
Post is either late or off-topic |
10 pts
Post is made on time and is focused on the prompt |
10 pts |
| Respond to at least two peers’ presentations |
0 pts
No response to peers |
2
pts
Responded to only one peer |
5 pts
Responded to two peers |
5 pts |
| Total: | 15 pts |


6.2: The Nature of Public Opinion
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Learning Objectives
By the end of this section, you will be able to:
- Define public opinion and political socialization
- Explain the process and role of political socialization in the U.S. political system
- Compare the ways in which citizens learn political information
- Explain how beliefs and ideology affect the formation of public opinion
The collection of public opinion through polling and interviews is a part of American political culture. Politicians want to know what the public thinks. Campaign managers want to know how citizens will vote. Media members seek to write stories about what Americans want. Every day, polls take the pulse of the people and report the results. And yet we have to wonder: Why do we care what people think?
What is Public Opinion?
Public opinion is a collection of popular views about something, perhaps a person, a local or national event, or a new idea. For example, each day, a number of polling companies call Americans at random to ask whether they approve or disapprove of the way the president is guiding the economy. 4 When situations arise internationally, polling companies survey whether citizens support U.S. intervention in places like Syria or Ukraine. These individual opinions are collected together to be analyzed and interpreted for politicians and the media. The analysis examines how the public feels or thinks, so politicians can use the information to make decisions about their future legislative votes, campaign messages, or propaganda.
But where do people’s opinions come from? Most citizens base their political opinions on their beliefs 5 and their attitudes, both of which begin to form in childhood. Beliefs are closely held ideas that support our values and expectations about life and politics. For example, the idea that we are all entitled to equality, liberty, freedom, and privacy is a belief most people in the United States share. We may acquire this belief by growing up in the United States or by having come from a country that did not afford these valued principles to its citizens.
Our attitudes are also affected by our personal beliefs and represent the preferences we form based on our life experiences and values. A person who has suffered racism or bigotry may have a skeptical attitude toward the actions of authority figures, for example.
Over time, our beliefs and our attitudes about people, events, and ideas will become a set of norms, or accepted ideas, about what we may feel should happen in our society or what is right for the government to do in a situation. In this way, attitudes and beliefs form the foundation for opinions.
Political Socialization
At the same time that our beliefs and attitudes are forming during childhood, we are also being socialized ; that is, we are learning from many information sources about the society and community in which we live and how we are to behave in it. Political socialization is the process by which we are trained to understand and join a country’s political world, and, like most forms of socialization, it starts when we are very young. We may first become aware of politics by watching a parent or guardian vote, for instance, or by hearing presidents and candidates speak on television or the Internet, or seeing adults honor the American flag at an event (Figure 6.2). As socialization continues, we are introduced to basic political information in school. We recite the Pledge of Allegiance and learn about the Founding Fathers, the Constitution, the two major political parties, the three branches of government, and the economic system.
By the time we complete school, we have usually acquired the information necessary to form political views and be contributing members of the political system. A young man may realize he prefers the Democratic Party because it supports his views on social programs and education, whereas a young woman may decide she wants to vote for the Republican Party because its platform echoes her beliefs about economic growth and family values.
Accounting for the process of socialization is central to our understanding of public opinion, because the beliefs we acquire early in life are unlikely to change dramatically as we grow older. 6 Our political ideology, made up of the attitudes and beliefs that help shape our opinions on political theory and policy, is rooted in who we are as individuals. Our ideology may change subtly as we grow older and are introduced to new circumstances or new information, but our underlying beliefs and attitudes are unlikely to change very much, unless we experience events that profoundly affect us. For example, family members of 9/11 victims became more Republican and more political following the terrorist attacks. 7 Similarly, young adults who attended political protest rallies in the 1960s and 1970s were more likely to participate in politics in general than their peers who had not protested. 8
If enough beliefs or attitudes are shattered by an event, such as an economic catastrophe or a threat to personal safety, ideology shifts may affect the way we vote. During the 1920s, the Republican Party controlled the House of Representatives and the Senate, sometimes by wide margins. 9 After the stock market collapsed and the nation slid into the Great Depression, many citizens abandoned the Republican Party. In 1932, voters overwhelmingly chose Democratic candidates, for both the presidency and Congress. The Democratic Party gained registered members and the Republican Party lost them. 10 Citizens’ beliefs had shifted enough to cause the control of Congress to change from one party to the other, and Democrats continued to hold Congress for several decades. Another sea change occurred in Congress in the 1994 elections when the Republican Party took control of both the House and the Senate for the first time in over forty years.
Today, polling agencies have noticed that citizens’ beliefs have become far more polarized, or widely opposed, over the last decade. 11 To track this polarization , Pew Research conducted a study of Republican and Democratic respondents over a twenty-five-year span. Every few years, Pew would poll respondents, asking them whether they agreed or disagreed with statements. These statements are referred to as “value questions” or “value statements,” because they measure what the respondent values. Examples of statements include “Government regulation of business usually does more harm than good,” “Labor unions are necessary to protect the working person,” and “Society should ensure all have equal opportunity to succeed.” After comparing such answers for twenty-five years, Pew Research found that Republican and Democratic respondents are increasingly answering these questions very differently. This is especially true for questions about the government and politics. In 1987, 58 percent of Democrats and 60 percent of Republicans agreed with the statement that the government controlled too much of our daily lives. In 2012, 47 percent of Democrats and 77 percent of Republicans agreed with the statement. This is an example of polarization, in which members of one party see government from a very different perspective than the members of the other party (Figure 6.3). 12 The gap between the parties on the role of government has grown further since 2012. While the 2012 gap was 30 percent, a 2019 Pew study put that number at 35 percent. 13
Political scientists noted this and other changes in beliefs following the 9/11 terrorist attacks on the United States, including an increase in the level of trust in government 14 and a new willingness to limit liberties for groups or citizens who “[did] not fit into the dominant cultural type.” 15 According to some scholars, these shifts led partisanship to become more polarized than in previous decades, as more citizens began thinking of themselves as conservative or liberal rather than moderate. 16 Some believe 9/11 caused a number of citizens to become more conservative overall, although it is hard to judge whether such a shift will be permanent. 17
Socialization Agents
An agent of political socialization is a source of political information intended to help citizens understand how to act in their political system and how to make decisions on political matters. The information may help a citizen decide how to vote, where to donate money, or how to protest decisions made by the government.
The most prominent agents of socialization are family and school. Other influential agents are social groups, such as religious institutions and friends, and the media. Political socialization is not unique to the United States. Many nations have realized the benefits of socializing their populations. China, for example, stresses nationalism in schools as a way to increase national unity. 18 In the United States, one benefit of socialization is that our political system enjoys diffuse support , which is support characterized by a high level of stability in politics, acceptance of the government as legitimate, and a common goal of preserving the system. 19 These traits keep a country steady, even during times of political or social upheaval. But diffuse support does not happen quickly, nor does it occur without the help of agents of political socialization.
For many children, family is the first introduction to politics. Children may hear adult conversations at home and piece together the political messages their parents support. They often know how their parents or grandparents plan to vote, which in turn can socialize them into political behavior such as political party membership. 20 Children who accompany their parents on Election Day in November are exposed to the act of voting and the concept of civic duty, which is the performance of actions that benefit the country or community. Families active in community projects or politics make children aware of community needs and politics.
Introducing children to these activities has an impact on their future behavior. Both early and recent findings suggest that children adopt some of the political beliefs and attitudes of their parents (Figure 6.4). 21 Children of Democratic parents often become registered Democrats, whereas children in Republican households often become Republicans. Children living in households where parents do not display a consistent political party loyalty are less likely to be strong Democrats or strong Republicans, and instead are often independents. 22
While family provides an informal political education, schools offer a more formal and increasingly important one. The early introduction is often broad and thematic, covering explorers, presidents, victories, and symbols, but generally the lessons are idealized and do not discuss many of the specific problems or controversies connected with historical figures and moments. George Washington’s contributions as our first president are highlighted, for instance, but teachers are unlikely to mention that he owned slaves. 23 Lessons will also try to personalize government and make leaders relatable to children. A teacher might discuss Abraham Lincoln’s childhood struggle to get an education despite the death of his mother and his family’s poverty. Children learn to respect government, follow laws, and obey the requests of police, firefighters, and other first responders. The Pledge of Allegiance becomes a regular part of the school day, as students learn to show respect to our country’s symbols such as the flag and to abstractions such as liberty and equality.
As students progress to higher grades, lessons will cover more detailed information about the history of the United States, its economic system, and the workings of the government. Complex topics such as the legislative process, checks and balances, and domestic policymaking are covered. Introductory economics classes teach about the various ways to build an economy, explaining how the capitalist system works. Many high schools have implemented civic volunteerism requirements as a way to encourage students to participate in their communities. Many offer Advanced Placement classes in U.S. government and history, or other honors-level courses, such as International Baccalaureate or dual-credit courses. These courses can introduce detail and realism, raise controversial topics, and encourage students to make comparisons and think critically about the United States in a global and historical context. College students may choose to pursue their academic study of the U.S. political system further, become active in campus advocacy or rights groups, or run for any of a number of elected positions on campus or even in the local community. Each step of the educational system’s socialization process will ready students to make decisions and be participating members of political society.
We are also socialized outside our homes and schools. When citizens attend religious ceremonies, as 64 percent of Americans in a recent survey claimed, 24 they are socialized to adopt beliefs that affect their politics. Religion leaders often teach on matters of life, death, punishment, and obligation, which translate into views on political issues such as abortion, euthanasia, social services, the death penalty, and military involvement abroad. Political candidates speak at religious centers and institutions in an effort to meet like-minded voters. For example, Senator Ted Cruz (R-TX) announced his 2016 presidential bid at Liberty University, an evangelical Christian institution. This university matched Cruz’s conservative and religious ideological leanings and was intended to give him a boost from the conservative evangelical community.
Friends and peers too have a socializing effect on citizens. Communication networks are based on trust and common interests, so when we receive information from friends and neighbors, we often readily accept it because we trust them. 25 Information transmitted through social media like Facebook is also likely to have a socializing effect. Friends “like” articles and information, sharing their political beliefs and information with one another. However, because of the self-selection of friends, social media outlets have the potential to introduce bias. If the social media platform does not properly police what is factual in posts and advertisements, then these outlets can become objects of manipulation, as in the 2016 election. 26
Media—newspapers, television, radio, and the Internet—also socialize citizens through the information they provide. For a long time, the media served as gatekeepers of our information, creating reality by choosing what to present. If the media did not cover an issue or event, it was as if it did not exist. With the rise of the Internet and social media, however, traditional media have become less powerful agents of this kind of socialization.
Another way the media socializes audiences is through framing , or choosing the way information is presented. Framing can affect the way an event or story is perceived. Candidates described with negative adjectives, for instance, may do poorly on Election Day. Consider the recent demonstrations over the deaths of Michael Brown in Ferguson, Missouri, and of Freddie Gray in Baltimore, Maryland. Both deaths were caused by police actions against unarmed African American men. Brown was shot to death by an officer on August 9, 2014. Gray died from spinal injuries sustained in transport to jail in April 2015. Following each death, family, friends, and sympathizers protested the police actions as excessive and unfair. While some television stations framed the demonstrations as riots and looting, other stations framed them as protests and fights against corruption. The demonstrations contained both riot and protest, but individuals’ perceptions were affected by the framing chosen by their preferred information sources (Figure 6.5). 27
Finally, media information presented as fact can contain covert or overt political material. Covert content is political information provided under the pretense that it is neutral. A magazine might run a story on climate change by interviewing representatives of only one side of the policy debate and downplaying the opposing view, all without acknowledging the one-sided nature of its coverage. In contrast, when the writer or publication makes clear to the reader or viewer that the information offers only one side of the political debate, the political message is overt content. Political commentators like Rush Limbaugh and publications like Mother Jones openly state their ideological viewpoints. While such overt political content may be offensive or annoying to a reader or viewer, all are offered the choice whether to be exposed to the material.
Socialization and Ideology
The socialization process leaves citizens with attitudes and beliefs that create a personal ideology. Ideologies depend on attitudes and beliefs, and on the way we prioritize each belief over the others. Most citizens hold a great number of beliefs and attitudes about government action. Many think government should provide for the common defense, in the form of a national military. They also argue that government should provide services to its citizens in the form of free education, unemployment benefits, and assistance for the poor.
When asked how to divide the national budget, Americans reveal priorities that divide public opinion. Should we have a smaller military and larger social benefits, or a larger military budget and limited social benefits? This is the guns versus butter debate, which assumes that governments have a finite amount of money and must choose whether to spend a larger part on the military or on social programs. The choice forces citizens into two opposing groups.
Divisions like these appear throughout public opinion. Assume we have four different people named Garcia, Chin, Smith, and Dupree. Garcia may believe that the United States should provide a free education for every citizen all the way through college, whereas Chin may believe education should be free only through high school. Smith might believe children should be covered by health insurance at the government’s expense, whereas Dupree believes all citizens should be covered. In the end, the way we prioritize our beliefs and what we decide is most important to us determines whether we are on the liberal or conservative end of the political spectrum, or somewhere in between.
Get Connected!
Express Yourself
You can volunteer to participate in public opinion surveys. Diverse respondents are needed across a variety of topics to give a reliable picture of what Americans think about politics, entertainment, marketing, and more. One polling group, Harris Interactive, maintains an Internet pool of potential respondents of varied ages, education levels, backgrounds, cultures, and more. When a survey is designed and put out into the field, Harris emails an invitation to the pool to find respondents. Respondents choose which surveys to complete based on the topics, time required, and compensation offered (usually small).
Harris Interactive is a subsidiary of Nielsen, a company with a long history of measuring television and media viewership in the United States and abroad. Nielsen ratings help television stations identify shows and newscasts with enough viewers to warrant being kept in production, and also to set advertising rates (based on audience size) for commercials on popular shows. Harris Interactive has expanded Nielsen’s survey methods by using polling data and interviews to better predict future political and market trends.
Harris polls cover the economy, lifestyles, sports, international affairs, and more. Which topic has the most surveys? Politics, of course.
Wondering what types of surveys you might get? The results of some of the surveys will give you an idea. They are available to the public on the Harris website. For more information, log in to Harris Poll Online .
Ideologies and the Ideological Spectrum
One useful way to look at ideologies is to place them on a spectrum that visually compares them based on what they prioritize. Liberal ideologies are traditionally put on the left and conservative ideologies on the right. (This placement dates from the French Revolution and is why liberals are called left-wing and conservatives are called right-wing.) The ideologies at the ends of the spectrum are the most extreme; those in the middle are moderate. Thus, people who identify with left- and right-wing ideologies identify with beliefs to the left and right ends of the spectrum, while moderates balance the beliefs at the extremes of the spectrum.
In the United States, ideologies at the right side of the spectrum prioritize government control over personal freedoms. They range from fascism to authoritarianism to conservatism. Ideologies on the left side of the spectrum prioritize equality and range from communism to socialism to liberalism (Figure 6.6). Moderate ideologies fall in the middle and try to balance the two extremes. When thinking about ideology and politics, it is important not to fall into thinking that involves false dichotomies. One such false dichotomy involves socialism versus capitalism. The two terms have been conflated into partisan categories that connote either collectivism or individualism. Political systems, including the U.S., can possess aspects of both socialism and capitalism. 28
Fascism promotes total control of the country by the ruling party or political leader. This form of government will run the economy, the military, society, and culture, and often tries to control the private lives of its citizens. Authoritarian leaders control the politics, military, and government of a country, and often the economy as well.
Conservative governments attempt to hold tight to the traditions of a nation by balancing individual rights with the good of the community. Traditional conservatism supports the authority of the monarchy and the church, believing government provides the rule of law and maintains a society that is safe and organized. Modern conservatism differs from traditional conservatism in assuming elected government will guard individual liberties and provide laws. Modern conservatives also prefer a smaller government that stays out of the economy, allowing the market and business to determine prices, wages, and supply.
Classical liberalism believes in individual liberties and rights. It is based on the idea of free will, that people are born equal with the right to make decisions without government intervention. It views government with suspicion, since history includes many examples of monarchs and leaders who limited citizens’ rights. Today, modern liberalism focuses on equality and supports government intervention in society and the economy if it promotes equality. Liberals expect government to provide basic social and educational programs to help everyone have a chance to succeed.
Under socialism, the government uses its authority to promote social and economic equality within the country. Socialists believe government should provide everyone with expanded services and public programs, such as health care, subsidized housing and groceries, childhood education, and inexpensive college tuition. Socialism sees the government as a way to ensure all citizens receive both equal opportunities and equal outcomes. Citizens with more wealth are expected to contribute more to the state’s revenue through higher taxes that pay for services provided to all. Socialist countries are also likely to have higher minimum wages than non-socialist countries.
In theory, communism promotes common ownership of all property, means of production, and materials. This means that the government, or states, should own the property, farms, manufacturing, and businesses. By controlling these aspects of the economy, Communist governments can prevent the exploitation of workers while creating an equal society. Extreme inequality of income, in which some citizens earn millions of dollars a year and other citizens merely hundreds, is prevented by instituting wage controls or by abandoning currency altogether. Communism presents a problem, however, because the practice differs from the theory. The theory assumes the move to communism is supported and led by the proletariat, or the workers and citizens of a country. 29 Human rights violations by governments of actual Communist countries make it appear the movement has been driven not by the people, but by leadership.
We can characterize economic variations on these ideologies by adding another dimension to the ideological spectrum above—whether we prefer that government control the state economy or stay out of it. The extremes are a command economy, such as existed in the former Soviet Russia, and a laissez-faire (“leave it alone”) economy, such as in the United States prior to the 1929 market crash, when banks and corporations were largely unregulated. Communism prioritizes control of both politics and economy, while libertarianism is its near-opposite. Libertarians believe in individual rights and limited government intervention in private life and personal economic decisions. Government exists to maintain freedom and life, so its main function is to ensure domestic peace and national defense. Libertarians also believe the national government should maintain a military in case of international threats, but that it should not engage in setting minimum wages or ruling in private matters, like same-sex marriage or the right to abortion. 30
The point where a person’s ideology falls on the spectrum gives us some insight to his or her opinions. Though people can sometimes be liberal on one issue and conservative on another, a citizen to the left of liberalism, near socialism, would likely be happy with the passage of the Raise the Wage Act of 2015, which would eventually increase the minimum wage from $7.25 to $12 an hour. A citizen falling near conservatism would believe the Patriot Act is reasonable, because it allows the FBI and other government agencies to collect data on citizens’ phone calls and social media communications to monitor potential terrorism (Figure 6.7). A citizen to the right of the spectrum is more likely to favor cutting social services like unemployment and Medicaid.
Link to Learning
Where do your beliefs come from? The Pew Research Center offers a typology quiz to help you find out. Ask a friend or family member to answer a few questions with you and compare results. What do you think about government regulation? The military? The economy? Now compare your results. Are you both liberal? Conservative? Moderate?


11.3: Three angles of triangle
Let \(\triangle ABC\) and \(\triangle A'B'C'\) be two triangles in the neutral plane such that \(AC = A'C'\) and \(BC = B'C'\). Then
\(AB < A'B'\) if and only if \(|\measuredangle ACB| < |\measuredangle A'C'B'|\).
- Proof
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Without loss of generality, we may assume that \(A = A'\), \(C = C'\), and \(\measuredangle ACB\), \(\measuredangle ACB' \ge 0\). In this case we need to show that
\(AB < AB' \Leftrightarrow \measuredangle ACB < \measuredangle ACB'.\)
Choose a point \(X\) so that
\(\measuredangle ACX = \dfrac{1}{2} \cdot (\measuredangle ACB + \measuredangle ACB').\)
Note that
- \((CX)\) bisects \(\angle BCB'\).
- \((CX)\) is the perpendicular bisector of \([BB']\).
- \(A\) and \(B\) lie on the same side of \((CX)\) if and only if
\(\measuredangle ACB < \measuredangle ACB'\).
From Exercise 5.2.1 , \(A\) and \(B\) lie on the same side of \((CX)\) if and only if \(AB < AB'\). Hence the result.
Let \(\triangle ABC\) be a triangle in the neutral plane. Then
\(|\measuredangle ABC| + |\measuredangle BCA| + |\measuredangle CAB| \le \pi.\)
The following proof is due to Legendre [ 12 ], earler proofs were due to Saccheri [ 16 ] and Lambert [ 11 ].
- Proof
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Set
\(\begin{array} {rclcrclcrcl} {a} & = & {BC,} & \ \ \ \ \ & {b} & = & {CA,} & \ \ \ \ \ & {c} & = & {AB,} \\ {\alpha} & = & {\measuredangle CAB,} & \ \ \ \ \ & {\beta} & = & {\measuredangle ABC,} & \ \ \ \ \ & {\gamma} & = & {\measuredangle BCA.} \end{array}\)
Without loss of generality, we may assume that \(\alpha, \beta, \gamma \ge 0\).
Fix a positive integer \(n\). Consider the points \(A_0, A_1, ..., A_n\) on the half-line \([BA)\), such that \(BA_i = i \cdot c\) for each \(i\). (In particular, \(A_0 = B\) and \(A_1 = A\).) Let us construct the points \(C_1, C_2, ..., C_n\), so that \(\measuredangle A_iA_{i-1}C_i = \beta\) and \(A_{i-1} C_i = a\) for each \(i\).
By SAS, we have constructed n congruent triangles
\(\triangle ABC = \triangle A_1A_0C_1 \cong \triangle A_2A_1C_2 \cong ... \cong \triangle A_nA_{n-1} C_n.\)
Set \(d = C_1C_2\) and \(\delta = \measuredangle C_2A_1C_1\). Note that
\[\alpha + \beta + \delta = \pi.\]
By Proposition 11.2.1 , we get that \(\Delta \ge 0\).
By construction
\(\triangle A_1C_1C_2 \cong \triangle A_2C_2C_3 \cong ... \cong \triangle A_{n - 1} C_{n - 1} C_n.\)
In particular, \(C_i C_{i + 1} = d\) for each \(i\).
By repeated application of the triangle inequality, we get that
\(\begin{array} {rcl} {n \cdot c} & = & {A_0A_n \le} \\ {} & \le & {A_0 C_1 + C_1 C_2 + \cdots + C_{n - 1} C_n + C_n A_n =} \\ {} & = & {a + (n - 1) \cdot d + b.} \end{array}\)
In particular,
\(c \le d + \dfrac{1}{n} \cdot (a + b - d).\)
Since \(n\) is arbitrary positive integer, the latter implies \(c \le d\). By Proposition \(\PageIndex{1}\), it is equivalent to
\(\gamma \le \delta.\)
From 11.3.1, the theorem follows.
Let \(ABCD\) be a quadrangle in the neutral plane. Suppose that the angles \(DAB\) and \(ABC\) are right. Show that \(AB \le CD\).
- Hint
-
Set \(a = AB, b = BC, c = CD\), and \(d = DA\); we nned to show that \(c \ge a\).
Mimic the proof of Theorem \(\PageIndex{1}\) for the shown fence made from copies of quadrangle \(ABCD\).


1.4.3: Electron Affinity
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- Kathryn Haas
- Duke University
Definitions of Electron Affinity
According to IUPAC, there are two different, but equivalent, definitions of electron affinity (EA) . 1
Definition: Electron Affinity defined as removal of an electron
Electron affinity can be defined as the energy required when an electron is removed from a gaseous anion. The reaction as shown in equation \(\ref{EA1}\) is endothermic (positive \(\Delta U\)) for elements except noble gases and alkaline earth metals. Under this definition, the more positive the EA value, the higher an atom's affinity for electrons.
\[A^{-}_{(g)} \longrightarrow A_{(g)} + e^- \hspace{1cm} EA = \Delta U \label{EA1} \]
The reaction shown in Equation \(\ref{EA1}\) is similar those that define ionization energy . For this reason, the EA is also described as the zeroth ionization energy .
Definition: Electron Affinity defined as addition of an electron
An alternate and more common definition is the microscopic reverse of Equation \(\ref{EA1}\). This more common definition states that electron affinity is the energy released when an electron is added to a gaseous atom, as shown in Equation \(\ref{EA2}\). The reaction as shown in equation \(\ref{EA2}\) is exothermic (negative \(\Delta U\)) for elements except noble gases and alkaline earth metals. The more negative this EA value, the higher an atom's affinity for electrons.
\[A_{(g)} + e^- \longrightarrow A^{-}_{(g)} \hspace{1cm} EA = \Delta U \label{EA2} \]
Conceptually, this second definition is quite similar to the concept of electronegativity; but unlike electronegativity, EA is a well-defined quantitative measurement.
Trends in Electron Affinity
For this discussion, we will use the definition of EA that is consistent with it being a zeroth ionization energy: a more positive (larger) value means that the EA is higher (meaning stronger affinity toward an electron).
- Across a period : Similar to ionization energy, EA generally increases across a row of the periodic table; this observation is consistent with the increase in effective nuclear charge (Z*) from left to right across a period. However, there are variations across a period that are similar to variations in ionization energy and that can be explained by shielding, penetration, and electron configuration.
- Down a group : Like the case of ionization energy trends, EA does not consistently decrease going down a column of the periodic table despite the fact that \(Z^*\) increases down a group.
The trend in EA follows a zig-zag pattern similar to the one seen with ionization energies, except that it is displaced by one unit from the trend in \(I_1\), two units from \(I_2\), and so on. For example, EA peaks at F, while \(I_1\) peaks at Ne, \(I_2\) peaks at Na, and \(I_3\) peaks at Mg. A plot of EA for the first 13 elements is shown overlaid on plots of \(I_1, I_2\) and \(I_3\) in Figure \(\PageIndex{1}\)., where the shifts in the peaks and valleys within each zig-zag trend are indicated.
Sources
- IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). Online version (2019-) created by S. J. Chalk. ISBN 0-9678550-9-8. doi.org/10.1351/goldbook.
- Electron Affinity (data page), Wikipedia. en.Wikipedia.org/wiki/Electron_affinity_(data_page) Accessed 12/3/19.


13.4: Shifting Equilibria - Le Chatelier’s Principle
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Learning Objectives
By the end of this section, you will be able to:
- Describe the ways in which an equilibrium system can be stressed
- Predict the response of a stressed equilibrium using Le Chatelier’s principle
A system at equilibrium is in a state of dynamic balance, with forward and reverse reactions taking place at equal rates. If an equilibrium system is subjected to a change in conditions that affects these reaction rates differently (a stress ), then the rates are no longer equal and the system is not at equilibrium. The system will subsequently experience a net reaction in the direction of greater rate (a shift ) that will re-establish the equilibrium. This phenomenon is summarized by Le Chatelier’s principle
Definition: Le Chatelier’s Principle
if an equilibrium system is stressed, the system will experience a shift in response to the stress that re-establishes equilibrium.
Reaction rates are affected primarily by concentrations, as described by the reaction’s rate law, and temperature, as described by the Arrhenius equation. Consequently, changes in concentration and temperature are the two stresses that can shift an equilibrium.
Effect of a Change in Concentration
If an equilibrium system is subjected to a change in the concentration of a reactant or product species, the rate of either the forward or the reverse reaction will change. As an example, consider the equilibrium reaction
\[\ce{H2(g) + I2(g) <=> 2 HI (g)} \quad K_c=50.0 \text { at } 400{ }^{\circ} C \nonumber \]
The rate laws for the forward and reverse reactions are
\[\begin{align*}
&\text{forward} \quad \ce{H2(g) + I2(g) \rightarrow 2HI(g)} \quad &&\text{rate}_f=k_f[ \ce{H2}]^m [ \ce{I2} ]^n \\[4pt]
&\text{reverse} \quad \ce{2 HI(g) \rightarrow H2(g) + I2(g)} \quad &&\text{rate}_r=k_r[\ce{HI}]^x
\end{align*} \nonumber \]
When this system is at equilibrium, the forward and reverse reaction rates are equal.
\[\text { rate }_f=\text { rate }_r \nonumber \]
If the system is stressed by adding reactant, either \(\ce{H_2}\) or \(\ce{I_2}\), the resulting increase in concentration causes the rate of the forward reaction to increase, exceeding that of the reverse reaction:
\[\operatorname{rate}_f>\text { rate }_r \nonumber \]
The system will experience a temporary net reaction in the forward direction to re-establish equilibrium (the equilibrium will shift right). This same shift will result if some product HI is removed from the system, which decreases the rate of the reverse reaction, again resulting in the same imbalance in rates.
The same logic can be used to explain the left shift that results from either removing reactant or adding product to an equilibrium system. These stresses both result in an increased rate for the reverse reaction
\[\text { rate }_f<\text { rate }_r \nonumber \]
and a temporary net reaction in the reverse direction to re-establish equilibrium.
As an alternative to this kinetic interpretation, the effect of changes in concentration on equilibria can be rationalized in terms of reaction quotients. When the system is at equilibrium,
\[Q_c=\dfrac{[ \ce{HI} ]^2}{\left[ \ce{H2} \right]\left[ \ce{I2} \right]}=K_c \nonumber \]
If reactant is added (increasing the denominator of the reaction quotient) or product is removed (decreasing the numerator), then Q c < K c and the equilibrium will shift right. Note that the three different ways of inducing this stress result in three different changes in the composition of the equilibrium mixture. If H 2 is added, the right shift will consume I 2 and produce HI as equilibrium is re-established, yielding a mixture with a greater concentrations of H 2 and HI and a lesser concentration of I 2 than was present before. If I 2 is added, the new equilibrium mixture will have greater concentrations of I 2 and HI and a lesser concentration of H 2 . Finally, if HI is removed, the concentrations of all three species will be lower when equilibrium is reestablished. Despite these differences in composition, the value of the equilibrium constant will be the same after the stress as it was before (per the law of mass action). The same logic may be applied for stresses involving removing reactants or adding product, in which case Q c > K c and the equilibrium will shift left.For gas-phase equilibria such as this one, some additional perspectives on changing the concentrations of reactants and products are worthy of mention. The partial pressure P of an ideal gas is proportional to its molar concentration M,
\[M=\dfrac{n}{V}=\dfrac{P}{R T} \nonumber \]
and so changes in the partial pressures of any reactant or product are essentially changes in concentrations and thus yield the same effects on equilibria. Aside from adding or removing reactant or product, the pressures (concentrations) of species in a gas-phase equilibrium can also be changed by changing the volume occupied by the system . Since
all species of a gas-phase equilibrium occupy the same volume, a given change in volume will cause the same change in concentration for both reactants and products. In order to discern what shift, if any, this type of stress will induce the stoichiometry of the reaction must be considered.
At equilibrium, the reaction
\[\ce{H2(g) + I2(g) <=> 2HI(g)} \nonumber \]
is described by the reaction quotient
\[Q_P=\dfrac{(P_{\ce{HI}})^2}{P_{\ce{H_2}} P_{\ce{I2}}}=K_p \nonumber \]
If the volume occupied by an equilibrium mixture of these species is decreased by a factor of 3, the partial pressures of all three species will be increased by a factor of 3:
\[\begin{align*}
Q_p{ }^{\prime} &=\dfrac{\left(3 P_{ HI }\right)^2}{3 P_{ H_2} 3 P_{ I_2}}=\dfrac{9 P_{ H^2}}{9 P_{ H_2} P_{ I_2}}=\dfrac{P_{ HI^2}}{P_{ H_2} P_{ I_2}} \\[4pt]
&=Q_P=K_P
\end{align*} \nonumber \]
And so, changing the volume of this gas-phase equilibrium mixture does not result in a shift of the equilibrium.
A similar treatment of a different system,
\[\ce{2NO2(g) ⇌ 2 NO(g) + O2(g)} \nonumber \]
however, yields a different result:
\[\begin{align*}
Q_P &=\dfrac{(P_{NO})^2} {(P_{O_2})}{(P_{NO_2})^2} \\[4pt]
Q_P^{\prime} &= \dfrac{(3 P_{ \ce{NO}})^2 (3 P_{ \ce{O2}}) }{3 (P_{ \ce{NO2} })^2} = \dfrac{9 (P_{ \ce{NO}})^2 (3 P_{\ce{O_2}}) }{9 (P_{\ce{NO2}})^2} = \dfrac{27 (P_{\ce{NO}})^2 P_{ \ce{O_2}}}{9 (P_{ \ce{NO_2} })^2} \\[4pt]
&=3 Q_P>K_P
\end{align*} \nonumber \]
In this case, the change in volume results in a reaction quotient greater than the equilibrium constant, and so the equilibrium will shift left.
These results illustrate the relationship between the stoichiometry of a gas-phase equilibrium and the effect of a volume-induced pressure (concentration) change. If the total molar amounts of reactants and products are equal, as in the first example, a change in volume does not shift the equilibrium. If the molar amounts of reactants and products are different, a change in volume will shift the equilibrium in a direction that better “accommodates” the volume change. In the second example, two moles of reactant (NO 2 ) yield three moles of product (2NO + O 2 ), and so decreasing the system volume causes the equilibrium to shift left since the reverse reaction produces less gas (2 mol) than the forward reaction (3 mol). Conversely, increasing the volume of this equilibrium system would result in a shift towards products.
Link to Learning
Check out this link to see a dramatic visual demonstration of how equilibrium changes with pressure changes.
Chemistry in Everyday Life: Equilibrium and Soft Drinks
The connection between chemistry and carbonated soft drinks goes back to 1767, when Joseph Priestley (1733–1804) developed a method of infusing water with carbon dioxide to make carbonated water. Priestley’s approach involved production of carbon dioxide by reacting oil of vitriol (sulfuric acid) with chalk (calcium carbonate).
The carbon dioxide was then dissolved in water, reacting to produce hydrogen carbonate, a weak acid that subsequently ionized to yield bicarbonate and hydrogen ions:
\[\begin{align*}
&\text{dissolution} \quad &\ce{CO2(g) <=> CO2(aq)} \\[4pt]
&\text{hydrolysis} \quad &\ce{CO2(aq) + H2O(l) <=> H2CO3(aq)} \\[4pt]
&\text{ionization} \quad &\ce{H2CO3(aq) <=> HCO3^{-}(aq) + H^{+}(aq)}
\end{align*} \nonumber \]
These same equilibrium reactions are the basis of today’s soft-drink carbonation process. Beverages are exposed to a high pressure of gaseous carbon dioxide during the process to shift the first equilibrium above to
the right, resulting in desirably high concentrations of dissolved carbon dioxide and, per similar shifts in the other two equilibria, its hydrolysis and ionization products. A bottle or can is then nearly filled with the carbonated beverage, leaving a relatively small volume of air in the container above the beverage surface (the headspace ) before it is sealed. The pressure of carbon dioxide in the container headspace is very low immediately after sealing, but it rises as the dissolution equilibrium is re-established by shifting to the left. Since the volume of the beverage is significantly greater than the volume of the headspace, only a relatively small amount of dissolved carbon dioxide is lost to the headspace.
When a carbonated beverage container is opened, a hissing sound is heard as pressurized CO 2 escapes from the headspace. This causes the dissolution equilibrium to shift left, resulting in a decrease in the concentration of dissolved CO 2 and subsequent left-shifts of the hydrolysis and ionization equilibria. Fortunately for the consumer, the dissolution equilibrium is usually re-established slowly, and so the beverage may be enjoyed while its dissolved carbon dioxide concentration remains palatably high. Once the equilibria are re-established, the CO 2 ( aq ) concentration will be significantly lowered, and the beverage acquires a characteristic taste referred to as “flat.”
Effect of a Change in Temperature
Consistent with the law of mass action, an equilibrium stressed by a change in concentration will shift to re-establish equilibrium without any change in the value of the
equilibrium constant, K. When an equilibrium shifts in response to a temperature change, however, it is re-established with a different relative composition that exhibits a different value for the equilibrium constant.
To understand this phenomenon, consider the elementary reaction
\[A \rightleftharpoons B \nonumber \]
Since this is an elementary reaction, the rates laws for the forward and reverse may be derived directly from the balanced equation’s stoichiometry:
\[\begin{aligned}
\operatorname{rate}_f & =k_f[ A ] \\[4pt]
\operatorname{rate}_r & =k_r[ B ]
\end{aligned} \nonumber \]
When the system is at equilibrium,
\[\text { rate }_r=\text { rate }_f \nonumber \]
Substituting the rate laws into this equality and rearranging gives
\[\begin{aligned}
& k_f[ A ]=k_r[ B ] \\[4pt]
& \dfrac{[B]}{[A]}=\dfrac{k_f}{k_r}=K_c
\end{aligned} \nonumber \]
The equilibrium constant is seen to be a mathematical function of the rate constants for the forward and reverse reactions. Since the rate constants vary with temperature as
described by the Arrhenius equation, is stands to reason that the equilibrium constant will likewise vary with temperature (assuming the rate constants are affected to different extents by the temperature change). For more complex reactions involving multistep reaction mechanisms, a similar but more complex mathematical relation exists between the equilibrium constant and the rate constants of the steps in the mechanism. Regardless of how complex the reaction may be, the temperature-dependence of its equilibrium constant persists.
Predicting the shift an equilibrium will experience in response to a change in temperature is most conveniently accomplished by considering the enthalpy change of the reaction. For example, the decomposition of dinitrogen tetroxide is an endothermic (heat-consuming) process:
\[\ce{N2O4(g) <=> 2 NO2(g)} \quad \Delta H=+57.20 ~\text{kJ} \nonumber \]
For purposes of applying Le Chatelier’s principle, heat (q) may be viewed as a reactant:
\[\ce{ heat + N2O4(g) <=> 2 NO2(g)} \nonumber \]
Raising the temperature of the system is akin to increasing the amount of a reactant, and so the equilibrium will shift to the right. Lowering the system temperature will likewise cause the equilibrium to shift left. For exothermic processes, heat is viewed as a product of the reaction and so the opposite temperature dependence is observed.
Effect of a Catalyst
The kinetics chapter of this text identifies a catalyst as a substance that enables a reaction to proceed via a different mechanism with an accelerated rate. The catalyzed reaction mechanism involves a lower energy transition state than the uncatalyzed reaction, resulting in a lower activation energy, E a , and a correspondingly greater rate constant.
To discern the effect of catalysis on an equilibrium system, consider the reaction diagram for a simple one-step (elementary) reaction shown in Figure \(\PageIndex{2}\). The lowered transition state energy of the catalyzed reaction results in lowered activation energies for both the forward and the reverse reactions. Consequently, both forward and reverse reactions are accelerated, and equilibrium is achieved more quickly but without a change in the equilibrium constant .
An interesting case study highlighting these equilibrium concepts is the industrial production of ammonia, NH 3 . This substance is among the “top 10” industrial chemicals with regard to production, with roughly two billion pounds produced annually in the US. Ammonia is used as a chemical feedstock to synthesize a wide range of commercially useful compounds, including fertilizers, plastics, dyes, and explosives.
Most industrial production of ammonia uses the Haber-Bosch process based on the following equilibrium reaction:
\[\ce{N2(g) + 3 H2(g) <=> 2 NH3(g)} \quad \Delta H=-92.2 ~\text{kJ} \nonumber \]
The traits of this reaction present challenges to its use in an efficient industrial process. The equilibrium constant is relatively small (Kp on the order of \(10^{−5}\) at 25 °C), meaning very little ammonia is present in an equilibrium mixture. Also, the rate of this reaction is relatively slow at low temperatures. To raise the yield of ammonia, the industrial process is designed to operate under conditions favoring product formation:
- High pressures (concentrations) of reactants are used, ~150−250 atm, to shift the equilibrium right, favoring product formation.
- Ammonia is continually removed (collected) from the equilibrium mixture during the process, lowering its concentration and also shifting the equilibrium right.
- Although low temperatures favor product formation for this exothermic process, the reaction rate at low temperatures is inefficiently slow. A catalyst is used to accelerate the reaction to reasonable rates at relatively moderate temperatures (400−500 °C).
A diagram illustrating a typical industrial setup for production of ammonia via the Haber-Bosch process is shown in Figure \(\PageIndex{3}\).