13.2: Arithmetic Sequences
- Find the common difference for an arithmetic sequence.
- Write terms of an arithmetic sequence.
- Use a recursive formula for an arithmetic sequence.
- Use an explicit formula for an arithmetic sequence.
Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.
As an example, consider a woman who starts a small contracting business. She purchases a new truck for \($25,000\). After five years, she estimates that she will be able to sell the truck for \($8,000\). The loss in value of the truck will therefore be \($17,000\), which is \($3,400\) per year for five years. The truck will be worth \($21,600\) after the first year; \($18,200\) after two years; \($14,800\) after three years; \($11,400\) after four years; and \($8,000\) at the end of five years. In this section, we will consider specific kinds of sequences that will allow us to calculate depreciation, such as the truck’s value.
Finding Common Differences
The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. Each term increases or decreases by the same constant value called the common difference of the sequence. For this sequence, the common difference is \(-3,400\).
The sequence below is another example of an arithmetic sequence. In this case, the constant difference is \(3\). You can choose any term of the sequence, and add \(3\) to find the subsequent term.
An arithmetic sequence is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the common difference . If \(a_1\) is the first term of an arithmetic sequence and \(d\) is the common difference, the sequence will be:
\[\{a_n\}=\{a_1,a_1+d,a_1+2d,a_1+3d,...\}\]
Is each sequence arithmetic? If so, find the common difference.
- \(\{1,2,4,8,16,...\}\)
- \(\{−3,1,5,9,13,...\}\)
Solution
Subtract each term from the subsequent term to determine whether a common difference exists.
- The sequence is not arithmetic because there is no common difference.
\(2-1={\color{red}1} \qquad 4-2={\color{red}2} \qquad 8-4={\color{red}4} \qquad 16-8={\color{red}8}\)
- The sequence is arithmetic because there is a common difference. The common difference is \(4\).
\(1-(-3)={\color{red}4} \qquad 5-1={\color{red}4} \qquad 9-5={\color{red}4} \qquad 13-9={\color{red}4}\)
Analysis
The graph of each of these sequences is shown in Figure \(\PageIndex{1}\). We can see from the graphs that, although both sequences show growth, (a) is not linear whereas (b) is linear. Arithmetic sequences have a constant rate of change so their graphs will always be points on a line.
Figure \(\PageIndex{1}\)
If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the common difference?
No. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract it from the subsequent term to find the common difference.
Is the given sequence arithmetic? If so, find the common difference.
\(\{18, 16, 14, 12, 10,…\}\)
- Answer
-
The sequence is arithmetic. The common difference is \(–2\).
Is the given sequence arithmetic? If so, find the common difference.
\(\{1, 3, 6, 10, 15,…\}\)
- Answer
-
The sequence is not arithmetic because \(3−1≠6−3\).
Writing Terms of Arithmetic Sequences
Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of \(n\) and \(d\) into formula below.
\[a_n=a_1+(n−1)d\]
- Add the common difference to the first term to find the second term.
- Add the common difference to the second term to find the third term.
- Continue until all of the desired terms are identified.
- Write the terms separated by commas within brackets.
Write the first five terms of the arithmetic sequence with \(a_1=17\) and \(d=−3\).
Solution
Adding \(−3\) is the same as subtracting \(3\). Beginning with the first term, subtract \(3\) from each term to find the next term.
The first five terms are \(\{17,14,11,8,5\}\)
Analysis
As expected, the graph of the sequence consists of points on a line as shown in Figure \(\PageIndex{2}\).
Figure \(\PageIndex{2}\)
List the first five terms of the arithmetic sequence with \(a_1=1\) and \(d=5\).
- Answer
-
\(\{1, 6, 11, 16, 21\}\)
- Substitute the values given for \(a_1\), \(a_n\), \(n\) into the formula \(a_n=a_1+(n−1)d\) to solve for \(d\).
- Find a given term by substituting the appropriate values for \(a_1\), \(n\), and \(d\) into the formula \(a_n=a_1+(n−1)d\).
Given \(a_1=8\) and \(a_4=14\), find \(a_5\).
Solution
The sequence can be written in terms of the initial term \(8\) and the common difference \(d\).
\(\{8,8+d,8+2d,8+3d\}\)
We know the fourth term equals \(14\); we know the fourth term has the form \(a_1+3d=8+3d\).
We can find the common difference \(d\).
\[\begin{align*} a_n&= a_1+(n-1)d \\ a_4&= a_1+3d \\ a_4&=8+3d\qquad \text{Write the fourth term of the sequence in terms of }a_1 \text{ and } d. \\ 14&=8+3d\qquad \text{Substitute }14 \text{ for } a_4. \\ d&=2\qquad \text{Solve for the common difference.} \end{align*}\]
Find the fifth term by adding the common difference to the fourth term.
\(a_5=a_4+2=16\)
Analysis
Notice that the common difference is added to the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by adding the common difference to the first term nine times or by using the equation \(a_n=a_1+(n−1)d\).
Given \(a_3=7\) and \(a_5=17\), find \(a_2\).
- Answer
-
\(a_2=2\)
Using Recursive Formulas for Arithmetic Sequences
Some arithmetic sequences are defined in terms of the previous term using a recursive formula . The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is \(5\), then each term is the previous term plus \(5\). As with any recursive formula, the first term must be given.
\(a_n=a_n−1+d\)
for \(n≥2\)
The recursive formula for an arithmetic sequence with common difference \(d\) is:
\[a_n=a_n−1+d\]
for \(n≥2\)
- Subtract any term from the subsequent term to find the common difference.
- State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.
Write a recursive formula for the arithmetic sequence .
\(\{−18, −7, 4, 15, 26, …\}\)
Solution
The first term is given as \(−18\). The common difference can be found by subtracting the first term from the second term.
\(d=−7−(−18)=11\)
Substitute the initial term and the common difference into the recursive formula for arithmetic sequences.
\(a_1=−18\)
\(a_n=a_{n−1}+11\)
for \(n≥2\)
Analysis
We see that the common difference is the slope of the line formed when we graph the terms of the sequence, as shown in Figure \(\PageIndex{3}\). The growth pattern of the sequence shows the constant difference of 11 units.
Figure \(\PageIndex{3}\)
Do we have to subtract the first term from the second term to find the common difference?
No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common difference.
Write a recursive formula for the arithmetic sequence.
\(\{25, 37, 49, 61, …\}\)
- Answer
-
\(\begin{align*}a_1 &= 25 \\ a_n &= a_{n−1}+12 , \text{ for }n≥2 \end{align*}\)
Using Explicit Formulas for Arithmetic Sequences
We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.
\(a_n=a_1+d(n−1)\)
To find the \(y\)-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.
The common difference is \(−50\), so the sequence represents a linear function with a slope of \(−50\). To find the \(y\)-intercept, we subtract \(−50\) from \(200\): \(200−(−50)=200+50=250\). You can also find the \(y\)-intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in Figure \(\PageIndex{4}\).
Figure \(\PageIndex{4}\)
Recall the slope-intercept form of a line is \(y=mx+b\). When dealing with sequences, we use \(a_n\) in place of \(y\) and \(n\) in place of \(x\). If we know the slope and vertical intercept of the function, we can substitute them for \(m\) and \(b\) in the slope-intercept form of a line. Substituting \(−50\) for the slope and \(250\) for the vertical intercept, we get the following equation:
\(a_n=−50n+250\)
We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is \(a_n=200−50(n−1)\), which simplifies to \(a_n=−50n+250\).
An explicit formula for the \(n^{th}\) term of an arithmetic sequence is given by
\[a_n=a_1+d(n−1)\]
- Find the common difference, \(a_2−a_1\).
- Substitute the common difference and the first term into \(a_n=a_1+d(n−1)\).
Write an explicit formula for the arithmetic sequence.
\(\{2, 12, 22, 32, 42, …\}\)
Solution
The common difference can be found by subtracting the first term from the second term.
\[\begin{align*} d &= a_2−a_1 \\ &= 12−2 \\ &= 10 \end{align*}\]
The common difference is \(10\). Substitute the common difference and the first term of the sequence into the formula and simplify.
\[\begin{align*}a_n &= 2+10(n−1) \\ a_n &= 10n−8 \end{align*}\]
Analysis
The graph of this sequence, represented in Figure \(\PageIndex{5}\), shows a slope of \(10\) and a vertical intercept of \(−8\).
Figure \(\PageIndex{5}\)
Write an explicit formula for the following arithmetic sequence.
\(\{50,47,44,41,…\}\)
- Answer
-
\(a_n=53−3n\)
Finding the Number of Terms in a Finite Arithmetic Sequence
Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.
- Find the common difference \(d\).
- Substitute the common difference and the first term into \(a_n=a_1+d(n–1)\).
- Substitute the last term for \(a_n\) and solve for \(n\).
Find the number of terms in the finite arithmetic sequence.
\(\{8, 1, –6, ..., –41\}\)
Solution
The common difference can be found by subtracting the first term from the second term.
\(1−8=−7\)
The common difference is \(−7\). Substitute the common difference and the initial term of the sequence into the nth term formula and simplify.
\[\begin{align*} a_n &= a_1+d(n−1) \\ a_n &= 8+−7(n−1) \\ a_n &= 15−7n \end{align*}\]
Substitute \(−41\) for \(a_n\) and solve for \(n\)
\[\begin{align*} -41&=15-7n\\ 8&=n \end{align*}\]
There are eight terms in the sequence.
Find the number of terms in the finite arithmetic sequence.
\(\{6, 11, 16, ..., 56\}\)
- Answer
-
There are \(11\) terms in the sequence.
Solving Application Problems with Arithmetic Sequences
In many application problems, it often makes sense to use an initial term of \(a_0\) instead of \(a_1\). In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:
\[a_n=a_0+dn\]
A five-year old child receives an allowance of \($1\) each week. His parents promise him an annual increase of \($2\) per week.
- Write a formula for the child’s weekly allowance in a given year.
- What will the child’s allowance be when he is \(16\) years old?
Solution
-
The situation can be modeled by an arithmetic sequence with an initial term of \(1\) and a common difference of \(2\).
Let \(A\) be the amount of the allowance and \(n\) be the number of years after age \(5\). Using the altered explicit formula for an arithmetic sequence we get:
\(A_n=1+2n\)
-
We can find the number of years since age \(5\) by subtracting.
\(16−5=11\)
We are looking for the child’s allowance after \(11\) years. Substitute \(11\) into the formula to find the child’s allowance at age \(16\).
\(A_{11}=1+2(11)=23\)
The child’s allowance at age \(16\) will be \($23\) per week.
A woman decides to go for a \(10\)-minute run every day this week and plans to increase the time of her daily run by \(4\) minutes each week. Write a formula for the time of her run after \(n\) weeks. How long will her daily run be \(8\) weeks from today?
- Answer
-
The formula is \(T_n=10+4n\), and it will take her \(42\) minutes.
Access this online resource for additional instruction and practice with arithmetic sequences.
Key Equations
| recursive formula for nth term of an arithmetic sequence | \(a_n=a_{n−1}+d\) \(n≥2\) |
| explicit formula for nth term of an arithmetic sequence | \(a_n=a_1+d(n−1)\) |
Key Concepts
- An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
- The constant between two consecutive terms is called the common difference.
- The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See Example \(\PageIndex{1}\).
- The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See Example \(\PageIndex{2}\) and Example \(\PageIndex{3}\).
- A recursive formula for an arithmetic sequence with common difference dd is given by \(a_n=a_{n−1}+d\), \(n≥2\). See Example \(\PageIndex{4}\).
- As with any recursive formula, the initial term of the sequence must be given.
- An explicit formula for an arithmetic sequence with common difference \(d\) is given by \(a_n=a_1+d(n−1)\). See Example \(\PageIndex{5}\).
- An explicit formula can be used to find the number of terms in a sequence. See Example \(\PageIndex{6}\).
- In application problems, we sometimes alter the explicit formula slightly to \(a_n=a_0+dn\). See Example \(\PageIndex{7}\).
Contributors and Attributions
-
Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .


Basic Definitions
Skills to Develop
- Understand the fundamental concepts of thermodynamics
Thermodynamics
Thermodynamics is the study of heat, energy, and work and how they move. This is important because we have to move or generate heat to stay comfortable in winter and summer; we need to do work by moving things for many different purposes; we need to generate chemical energy to live and grow our bodies, etc.
System and Surroundings
In thermodynamics, we often separate the universe (that is, everything that exists) into 2 parts: the system , which is the small part we are interested in, and the surroundings which is everything outside the system. This will help us think about how heat, energy and work move between parts of the universe.
Open, Closed, and Isolated Systems
Open systems allow energy and matter (stuff) to enter and leave the system. A pan on the stove is an open system because water can evaporate or be poured in, and heat can enter the pan if the stove is turned on, and leave the pan also. A closed system does not allow matter to enter or leave, but does allow energy to enter or leave. A covered pot on the stove is approximately a closed system. An isolated system does not allow either matter or energy to enter or leave. A thermos or cooler is approximately an isolated system. There are no truly isolated systems.
State Functions
State functions are quantities that don't depend on path. Your bank balance is a good example. It doesn't matter how the money entered your bank account, the total amount there at any given time is what it is and you can measure it easily. It doesn't matter if you put it in all at once, or a little bit every month, or put a lot in then spent it slowly... any time you want to know how much is there, you just check. Most of the quantities you know are state functions, like pressure, volume, temperature, location, etc. But some quantities that are important in thermodynamics, like heat and work, are only defined by a process, so they aren't state functions.
Extensive vs. Intensive
Extensive refers to properties that depend on how much stuff there is. For instance, the volume or pressure created by a sample of gas depend on how much gas is in the sample, so they are extensive. Intensive quantities don't depend on how much there is. For instance, temperature, density, etc. If you divide a sample in 2, it does not change temperature. Density is a ratio of 2 extensive properties, so it is intensive.


7.3: Double Angle Identities
Two special cases of the sum of angles identities arise often enough that we choose to state these identities separately.
IDENTITIES
The double angle identities
\[\sin (2\alpha )=2\sin (\alpha )\cos (\alpha )\]
\[\begin{array}{ccc} {\cos (2\alpha )} & {=} & {\cos ^{2} (\alpha )-\sin ^{2} (\alpha )} \\ {} & {=} & {1-2\sin ^{2} (\alpha )} \\ {} & {=} & {2\cos ^{2} (\alpha )-1} \end{array}\]
These identities follow from the sum of angles identities.
Proof of the sine double angle identity
\[\sin (2\alpha )\nonumber\]
\[=\sin (\alpha +\alpha )\nonumber\]Apply the sum of angles identity
\[=\sin (\alpha )\cos (\alpha )+\cos (\alpha )\sin (\alpha )\nonumber\]Simplify
\[=2\sin (\alpha )\cos (\alpha )\nonumber\]Establishing the identity
Exercise \(\PageIndex{1}\)
Show \(\cos (2\alpha )=\cos ^{2} (\alpha )-\sin ^{2} (\alpha )\) by using the sum of angles identity for cosine.
- Answer
-
\[\begin{array}{l} {\cos \left(2\alpha \right)=\cos (\alpha +\alpha )} \\ {\cos (\alpha )\cos (\alpha )-\sin (\alpha )\sin (\alpha )} \\ {\cos ^{2} (\alpha )-\sin ^{2} (\alpha )} \end{array}\nonumber\]
For the cosine double angle identity, there are three forms of the identity stated because the basic form, \(\cos (2\alpha )=\cos ^{2} (\alpha )-\sin ^{2} (\alpha )\), can be rewritten using the Pythagorean Identity. Rearranging the Pythagorean Identity results in the equality \(\cos ^{2} (\alpha )=1-\sin ^{2} (\alpha )\), and by substituting this into the basic double angle identity, we obtain the second form of the double angle identity.
\[\cos (2\alpha )=\cos ^{2} (\alpha )-\sin ^{2} (\alpha )\nonumber\]Substituting using the Pythagorean identity
\[\cos (2\alpha )=1-\sin ^{2} (\alpha )-\sin ^{2} (\alpha )\nonumber\]Simplifying
\[\cos (2\alpha )=1-2\sin ^{2} (\alpha )\nonumber\]
Example \(\PageIndex{1}\)
If \(\sin (\theta )=\dfrac{3}{5}\) and \(\theta\) is in the second quadrant, find exact values for \(\sin (2\theta )\) and \(\cos (2\theta )\).
Solution
To evaluate \(\cos (2\theta )\), since we know the value for \(\sin (\theta )\) we can use the version of the double angle that only involves sine.
\[\cos (2\theta )=1-2\sin ^{2} (\theta )=1-2\left(\dfrac{3}{5} \right)^{2} =1-\dfrac{18}{25} =\dfrac{7}{25}\nonumber\]
Since the double angle for sine involves both sine and cosine, we’ll need to first find \(\cos (\theta )\), which we can do using the Pythagorean Identity.
\[\sin ^{2} (\theta )+\cos ^{2} (\theta )=1\nonumber\]
\[\left(\dfrac{3}{5} \right)^{2} +\cos ^{2} (\theta )=1\nonumber\]
\[\cos ^{2} (\theta )=1-\dfrac{9}{25}\nonumber\]
\[\cos (\theta )=\pm \sqrt{\dfrac{16}{25} } =\pm \dfrac{4}{5}\nonumber\]
Since \(\theta\) is in the second quadrant, we know that cos( \(\theta\) ) \(\mathrm{<}\) 0, so
\[\cos (\theta )=-\dfrac{4}{5}\nonumber\]
Now we can evaluate the sine double angle
\[\sin (2\theta )=2\sin (\theta )\cos (\theta )=2\left(\dfrac{3}{5} \right)\left(-\dfrac{4}{5} \right)=-\dfrac{24}{25}\nonumber\]
Example \(\PageIndex{2}\)
Simplify the expressions
a) \(2\cos ^{2} \left(12{}^\circ \right)-1\)
b) \(8\sin \left(3x\right)\cos \left(3x\right)\)
Solution
a) Notice that the expression is in the same form as one version of the double angle identity for cosine: \(\cos (2\theta )=2\cos ^{2} (\theta )-1\). Using this,
\[2\cos ^{2} \left(12{}^\circ \right)-1=\cos \left(2\cdot 12{}^\circ \right)=\cos \left(24{}^\circ \right)\nonumber\]
b) This expression looks similar to the result of the double angle identity for sine.
\[8\sin \left(3x\right)\cos \left(3x\right)\nonumber\]Factoring a 4 out of the original expression
\[4\cdot 2\sin \left(3x\right)\cos \left(3x\right)\nonumber\]Applying the double angle identity
\[4\sin (6x)\nonumber\]
We can use the double angle identities to simplify expressions and prove identities.
Example \(\PageIndex{3}\)
Simplify \(\dfrac{\cos (2t)}{\cos (t)-\sin (t)}\).
Solution
With three choices for how to rewrite the double angle, we need to consider which will be the most useful. To simplify this expression, it would be great if the denominator would cancel with something in the numerator, which would require a factor of \(\cos (t)-\sin (t)\) in the numerator, which is most likely to occur if we rewrite the numerator with a mix of sine and cosine.
\[\dfrac{\cos (2t)}{\cos (t)-\sin (t)}\nonumber\]Apply the double angle identity
\[=\dfrac{\cos ^{2} (t)-\sin ^{2} (t)}{\cos (t)-\sin (t)}\nonumber\]Factor the numerator
\[=\dfrac{\left(\cos (t)-\sin (t)\right)\left(\cos (t)+\sin (t)\right)}{\cos (t)-\sin (t)}\nonumber\]Cancelling the common factor
\[=\cos (t)+\sin (t)\nonumber\]Resulting in the most simplified form
Example \(\PageIndex{4}\)
Prove \(\sec (2\alpha )=\dfrac{\sec ^{2} (\alpha )}{2-\sec ^{2} (\alpha )}\).
Solution
Since the right side seems a bit more complicated than the left side, we begin there.
\[\dfrac{\sec ^{2} (\alpha )}{2-\sec ^{2} (\alpha )}\nonumber\]Rewrite the secants in terms of cosine
\[=\dfrac{\dfrac{1}{\cos ^{2} (\alpha )} }{2-\dfrac{1}{\cos ^{2} (\alpha )} }\nonumber\]
At this point, we could rewrite the bottom with common denominators, subtract the terms, invert and multiply, then simplify. Alternatively, we can multiple both the top and bottom by \(\cos ^{2} (\alpha )\), the common denominator:
\[=\dfrac{\dfrac{1}{\cos ^{2} (\alpha )} \cdot \cos ^{2} (\alpha )}{\left(2-\dfrac{1}{\cos ^{2} (\alpha )} \right)\cdot \cos ^{2} (\alpha )}\nonumber\]Distribute on the bottom
\[=\dfrac{\dfrac{\cos ^{2} (\alpha )}{\cos ^{2} (\alpha )} }{2\cos ^{2} (\alpha )-\dfrac{\cos ^{2} (\alpha )}{\cos ^{2} (\alpha )} \cdot }\nonumber\]Simplify
\[=\dfrac{1}{2\cos ^{2} (\alpha )-1}\nonumber\]Rewrite the denominator as a double angle
\[=\dfrac{1}{\cos (2\alpha )}\nonumber\]Rewrite as a secant
\[=\sec (2\alpha )\nonumber\]Establishing the identity
Exercise \(\PageIndex{2}\)
Use an identity to find the exact value of \(\cos ^{2} \left(75{}^\circ \right)-\sin ^{2} \left(75{}^\circ \right)\).
- Answer
-
\[\cos ^{2} \left(75{}^\circ \right)-\sin ^{2} \left(75{}^\circ \right)=\cos (2\cdot 75{}^\circ )\nonumber\] \[ = \cos (150{}^\circ )=\dfrac{-\sqrt{3} }{2}\nonumber\]
As with other identities, we can also use the double angle identities for solving equations.
Example \(\PageIndex{5}\)
Solve \(\cos (2t)=\cos (t)\) for all solutions with \(0\le t<2\pi\).
Solution
In general when solving trig equations, it makes things more complicated when we have a mix of sines and cosines and when we have a mix of functions with different periods. In this case, we can use a double angle identity to rewrite the cos(2 t ). When choosing which form of the double angle identity to use, we notice that we have a cosine on the right side of the equation. We try to limit our equation to one trig function, which we can do by choosing the version of the double angle formula for cosine that only involves cosine.
\[\cos (2t)=\cos (t)\nonumber\]Apply the double angle identity
\[2\cos ^{2} (t)-1=\cos (t)\nonumber\]This is quadratic in cosine, so make one side 0
\[2\cos ^{2} (t)-\cos (t)-1=0\nonumber\]Factor
\[\left(2\cos (t)+1\right)\left(\cos (t)-1\right)=0\nonumber\]Break this apart to solve each part separately
\[2\cos (t)+1=0\text{ or }\cos (t)-1=0\nonumber\]
\[\cos (t)=-\dfrac{1}{2}\text{ or }\cos (t)=1\nonumber\]
\[t=\dfrac{2\pi }{3}\text{ or }t=\dfrac{4\pi }{3}\text{ or }t=0\nonumber\]
Looking at a graph of cos(2 t ) and cos( t ) shown together, we can verify that these three solutions on [0, 2 \(\pi\)) seem reasonable.
Example \(\PageIndex{6}\)
A cannonball is fired with velocity of 100 meters per second. If it is launched at an angle of \(\theta\) , the vertical component of the velocity will be \(100\sin (\theta )\) and the horizontal component will be \(100\cos (\theta )\). Ignoring wind resistance, the height of the cannonball will follow the equation \(h(t)=-4.9t^{2} +100\sin (\theta )t\) and horizontal position will follow the equation \(x(t)=100\cos (\theta )t\). If you want to hit a target 900 meters away, at what angle should you aim the cannon?
Solution
To hit the target 900 meters away, we want \(x(t)=900\)at the time when the cannonball hits the ground, when \(h(t)=0\). To solve this problem, we will first solve for the time, \(t\), when the cannonball hits the ground. Our answer will depend upon the angle \(\theta\).
\[h(t)=0\nonumber\]
\[-4.9t^{2} +100\sin (\theta )t=0\nonumber\]Factor
\[t\left(-4.9t+100\sin (\theta )\right)=0\nonumber\]Break this apart to find two solutions
\[t=0\text{ or }-4.9t+100\sin (\theta )=0\nonumber\] Solve for \(t\)
\[-4.9t=-100\sin (\theta )\nonumber\]
\[t=\dfrac{100\sin (\theta )}{4.9}\nonumber\]
This shows that the height is 0 twice, once at \(t = 0\) when the cannonball is fired, and again when the cannonball hits the ground after flying through the air. This second value of \(t\) gives the time when the ball hits the ground in terms of the angle \(\theta\). We want the horizontal distance \(x(t)\) to be 900 when the ball hits the ground, in other words when \(t=\dfrac{100\sin (\theta )}{4.9}\).
Since the target is 900 m away we start with
\[x(t)=900\nonumber\]Use the formula for \(x(t)\)
\[100\cos (\theta )t=900\nonumber\]Substitute the desired time, \(t\) from above
\[100\cos (\theta )\dfrac{100\sin (\theta )}{4.9} =900\nonumber\]Simplify
\[\dfrac{100^{2} }{4.9} \cos (\theta )\sin (\theta )=900\nonumber\]Isolate the cosine and sine product
\[\cos (\theta )\sin (\theta )=\dfrac{900(4.9)}{100^{2} }\nonumber\]
The left side of this equation almost looks like the result of the double angle identity for sine: \[\sin (2\theta )=2\sin \left(\theta \right)\cos \left(\theta \right)\nonumber\]
Multiplying both sides of our equation by 2,
\[2\cos (\theta )\sin (\theta )=\dfrac{2(900)(4.9)}{100^{2} }\nonumber\]Using the double angle identity on the left
\[\sin (2\theta )=\dfrac{2(900)(4.9)}{100^{2} }\nonumber\]Use the inverse sine
\[2\theta =\sin ^{-1} \left(\dfrac{2(900)(4.9)}{100^{2} } \right)\approx 1.080\nonumber\]Divide by 2
\[\theta =\dfrac{1.080}{2} =0.540\nonumber\], or about 30.94 degrees
Power Reduction and Half Angle Identities
Another use of the cosine double angle identities is to use them in reverse to rewrite a squared sine or cosine in terms of the double angle. Starting with one form of the cosine double angle identity:
\[\cos (2\alpha )=2\cos ^{2} (\alpha )-1\nonumber\]Isolate the cosine squared term
\[\cos (2\alpha )+1=2\cos ^{2} (\alpha )\nonumber\] Add 1
\[\cos ^{2} (\alpha )=\dfrac{\cos (2\alpha )+1}{2}\nonumber\]Divide by 2
\[\cos ^{2} (\alpha )=\dfrac{\cos (2\alpha )+1}{2}\nonumber\] This is called a
power reduction identity
Exercise \(\PageIndex{3}\)
Use another form of the cosine double angle identity to prove the identity \(\sin ^{2} (\alpha )=\dfrac{1-\cos (2\alpha )}{2}\).
- Answer
-
\[\begin{array}{l} {\dfrac{1-\cos (2\alpha )}{2} } \\ {\dfrac{1-\left(\cos ^{2} (\alpha )-\sin ^{2} (\alpha )\right)}{2} } \\ {\dfrac{1-\cos ^{2} (\alpha )+\sin ^{2} (\alpha )}{2} } \\ {\dfrac{\sin ^{2} (\alpha )+\sin ^{2} (\alpha )}{2} } \\ {\dfrac{2\sin ^{2} (\alpha )}{2} =\sin ^{2} (\alpha )} \end{array}\nonumber\]
The cosine double angle identities can also be used in reverse for evaluating angles that are half of a common angle. Building from our formula \(\cos ^{2} (\alpha )=\dfrac{\cos (2\alpha )+1}{2}\), if we let \(\theta =2\alpha\), then \(\alpha =\dfrac{\theta }{2}\) this identity becomes \(\cos ^{2} \left(\dfrac{\theta }{2} \right)=\dfrac{\cos (\theta )+1}{2}\). Taking the square root, we obtain
\[\cos \left(\dfrac{\theta }{2} \right)=\pm \sqrt{\dfrac{\cos (\theta )+1}{2} }\nonumber\]where the sign is determined by the quadrant.
This is called a half-angle identity .
Exercise \(\PageIndex{4}\)
Use your results from the last Try it Now to prove the identity \(\sin \left(\dfrac{\theta }{2} \right)=\pm \sqrt{\dfrac{1-\cos (\theta )}{2} }\).
- Answer
-
\[\begin{array}{l} {\sin ^{2} (\alpha )=\dfrac{1-\cos (2\alpha )}{2} } \\ {\sin (\alpha )=\pm \sqrt{\dfrac{1-\cos (2\alpha )}{2} } } \\ {\alpha =\dfrac{\theta }{2} } \\ {\sin \left(\dfrac{\theta }{2} \right)=\pm \sqrt{\dfrac{1-\cos \left(2\left(\dfrac{\theta }{2} \right)\right)}{2} } } \\ {\sin \left(\dfrac{\theta }{2} \right)=\pm \sqrt{\dfrac{1-\cos (\theta )}{2} } } \end{array}\nonumber\]
IDENTITIES
Half-Angle Identities
\[\cos \left(\dfrac{\theta }{2} \right)=\pm \sqrt{\dfrac{\cos (\theta )+1}{2} }\]
\[\sin \left(\dfrac{\theta }{2} \right)=\pm \sqrt{\dfrac{1-\cos (\theta )}{2} }\]
Power Reduction Identities
\[\cos ^{2} (\alpha )=\dfrac{\cos (2\alpha )+1}{2}\]
\[\sin ^{2} (\alpha )=\dfrac{1-\cos (2\alpha )}{2}\]
Since these identities are easy to derive from the double-angle identities, the power reduction and half-angle identities are not ones you should need to memorize separately.
Example \(\PageIndex{7}\)
Rewrite \(\cos ^{4} (x)\) without any powers.
Solution
\[\cos ^{4} (x)=\left(\cos ^{2} (x)\right)^{2}\nonumber\]Using the power reduction formula
\[=\left(\dfrac{\cos (2x)+1}{2} \right)^{2}\nonumber\]Square the numerator and denominator
\[=\dfrac{\left(\cos (2x)+1\right)^{2} }{4}\nonumber\]Expand the numerator
\[=\dfrac{\cos ^{2} (2x)+2\cos (2x)+1}{4}\nonumber\]Split apart the fraction
\[=\dfrac{\cos ^{2} (2x)}{4} +\dfrac{2\cos (2x)}{4} +\dfrac{1}{4}\nonumber\]Apply the formula above to \(\cos ^{2} (2x)\)
\[\cos ^{2} (2x)=\dfrac{\cos (2\cdot 2x)+1}{2}\nonumber\]
\[=\dfrac{\left(\dfrac{\cos (4x)+1}{2} \right)}{4} +\dfrac{2\cos (2x)}{4} +\dfrac{1}{4}\nonumber\]Simplify
\[=\dfrac{\cos (4x)}{8} +\dfrac{1}{8} +\dfrac{1}{2} \cos (2x)+\dfrac{1}{4}\nonumber\]Combine the constants
\[=\dfrac{\cos (4x)}{8} +\dfrac{1}{2} \cos (2x)+\dfrac{3}{8}\nonumber\]
Example \(\PageIndex{8}\)
Find an exact value for \(\cos \left(15{}^\circ \right)\).
Solution
Since 15 degrees is half of 30 degrees, we can use our result from above:
\[\cos (15{}^\circ )=\cos \left(\dfrac{30{}^\circ }{2} \right)=\pm \sqrt{\dfrac{\cos (30{}^\circ )+1}{2} }\nonumber\]
We can evaluate the cosine. Since 15 degrees is in the first quadrant, we need the positive result.
\[\sqrt{\dfrac{\cos (30{}^\circ )+1}{2} } =\sqrt{\dfrac{\dfrac{\sqrt{3} }{2} +1}{2} }\nonumber\]
\[=\sqrt{\dfrac{\sqrt{3} }{4} +\dfrac{1}{2} }\nonumber\]
Important Topics of This Section
- Double angle identity
- Power reduction identity
- Half angle identity
- Using identities
- Simplify equations
- Prove identities
- Solve equations


13.3: Brønsted-Lowry Definition of Acids and Bases
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Learning Objectives
- Recognize a compound as a Brønsted-Lowry acid or a Brønsted-Lowry base.
- Illustrate the proton transfer process that defines a Brønsted-Lowry acid-base reaction.
Ammonia (NH 3 ) increases the hydroxide ion concentration in aqueous solution by reacting with water rather than releasing hydroxide ions directly. In fact, the Arrhenius definitions of an acid and a base focus on hydrogen ions and hydroxide ions. Are there more fundamental definitions for acids and bases?
In 1923, the Danish scientist Johannes Brønsted and the English scientist Thomas Lowry independently proposed new definitions for acids and bases. Rather than considering both hydrogen and hydroxide ions, they focused on the hydrogen ion only. A Brønsted-Lowry acid is a compound that supplies a hydrogen ion in a reaction. A Brønsted-Lowry base, conversely, is a compound that accepts a hydrogen ion in a reaction. Thus, the Brønsted-Lowry definitions of an acid and a base focus on the movement of hydrogen ions in a reaction, rather than on the production of hydrogen ions and hydroxide ions in an aqueous solution.
Let us use the reaction of ammonia in water to demonstrate the Brønsted-Lowry definitions of an acid and a base. Ammonia and water molecules are reactants, while the ammonium ion and the hydroxide ion are products:
\[NH_{3}(aq) + H_2O(ℓ) \rightarrow NH^+_{4}(aq) + OH^−(aq) \label{Eq1} \]
What has happened in this reaction is that the original water molecule has donated a hydrogen ion to the original ammonia molecule, which in turn has accepted the hydrogen ion. We can illustrate this as follows:
Because the water molecule donates a hydrogen ion to the ammonia, it is the Brønsted-Lowry acid, while the ammonia molecule—which accepts the hydrogen ion—is the Brønsted-Lowry base. Thus, ammonia acts as a base in both the Arrhenius sense and the Brønsted-Lowry sense.
Is an Arrhenius acid like hydrochloric acid still an acid in the Brønsted-Lowry sense? Yes, but it requires us to understand what really happens when HCl is dissolved in water. Recall that the hydrogen atom is a single proton surrounded by a single electron. To make the hydrogen ion , we remove the electron, leaving a bare proton. Do we really have bare protons floating around in aqueous solution? No, we do not. What really happens is that the H + ion attaches itself to H 2 O to make H 3 O + , which is called the hydronium ion . For most purposes, H + and H 3 O + represent the same species, but writing H 3 O + instead of H + shows that we understand that there are no bare protons floating around in solution. Rather, these protons are actually attached to solvent molecules.
A proton in aqueous solution may be surrounded by more than one water molecule, leading to formulas like H 5 O 2 + or H 9 O 4 + rather than H 3 O + . It is simpler, however, to use H 3 O + .
With this in mind, how do we define HCl as an acid in the Brønsted-Lowry sense? Consider what happens when HCl is dissolved in H 2 O:
\[HCl(g) + H_2O(ℓ) \rightarrow H_3O^+(aq) + Cl^−(aq) \label{Eq2} \]
We can depict this process using Lewis electron dot diagrams:
Now we see that a hydrogen ion is transferred from the HCl molecule to the H 2 O molecule to make chloride ions and hydronium ions. As the hydrogen ion donor, HCl acts as a Brønsted-Lowry acid ; as a hydrogen ion acceptor, H 2 O is a Brønsted-Lowry base . So HCl is an acid not just in the Arrhenius sense but also in the Brønsted-Lowry sense. Moreover, by the Brønsted-Lowry definitions, H 2 O is a base in the formation of aqueous HCl. So the Brønsted-Lowry definitions of an acid and a base classify the dissolving of HCl in water as a reaction between an acid and a base—although the Arrhenius definition would not have labeled H 2 O a base in this circumstance.
All Arrhenius acids and bases are Brønsted-Lowry acids and bases as well. However, not all Brønsted-Lowry acids and bases are Arrhenius acids and bases.
Example \(\PageIndex{1}\)
Aniline (C 6 H 5 NH 2 ) is slightly soluble in water. It has a nitrogen atom that can accept a hydrogen ion from a water molecule just like the nitrogen atom in ammonia does. Write the chemical equation for this reaction and identify the Brønsted-Lowry acid and base.
Solution
C 6 H 5 NH 2 and H 2 O are the reactants. When C 6 H 5 NH 2 accepts a proton from H 2 O, it gains an extra H and a positive charge and leaves an OH − ion behind. The reaction is as follows:
C 6 H 5 NH 2 (aq) + H 2 O(ℓ) → C 6 H 5 NH 3 + (aq) + OH − (aq)
Because C 6 H 5 NH 2 accepts a proton, it is the Brønsted-Lowry base. The H 2 O molecule, because it donates a proton, is the Brønsted-Lowry acid.
Exercise \(\PageIndex{1}\)
Caffeine (C 8 H 10 N 4 O 2 ) is a stimulant found in coffees and teas. When dissolved in water, it can accept a proton from a water molecule. Write the chemical equation for this process and identify the Brønsted-Lowry acid and base.
- Answer
-
C 8 H 10 N 4 O 2 (aq) + H 2 O(ℓ) → C 8 H 11 N 4 O 2 + (aq) + OH − (aq)
B-L base B-L acid
The Brønsted-Lowry definitions of an acid and a base can be applied to chemical reactions that occur in solvents other than water. The following example illustrates.
Example \(\PageIndex{2}\)
Sodium amide (NaNH 2 ) dissolves in methanol (CH 3 OH) and separates into sodium ions and amide ions (NH 2 − ). The amide ions react with methanol to make ammonia and the methoxide ion (CH 3 O − ). Write a balanced chemical equation for this process and identify the Brønsted-Lowry acid and base.
Solution
The equation for the reaction is between NH 2 − and CH 3 OH to make NH 3 and CH 3 O − is as follows:
NH 2 − (solv) + CH 3 OH(ℓ) → NH 3 (solv) + CH 3 O − (solv)
The label (solv) indicates that the species are dissolved in some solvent, in contrast to (aq) , which specifies an aqueous (H 2 O) solution. In this reaction, we see that the NH 2 − ion accepts a proton from a CH 3 OH molecule to make an NH 3 molecule. Thus, as the proton acceptor, NH 2 − is the Brønsted-Lowry base. As the proton donor, CH 3 OH is the Brønsted-Lowry acid.
Exercise \(\PageIndex{2}\)
Pyridinium chloride (C 5 H 5 NHCl) dissolves in ethanol (C 2 H 5 OH) and separates into pyridinium ions (C 5 H 5 NH + ) and chloride ions. The pyridinium ion can transfer a hydrogen ion to a solvent molecule. Write a balanced chemical equation for this process and identify the Brønsted-Lowry acid and base.
- Answer
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C 5 H 5 NH + (solv) + C 2 H 5 OH (ℓ) → C 5 H 5 N (solv) + C 2 H 5 OH 2 + (solv)
B-L acid B-L base
Application in Everyday Life
Many people like to put lemon juice or vinegar, both of which are acids, on cooked fish (Figure \(\PageIndex{1}\)). It turns out that fish have volatile amines (bases) in their systems, which are neutralized by the acids to yield involatile ammonium salts. This reduces the odor of the fish, and also adds a “sour” taste that we seem to enjoy.
Pickling is a method used to preserve vegetables using a naturally produced acidic environment. The vegetable, such as a cucumber, is placed in a sealed jar submerged in a brine solution. The brine solution favors the growth of beneficial bacteria and suppresses the growth of harmful bacteria. The beneficial bacteria feed on starches in the cucumber and produce lactic acid as a waste product in a process called fermentation. The lactic acid eventually increases the acidity of the brine to a level that kills any harmful bacteria, which require a basic environment. Without the harmful bacteria consuming the cucumbers they are able to last much longer than if they were unprotected. A byproduct of the pickling process changes the flavor of the vegetables with the acid making them taste sour.
To Your Health: Brønsted-Lowry Acid-Base Reactions in Pharmaceuticals
There are many interesting applications of Brønsted-Lowry acid-base reactions in the pharmaceutical industry. For example, drugs often need to be water soluble for maximum effectiveness. However, many complex organic compounds are not soluble or are only slightly soluble in water. Fortunately, those drugs that contain proton-accepting nitrogen atoms (and there are a lot of them) can be reacted with dilute hydrochloric acid [HCl(aq)]. The nitrogen atoms—acting as Brønsted-Lowry bases—accept the hydrogen ions from the acid to make an ion, which is usually much more soluble in water. The modified drug molecules can then be isolated as chloride salts:
\[\mathrm{RN(sl\: aq) + H^+(aq) \rightarrow RN H^+(aq) \xrightarrow{Cl^-(aq)} RNHCl(s)} \label{Eq3} \]
where RN represents some organic compound containing nitrogen. The label (sl aq) means “slightly aqueous,” indicating that the compound RN is only slightly soluble. Drugs that are modified in this way are called hydrochloride salts . Examples include the powerful painkiller codeine, which is commonly administered as codeine hydrochloride. Acids other than hydrochloric acid are also used. Hydrobromic acid, for example, gives hydrobromide salts . Dextromethorphan, an ingredient in many cough medicines, is dispensed as dextromethorphan hydrobromide. The accompanying figure shows another medication (lidocaine) as a hydrochloride salt.
Conjugate Acid-Base Pairs
According to the Bronsted-Lowry theory of acids and bases, an acid is a proton donor and a base is a proton acceptor. Once an acid has given up a proton, the part that remains is called the acid's conjugate base . This species is a base because it can accept a proton (to re-form the acid). The conjugate base of HF (first example below) is fluoride ion, F - .
\(\mathrm{{\color{Red} Acid} = H^+ + {\color{Blue} Conjugate\: base\: of\: Acid}^-}\)
\({\color{Red} \mathrm{HF}} \rightleftharpoons \mathrm{H^+} + {\color{Blue} \mathrm{F^-}}\)
\({\color{Red} \mathrm{H_2O}} \rightleftharpoons \mathrm{H^+} + {\color{Blue} \mathrm{OH^-}}\)
\({\color{Red} \mathrm{NH_4^+}} \rightleftharpoons \mathrm{H^+} + {\color{Blue} \mathrm{NH_3}}\)
Similarly, the part of the base that remains after a base accepts a proton is called the base's conjugate acid . This species is an acid because it can give up a proton (and thus re-form the base). The conjugate acid of fluoride ion, F - (first example below) is HF.
\(\mathrm{H^+ + {\color{Blue} Base} = {\color{Red} Conjugate\: acid\: of\: Base}^+}\)
\(\mathrm{H^+ + {\color{Blue} F^- } \rightleftharpoons {\color{Red} HF}}\)
\(\mathrm{H^+ + {\color{Blue} OH^- } \rightleftharpoons {\color{Red} H_2O}}\)
\(\mathrm{H^+ + {\color{Blue} H_2O } \rightleftharpoons {\color{Red} H_3O^+}}\)
\(\mathrm{H^+ + {\color{Blue} NH_3 } \rightleftharpoons {\color{Red} NH_4^+}}\)
To summarize, the conjugate base of HF is fluoride ion, F - , and the conjugate acid of fluoride ion, F - , is HF. The HF/F- pair is referred to as a conjugate acid-base pair. The difference in the formulas of a conjugate acid-base pair (example: HF and F - ) is H + . The table below lists conjugate acid-base pairs for your reference so that you can figure out the strategy of identifying them. For any given acid or base, you should be able to give its conjugate base or conjugate acid. The formula of an acid's conjugate base is generated by removing a proton (H + ) from the acid formula. The formula of the base's conjugate acid is formed by adding a proton (H + ) to the formula of the base.
| Conjugate Acid | Conjugate Base |
|---|---|
| \(\ce{H3O+}\) | \(\ce{H2O}\) |
| \(\ce{H2O}\) | \(\ce{OH-}\) |
| \(\ce{H2SO4}\) | \(\ce{HSO4-}\) |
| \(\ce{HSO4-}\) | \(\ce{SO4^2-}\) |
| \(\ce{NH4+}\) | \(\ce{NH3}\) |
| \(\ce{NH3}\) | \(\ce{NH2-}\) |
| \(\ce{CH3COOH}\) | \(\ce{CH3COO-}\) |
| \(\ce{CH3NH3+}\) | \(\ce{CH3NH2}\) |
Example \(\PageIndex{3}\) : Conjugate Pairs
Write the formula of the conjugate base of (a) HCl and (b) HCO 3 – .
Write the formula of the conjugate acid of (c) CH 3 NH 2 and (d) OH – .
Solution:
A conjugate base is formed by removing a proton (H + ). A conjugate acid is formed by accepting a proton (H + ).
- After HCl donates a proton, a Cl – ion is produced, and so Cl – is the conjugate base.
- After hydrogen carbonate ion, HCO 3 – , donates a proton, its conjugate base, CO 3 2 – is produced.
- After accepting a proton (H + ), CH 3 NH 2 is converted to CH 3 NH 3 + , its conjugate acid.
- After accepting a proton (H + ), OH – is converted to H 2 O, its conjugate acid.
Exercise \(\PageIndex{3}\): Conjugate Pairs
Write the formula of the conjugate base of (a) HNO 2 and (b) H 2 CO 3 .
Write the formula of the conjugate acid of (c) C 6 H 5 NH 2 and (d) HCO 3 – .
- Answer
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a. NO 2 – is the conjugate base of HNO 2 .
b. HCO 3 – is the conjugate base of H 2 CO 3
c. C 6 H 5 NH 3 + is the conjugate acid of C 6 H 5 NH 2 .
d. H 2 CO 3 is the conjugate acid of HCO 3 –
In the reaction of ammonia with water to give ammonium ions and hydroxide ions, ammonia acts as a base by accepting a proton from a water molecule, which in this case means that water is acting as an acid. In the reverse reaction, an ammonium ion acts as an acid by donating a proton to a hydroxide ion, and the hydroxide ion acts as a base. The conjugate acid–base pairs for this reaction are \(NH_4^+/NH_3\) and \(H_2O/OH^−\). This means that the conjugate acid of the base NH 3 is NH 4 + while the conjugate base of the acid NH 4 + is NH 3 . Similarly, the conjugate base of the acid H 2 O is OH - , and the conjugate acid of the base OH - is H 2 O.
In the forward reaction, the parent acid is H 2 O and and the parent base is NH 3 (shown in the illustration below). The acid H 2 O loses a proton (H + ) to form its conjugate base OH - . The base NH 3 gains a proton, to produce its conjugate acid NH 4 + . In the reverse reaction, the acid NH 4 + loses a proton (H + ) to form its conjugate base NH 3 . The base OH - gains a proton, to produce its conjugate acid H 2 O.
When hydrogen fluoride (HF) dissolves in water and ionizes, protons are transferred from hydrogen fluoride (parent acid) molecules to water (parent base) molecules, yielding hydronium ions (conjugate acid of water) and fluoride ions (conjugate base of HF):
\({\color{Red} \mathrm{HF}} + {\color{Blue} H_2O } \rightleftharpoons {\color{Blue} H_3O^+} + {\color{Red} \mathrm{F^-}}\)
Example \(\PageIndex{3}\)
Identify the conjugate acid-base pairs in this equilibrium.
\[\ce{CH3CO2H + H2O <=> H3O^{+} + CH3CO2^{-}} \nonumber \]
Solution
Similarly, in the reaction of acetic acid with water, acetic acid donates a proton to water, which acts as the base. In the reverse reaction, \(H_3O^+\) is the acid that donates a proton to the acetate ion, which acts as the base.
Once again, we have two conjugate acid–base pairs:
- the parent acid and its conjugate base (\(CH_3CO_2H/CH_3CO_2^−\)) and
- the parent base and its conjugate acid (\(H_3O^+/H_2O\)).
Example \(\PageIndex{4}\)
Identify the conjugate acid-base pairs in this equilibrium.
\[(CH_{3})_{3}N + H_{2}O\rightleftharpoons (CH_{3})_{3}NH^{+} + OH^{-} \nonumber \]
Solution
One pair is H 2 O and OH − , where H 2 O has one more H + and is the conjugate acid, while OH − has one less H + and is the conjugate base.
The other pair consists of (CH 3 ) 3 N and (CH 3 ) 3 NH + , where (CH 3 ) 3 NH + is the conjugate acid (it has an additional proton) and (CH 3 ) 3 N is the conjugate base.
Exercise \(\PageIndex{4}\)
Identify the conjugate acid-base pairs in this equilibrium.
\[\ce{NH2^{-} + H2O\rightleftharpoons NH3 + OH^{-}} \nonumber \]
- Answer:
- H 2 O (acid) and OH − (base); NH 2 − (base) and NH 3 (acid)
The use of conjugate acid-base pairs allows us to make a very simple statement about relative strengths of acids and bases. The stronger an acid, the weaker its conjugate base , and, conversely, the stronger a base, the weaker its conjugate acid.

- Brønsted-Lowry acid-base reactions are essentially proton transfer reactions.


<IP_ADDRESS>: from Preface to Lyrical Ballads
The first Volume of these Poems has already been submitted to general perusal. It was published, as an experiment, which, I hoped, might be of some use to ascertain, how far, by fitting to metrical arrangement a selection of the real language of men in a state of vivid sensation, that sort of pleasure and that quantity of pleasure may be imparted, which a Poet may rationally endeavour to impart.
I had formed no very inaccurate estimate of the probable effect of those Poems: I flattered myself that they who should be pleased with them would read them with more than common pleasure: and, on the other band, I was well aware, that by those who should dislike them they would be read with more than common dislike. The result has differed from my expectation in this only, that I have pleased a greater number, than I ventured to hope I should please. For the sake of variety, and from a consciousness of my own weakness, I was induced to request the assistance of a Friend, who furnished me with the Poems of the ANCIENT MARINER, the FOSTER-MOTHER'S TALE:, the NIGHTINGALE, and the Poem entitled LOVE. I should not, however, have requested this assistance, had I not believed that the Poems of my Friend would in a great measure have the same tendency as my own, and that, though there would be found a difference, there would be found no discordance in the colours of our style; as our opinions on the subject of poetry do almost entirely coincide.
Several of my Friends are anxious for the success of these Poems from a belief, that, if the views with which they were composed were indeed realized, a class of Poetry would be produced, well adapted to interest mankind permanently, and not unimportant in the multiplicity, and in the quality of its moral relations: and on this account they have advised me to prefix a systematic defence of the theory, upon which the poems were written. But I was unwilling to undertake the task, because I knew that on this occasion the Reader would look coldly upon my arguments, since I might be suspected of having been principally influenced by the selfish and foolish hope of *reasoning* him into an approbation of these particular Poems: and I was still more unwilling to undertake the task, because, adequately to display my opinions, and fully to enforce my arguments, would require a space wholly disproportionate to the nature of a preface. For to treat the subject with the clearness and coherence, of which I believe it susceptible, it would be necessary to give a full account of the present state of the public taste in this country, and to determine how far this taste is healthy or depraved; which, again, could not be determined, without pointing out, in what manner language and the human mind act and re-act on each other and without retracing the revolutions, not of literature alone, but likewise of society itself. I have therefore altogether declined to enter regularly upon this defence; yet I am sensible, that there would be some impropriety in abruptly obtruding upon the Public, without a few words of introduction, Poems so materially different from those, upon which general approbation is at present bestowed.
It is supposed, that by the act of writing in verse an Author makes a formal engagement that he will gratify certain known habits of association; that he not only thus apprizes the Reader that certain classes of ideas and expressions will be found in his book, but that others will be carefully excluded. This exponent or symbol held forth by metrical language must in different eras of literature have excited very different expectations: for example, in the age of Catullus, Terence, and Lucretius and that of Statius or Claudian; and in our own country, in the age of Shakespeare and Beaumont and Fletcher, and that of Donne and Cowley, or Dryden, or Pope. I will not take upon me to determine the exact import of the promise which by the act of writing in verse an Author, in the present day, makes to his Reader; but I am certain, it will appear to many persons that I have not fulfilled the terms of an engagement thus voluntarily contracted. They who have been accustomed to the gaudiness and inane phraseology of many modern writers, if they persist in reading this book to its conclusion, will, no doubt, frequently have to struggle with feelings of strangeness and aukwardness: they will look round for poetry, and will be induced to inquire by what species of courtesy these attempts can be permitted to assume that title. I hope therefore the Reader will not censure me, if I attempt to state what I have proposed to myself to perform; and also, (as far as the limits of a preface will permit) to explain some of the chief reasons which have determined me in the choice of my purpose: that at least he may be spared any unpleasant feeling of disappointment, and that I myself may be protected from the most dishonorable accusation which can be brought against an Author, namely, that of an indolence which prevents him from endeavouring to ascertain what is his duty, or, when his duty is ascertained, prevents him from performing it.
The principal object, then, which I proposed to myself in these Poems was to chuse incidents and situations from common life, and to relate or describe them, throughout, as far as was possible, in a selection of language really used by men; and, at the same time, to throw over them a certain colouring of imagination, whereby ordinary things should be presented to the mind in an unusual way; and, further, and above all, to make these incidents and situations interesting by tracing in them, truly though not ostentatiously, the primary laws of our nature: chiefly, as far as regards the manner in which we associate ideas in a state of excitement. Low and rustic life was generally chosen, because in that condition, the essential passions of the heart find a better soil in which they can attain their maturity, are less under restraint, and speak a plainer and more emphatic language; because in that condition of life our elementary feelings co-exist in a state of greater simplicity, and, consequently, may be more accurately contemplated, and more forcibly communicated; because the manners of rural life germinate from those elementary feelings; and, from the necessary character of rural occupations, are more easily comprehended, and are more durable; and lastly, because in that condition the passions of men are incorporated with the beautiful and permanent forms of nature. The language, too, of these men is adopted (purified indeed from what appear to be its real defects, from all lasting and rational causes of dislike or disgust) because such men hourly communicate with the best objects from which the best part of language is originally derived; and because, from their rank in society and the sameness and narrow circle of their intercourse, being less under the influence of social vanity they convey their feelings and notions in simple and unelaborated expressions. Accordingly, such a language, arising out of repeated experience and regular feelings, is a more permanent, and a far more philosophical language, than that which is frequently substituted for it by Poets, who think that they are conferring honour upon themselves and their art, in proportion as they separate themselves from the sympathies of men, and indulge in arbitrary and capricious habits of expression, in order to furnish food for fickle tastes, and fickle appetites, of their own creation.
I cannot, however, be insensible of the present outcry against the triviality and meanness both of thought and language, which some of my contemporaries have occasionally introduced into their metrical compositions; and I acknowledge, that this defect, where it exists, is more dishonorable to the Writer's own character than false refinement or arbitrary innovation, though I should contend at the same time that it is far less pernicious in the sum of its consequences. From such verses the Poems in these volumes will be found distinguished at least by one mark of difference, that each of them has a worthy purpose. Not that I mean to say, that I always began to write with a distinct purpose formally conceived; but I believe that my habits of meditation have so formed my feelings, as that my descriptions of such objects as strongly excite those feelings, will be found to carry along with them a purpose. If in this opinion I am mistaken, I can have little right to the name of a Poet. For all good poetry is the spontaneous overflow of powerful feelings: but though this be true, Poems to which any value can be attached, were never produced on any variety of subjects but by a man, who being possessed of more than usual organic sensibility, had also thought long and deeply. For our continued influxes of feeling are modified and directed by our thoughts, which are indeed the representatives of all our past feelings; and, as by contemplating the relation of these general representatives to each other we discover what is really important to men, so, by the repetition and continuance of this act, our feelings will be connected with important subjects, till at length, if we be originally possessed of much sensibility, such habits of mind will be produced, that, by obeying blindly and mechanically the impulses of those habits, we shall describe objects, and utter sentiments, of such a nature and in such connection with each other, that the understanding of the being to whom we address ourselves, if he be in a healthful state of association, must necessarily be in some degree enlightened, and his affections ameliorated.
I have said that each of these poems has a purpose. I have also informed my Reader what this purpose will be found principally to be: namely to illustrate the manner in which our feelings and ideas are associated in a state of excitement. But, speaking in language somewhat more appropriate, it is to follow the fluxes and refluxes of the mind when agitated by the great and simple affections of our nature. This object I have endeavoured in these short essays to attain by various means; by tracing the maternal passion through many of its more subtle windings, as in the poems of the IDIOT BOY and the MAD MOTHER; by accompanying the last struggles of a human being, at the approach of death, cleaving in solitude to life and society, as in the Poem of the FORSAKEN INDIAN; by shewing, as in the Stanzas entitled WE ARE SEVEN, the perplexity and obscurity which in childhood attend our notion of death, or rather our utter inability to admit that notion; or by displaying the strength of fraternal, or to speak more philosophically, of moral attachment when early associated with the great and beautiful objects of nature, as in THE BROTHERS; or, as in the Incident of SIMON LEE, by placing my Reader in the way of receiving from ordinary moral sensations another and more salutary impression than we are accustomed to receive from them. It has also been part of my general purpose to attempt to sketch characters under the influence of less impassioned feelings, as in the TWO APRIL MORNINGS, THE FOUNTAIN, THE OLD MAN TRAVELLING, THE TWO THIEVES, &c. characters of which the elements are simple, belonging rather to nature than to manners, such as exist now, and will probably always exist, and which from their constitution may be distinctly and profitably contemplated. I will not abuse the indulgence of my Reader by dwelling longer upon this subject; but it is proper that I should mention one other circumstance which distinguishes these Poems from the popular Poetry of the day; it is this, that the feeling therein developed gives importance to the action and situation, and not the action and situation to the feeling. My meaning will be rendered perfectly intelligible by referring my Reader to the Poems entitled POOR SUSAN and the CHILDLESS FATHER, particularly to the last Stanza of the latter Poem.
I will not suffer a sense of false modesty to prevent me from asserting, that I point my Reader's attention to this mark of distinction, far less for the sake of these particular Poems than from the general importance of the subject. The subject is indeed important! For the human mind is capable of being excited without the application of gross and violent stimulants; and he must have a very faint perception of its beauty and dignity who does not know this, and who does not further know, that one being is elevated above another, in proportion as he possesses this capability. It has therefore appeared to me, that to endeavour to produce or enlarge this capability is one of the best services in which, at any period, a Writer can be engaged; but this service, excellent at all times, is especially so at the present day. For a multitude of causes, unknown to former times, are now acting with a combined force to blunt the discriminating powers of the mind, and unfitting it for all voluntary exertion to reduce it to a state of almost savage torpor. The most effective of these causes are the great national events which are daily taking place, and the encreasing accumulation of men in cities, where the uniformity of their occupations produces a craving for extraordinary incident, which the rapid communication of intelligence hourly gratifies. To this tendency of life and manners the literature and theatrical exhibitions of the country have conformed themselves. The invaluable works of our elder writers, I had almost said the works of Shakespeare and Milton, are driven into neglect by frantic novels, sickly and stupid German Tragedies, and deluges of idle and extravagant stories in verse. When I think upon this degrading thirst after outrageous stimulation, I am almost ashamed to have spoken of the feeble effort with which I have endeavoured to counteract it; and, reflecting upon the magnitude of the general evil, I should be oppressed with no dishonorable melancholy, had I not a deep impression of certain inherent and indestructible qualities of the human mind, and likewise of certain powers in the great and permanent objects that act upon it which are equally inherent and indestructible; and did I not further add to this impression a belief, that the time is approaching when the evil will be systematically opposed, by men of greater powers, and with far more distinguished success.
Having dwelt thus long on the subjects and aim of these Poems, I shall request the Reader's permission to apprize him of a few circumstances relating to their style, in order, among other reasons, that I may not be censured for not having performed what I never attempted. The Reader will find that personifications of abstract ideas rarely occur in these volumes; and, I hope, are utterly rejected as an ordinary device to elevate the style, and raise it above prose. I have proposed to myself to imitate, and, as far as is possible, to adopt the very language of men; and assuredly such personifications do not make any natural or regular part of that language. They are, indeed, a figure of speech occasionally prompted by passion, and I have made use of them as such; but I have endeavoured utterly to reject them as a mechanical device of style, or as a family language which Writers in metre seem to lay claim to by prescription. I have wished to keep my Reader in the company of flesh and blood, persuaded that by so doing I shall interest him. I am, however, well aware that others who pursue a different track may interest him likewise; I do not interfere with their claim, I only wish to prefer a different claim of my own. There will also be found in these volumes little of what is usually called poetic diction; I have taken as much pains to avoid it as others ordinarily take to produce it; this I have done for the reason already alleged, to bring my language near to the language of men, and further, because the pleasure which I have proposed to myself to impart is of a kind very different from that which is supposed by many persons to be the proper object of poetry. I do not know how without being culpably particular I can give my Reader a more exact notion of the style in which I wished these poems to be written than by informing him that I have at all times endeavoured to look steadily at my subject, consequently, I hope that there is in these Poems little falsehood of description, and that my ideas are expressed in language fitted to their respective importance. Something I must have gained by this practice, as it is friendly to one property of all good poetry, namely, good sense; but it has necessarily cut me off from a large portion of phrases and figures of speech which from father to son have long been regarded as the common inheritance of Poets. I have also thought it expedient to restrict myself still further, having abstained from the use of many expressions, in themselves proper and beautiful, but which have been foolishly repeated by bad Poets, till such feelings of disgust are connected with them as it is scarcely possible by any art of association to overpower.
If in a Poem there should be found a series of lines, or even a single line, in which the language, though naturally arranged and according to the strict laws of metre, does not differ from that of prose, there is a numerous class of critics, who, when they stumble upon these prosaisms as they call them, imagine that they have made a notable discovery, and exult over the Poet as over a man ignorant of his own profession. Now these men would establish a canon of criticism which the Reader will conclude he must utterly reject, if he wishes to be pleased with these volumes. And it would be a most easy task to prove to him, that not only the language of a large portion of every good poem, even of the most elevated character, must necessarily, except with reference to the metre, in no respect differ from that of good prose, but likewise that some of the most interesting parts of the best poems will be found to be strictly the language of prose, when prose is well written. The truth of this assertion might be demonstrated by innumerable passages from almost all the poetical writings, even of Milton himself. I have not space for much quotation; but, to illustrate the subject in a general manner, I will here adduce a short composition of Gray, who was at the head of those who by their reasonings have attempted to widen the space of separation betwixt Prose and Metrical composition, and was more than any other man curiously elaborate in the structure of his own poetic diction.
It will easily be perceived that the only part of this Sonnet which is of any value is the lines printed in Italics: it is equally obvious, that, except in the rhyme, and in the use of the single word "fruitless" for fruitlessly, which is so far a defect, the language of these lines does in no respect differ from that of prose.
By the foregoing quotation I have shewn that the language of Prose may yet be well adapted to Poetry; and I have previously asserted that a large portion of the language of every good poem can in no respect differ from that of good Prose. I will go further. I do not doubt that it may be safely affirmed, that there neither is, nor can be, any essential difference between the language of prose and metrical composition. We are fond of tracing the resemblance between Poetry and Painting, and, accordingly, we call them Sisters: but where shall we find bonds of connection sufficiently strict to typify the affinity betwixt metrical and prose composition? They both speak by and to the same organs; the bodies in which both of them are clothed may be said to be of the same substance, their affections are kindred and almost identical, not necessarily differing even in degree; Poetry* sheds no tears "such as Angels weep," but natural and human tears; she can boast of no celestial Ichor that distinguishes her vital juices from those of prose; the same human blood circulates through the veins of them both.
* * * "What Is a Poet?"
Taking up the subject, then, upon general grounds, I ask what is meant by the word Poet? What is a Poet? To whom does he address himself? And what language is to be expected from him? He is a man speaking to men: a man, it is true, endued with more lively sensibility, more enthusiasm and tenderness, who has a greater knowledge of human nature, and a more comprehensive soul, than are supposed to be common among mankind; a man pleased with his own passions and volitions, and who rejoices more than other men in the spirit of life that is in him; delighting to contemplate similar volitions and passions as manifested in the goings-on of the Universe, and habitually impelled to create them where he does not find them. To these qualities he has added a disposition to be affected more than other men by absent things as if they were present; an ability of conjuring up in himself passions, which are indeed far from being the same as those produced by real events, yet (especially in those parts of the general sympathy which are pleasing and delightful) do more nearly resemble the passions produced by real events, than any thing which, from the motions of their own minds merely, other men are accustomed to feel in themselves; whence, and from practice, he has acquired a greater readiness and power in expressing what he thinks and feels, and especially those thoughts and feelings which, by his own choice, or from the structure of his own mind, arise in him without immediate external excitement.
But, whatever portion of this faculty we may suppose even the greatest Poet to possess, there cannot be a doubt but that the language which it will suggest to him, must, in liveliness and truth, fall far short of that which is uttered by men in real life, under the actual pressure of those passions, certain shadows of which the Poet thus produces, or feels to be produced, in himself. However exalted a notion we would wish to cherish of the character of a Poet, it is obvious, that, while he describes and imitates passions, his situation is altogether slavish and mechanical, compared with the freedom and power of real and substantial action and suffering. So that it will be the wish of the Poet to bring his feelings near to those of the persons whose feelings he describes, nay, for short spaces of time perhaps, to let himself slip into an entire delusion, and even confound and identify his own feelings with theirs; modifying only the language which is thus suggested to him, by a consideration that he describes for a particular purpose, that of giving pleasure. Here, then, he will apply the principle on which I have so much insisted, namely, that of selection; on this he will depend for removing what would otherwise be painful or disgusting in the passion; he will feel that there is no necessity to trick out or to elevate nature: and, the more industriously he applies this principle, the deeper will be his faith that no words, which his fancy or imagination can suggest, will be to be compared with those which are the emanations of reality and truth.
But it may be said by those who do not object to the general spirit of these remarks, that, as it is impossible for the Poet to produce upon all occasions language as exquisitely fitted for the passion as that which the real passion itself suggests, it is proper that he should consider himself as in the situation of a translator, who deems himself justified when he substitutes excellences of another kind for those which are unattainable by him; and endeavours occasionally to surpass his original, in order to make some amends for the general inferiority to which he feels that he must submit. But this would be to encourage idleness and unmanly despair. Further, it is the language of men who speak of what they do not understand; who talk of Poetry as of a matter of amusement and idle pleasure; who will converse with us as gravely about a taste for Poetry, as they express it, as if it were a thing as indifferent as a taste for Rope-dancing, or Frontiniac or Sherry. Aristotle, I have been told, hath said, that Poetry is the most philosophic of all writing: it is so: its object is truth, not individual and local, but general, and operative; not standing upon external testimony, but carried alive into the heart by passion; truth which is its own testimony, which gives strength and divinity to the tribunal to which it appeals, and receives them from the same tribunal. Poetry is the image of man and nature. The obstacles which stand in the way of the fidelity of the Biographer and Historian, and of their consequent utility, are incalculably greater than those which are to be encountered by the Poet, who has an adequate notion of the dignity of his art. The Poet writes under one restriction only, namely, that of the necessity of giving immediate pleasure to a human Being possessed of that information which may be expected from him, not as a lawyer, a physician, a mariner, an astronomer or a natural philosopher, but as a Man. Except this one restriction, there is no object standing between the Poet and the image of things; between this, and the Biographer and Historian there are a thousand.
Nor let this necessity of producing immediate pleasure be considered as a degradation of the Poet's art. It is far otherwise. It is an acknowledgment of the beauty of the universe, an acknowledgment the more sincere because it is not formal, but indirect; it is a task light and easy to him who looks at the world in the spirit of love: further, it is a homage paid to the native and naked dignity of man, to the grand elementary principle of pleasure, by which he knows, and feels, and lives, and moves. We have no sympathy but what is propagated by pleasure: I would not be misunderstood; but wherever we sympathize with pain it will be found that the sympathy is produced and carried on by subtle combinations with pleasure. We have no knowledge, that is, no general principles drawn from the contemplation of particular facts, but what has been built up by pleasure, and exists in us by pleasure alone. The Man of Science, the Chemist and Mathematician, whatever difficulties and disgusts they may have had to struggle with, know and feel this. However painful may be the objects with which the Anatomist's knowledge is connected, he feels that his knowledge is pleasure; and where he has no pleasure he has no knowledge. What then does the Poet? He considers man and the objects that surround him as acting and re-acting upon each other, so as to produce an infinite complexity of pain and pleasure; he considers man in his own nature and in his ordinary life as contemplating this with a certain quantity of immediate knowledge, with certain convictions, intuitions, and deductions which by habit become of the nature of intuitions; he considers him as looking upon this complex scene of ideas and sensations, and finding every where objects that immediately excite in him sympathies which, from the necessities of his nature, are accompanied by an overbalance of enjoyment.
To this knowledge which all men carry about with them, and to these sympathies in which without any other discipline than that of our daily life we are fitted to take delight, the Poet principally directs his attention. He considers man and nature as essentially adapted to each other, and the mind of man as naturally the mirror of the fairest and most interesting qualities of nature. And thus the Poet, prompted by this feeling of pleasure which accompanies him through the whole course of his studies, converses with general nature with affections akin to those, which, through labour and length of time, the Man of Science has raised up in himself, by conversing with those particular parts of nature which are the objects of his studies. The knowledge both of the Poet and the Man of Science is pleasure; but the knowledge of the one cleaves to us as a necessary part of our existence, our natural and unalienable inheritance; the other is a personal and individual acquisition, slow to come to us, and by no habitual and direct sympathy connecting us with our fellow- beings. The Man of Science seeks truth as a remote and unknown benefactor; he cherishes and loves it in his solitude: the Poet, singing a song in which all human beings join with him, rejoices in the presence of truth as our visible friend and hourly companion. Poetry is the breath and finer spirit of all knowledge; it is the impassioned expression which is in the countenance of all Science. Emphatically may it be said of the Poet, as Shakespeare hath said of man, "that he looks before and after." He is the rock of defence of human nature; an upholder and preserver, carrying every where with him relationship and love. In spite of difference of soil and climate, of language and manners, of laws and customs, in spite of things silently gone out of mind and things violently destroyed, the Poet binds together by passion and knowledge the vast empire of human society, as it is spread over the whole earth, and over all time. The objects of the Poet's thoughts are every where; though the eyes and senses of man are, it is true, his favorite guides, yet he will follow wheresoever he can find an atmosphere of sensation in which to move his wings. Poetry is the first and last of all knowledge--it is as immortal as the heart of man. If the labours of men of Science should ever create any material revolution, direct or indirect, in our condition, and in the impressions which we habitually receive, the Poet will sleep then no more than at present, but he will be ready to follow the steps of the man of Science, not only in those general indirect effects, but he will be at his side, carrying sensation into the midst of the objects of the Science itself. The remotest discoveries of the Chemist, the Botanist, or Mineralogist, will be as proper objects of the Poet's art as any upon which it can be employed, if the time should ever come when these things shall be familiar to us, and the relations under which they are contemplated by the followers of these respective Sciences shall be manifestly and palpably material to us as enjoying and suffering beings. If the time should ever come when what is now called Science, thus familiarized to men, shall be ready to put on, as it were, a form of flesh and blood, the Poet will lend his divine spirit to aid the transfiguration, and will welcome the Being thus produced, as a dear and genuine inmate of the household of man. It is not, then, to be supposed that any one, who holds that sublime notion of Poetry which I have attempted to convey, will break in upon the sanctity and truth of his pictures by transitory and accidental ornaments, and endeavour to excite admiration of himself by arts, the necessity of which must manifestly depend upon the assumed meanness of his subject.
What I have thus far said applies to Poetry in general; but especially to those parts of composition where the Poet speaks through the mouths of his characters; and upon this point it appears to have such weight that I will conclude, there are few persons, of good sense, who would not allow that the dramatic parts of composition are defective, in proportion as they deviate from the real language of nature, and are coloured by a diction of the Poet's own, either peculiar to him as an individual Poet, or belonging simply to Poets in general, to a body of men who, from the circumstance of their compositions being in metre, it is expected will employ a particular language.
It is not, then, in the dramatic parts of composition that we look for this distinction of language; but still it may be proper and necessary where the Poet speaks to us in his own person and character. To this I answer: by referring my Reader to the description which I have before given of a Poet. Among the qualities which I have enumerated as principally conducting to form a Poet, is implied nothing differing in kind from other men, but only in degree. The sum of what I have there said is, that the Poet is chiefly distinguished from other men by a greater promptness to think and feel without immediate external excitement, and a greater power in expressing such thoughts and feelings as are produced in him in that manner. But these passions and thoughts and feelings are the general passions and thoughts and feelings of men. And with what are they connected? Undoubtedly with our moral sentiments and animal sensations, and with the causes which excite these; with the operations of the elements and the appearances of the visible universe; with storm and sun-shine, with the revolutions of the seasons, with cold and heat, with loss of friends and kindred, with injuries and resentments, gratitude and hope, with fear and sorrow. These, and the like, are the sensations and objects which the Poet describes, as they are the sensations of other men, and the objects which interest them. The Poet thinks and feels in the spirit of the passions of men. How, then, can his language differ in any material degree from that of all other men who feel vividly and see clearly? It might be proved that it is impossible. But supposing that this were not the case, the Poet might then be allowed to use a peculiar language, when expressing his feelings for his own gratification, or that of men like himself. But Poets do not write for Poets alone, but for men. Unless therefore we are advocates for that admiration which depends upon ignorance, and that pleasure which arises from hearing what we do not understand, the Poet must descend from this supposed height, and, in order to excite rational sympathy, he must express himself as other men express themselves.
* * * "Emotion Recollected in Tranquillity"
I have said that Poetry is the spontaneous overflow of powerful feelings: it takes its origin from emotion recollected in tranquillity: the emotion is contemplated till by a species of reaction the tranquillity gradually disappears, and an emotion, kindred to that which was before the subject of contemplation, is gradually produced, and does itself actually exist in the mind. In this mood successful composition generally begins, and in a mood similar to this it is carried on; but the emotion, of whatever kind and in whatever degree, from various causes is qualified by various pleasures, so that in describing any passions whatsoever, which are voluntarily described, the mind will upon the whole be in a state of enjoyment. Now, if Nature be thus cautious in preserving in a state of enjoyment a being thus employed, the Poet ought to profit by the lesson thus held forth to him, and ought especially to take care, that whatever passions he communicates to his Reader, those passions, if his Reader's mind be sound and vigorous, should always be accompanied with an overbalance of pleasure. Now the music of harmonious metrical language, the sense of difficulty overcome, and the blind association of pleasure which has been previously received from works of rhyme or metre of the same or similar construction, an indistinct perception perpetually renewed of language closely resembling that of real life, and yet, in the circumstance of metre, differing from it so widely, all these imperceptibly make up a complex feeling of delight, which is of the most important use in tempering the painful feeling, which will always be found intermingled with powerful descriptions of the deeper passions. This effect is always produced in pathetic and impassioned poetry; while, in lighter compositions, the ease and gracefulness with which the Poet manages his numbers are themselves confessedly a principal source of the gratification of the Reader. I might perhaps include all which it is necessary to say upon this subject by affirming, what few persons will deny, that, of two descriptions, either of passions, manners, or characters, each of them equally well executed, the one in prose and the other in verse, the verse will be read a hundred times where the prose is read once.
* * *
I know that nothing would have so effectually contributed to further the end which I have in view as to have shewn of what kind the pleasure is, and how that pleasure is produced, which is confessedly produced by metrical composition essentially different from that which I have here endeavoured to recommend: for the Reader will say that he has been pleased by such composition; and what can I do more for him? The power of any art is limited; and he will suspect, that, if I propose to furnish him with new friends, it is only upon condition of his abandoning his old friends. Besides, as I have said, the Reader is himself conscious of the pleasure which he has received from such composition, composition to which he has peculiarly attached the endearing name of Poetry; and all men feel an habitual gratitude, and something of an honorable bigotry for the objects which have long continued to please them: we not only wish to be pleased, but to be pleased in that particular way in which we have been accustomed to be pleased. There is a host of arguments in these feelings; and I should be the less able to combat them successfully, as I am willing to allow, that, in order entirely to enjoy the Poetry which I am recommending, it would be necessary to give up much of what is ordinarily enjoyed. But, would my limits have permitted me to point out how this pleasure is produced, I might have removed many obstacles, and assisted my Reader in perceiving that the powers of language are not so limited as he may suppose; and that it is possible that poetry may give other enjoyments, of a purer, more lasting, and more exquisite nature. This part of my subject I have not altogether neglected; but it has been less my present aim to prove, that the interest excited by some other kinds of poetry is less vivid, and less worthy of the nobler powers of the mind, than to offer reasons for presuming, that, if the object which I have proposed to myself were adequately attained, a species of poetry would be produced, which is genuine poetry; in its nature well adapted to interest mankind permanently, and likewise important in the multiplicity and quality of its moral relations.
From what has been said, and from a perusal of the Poems, the Reader will be able clearly to perceive the object which I have proposed to myself: he will determine how far I have attained this object; and, what is a much more important question, whether it be worth attaining; and upon the decision of these two questions will rest my claim to the approbation of the public.
 But much confusion has been introduced into criticism by this contradistinction of Poetry and Prose, instead of the more philosophical one of Poetry and Matter of Fact, or Science. The only strict antithesis to Prose is Metre; nor is this, in truth, a strict antithesis; because lines and passages of metre so naturally occur in writing prose, that it would be scarcely possible to avoid them, even were it desirable.


6.2: Psychological Constructs
Maslow’s Hierarchy of Needs
- Maslow’s hierarchy of needs is often portrayed in the shape of a pyramid, with the greatest and most fundamental levels of needs at the bottom, and the need for self-actualization at the top.
- The order of needs as categorized by Maslow are physiological; safety; love and belonging; esteem; and self-actualization.
- Maslow acknowledged that many different levels of motivation are likely to be present in a human all at once. His focus in discussing the hierarchy was to identify the basic types of motivation and the order that they generally progress as lower needs are reasonably well met.
Physiological Needs
Physiological needs are generally obvious because they are required for survival. If requirements are not met, the body cannot continue to function. Air, water, food, clothing, and shelter are the basic physiological needs.
Safety Needs
Once physical needs are satisfied, individual safety takes precedence. Safety and Security needs include:
- Personal and family safety
- Financial security
- Health and well-being
Love/belonging Needs
After physiological and safety needs are fulfilled, the third layer of human needs are interpersonal. This involves feelings of belongingness. Deficiencies in interpersonal needs, due to neglect, shunning, ostracism, etc., can impact an individual’s ability to form and maintain emotionally significant relationships in general, such as:
- Friendship
- Intimacy
- Family
Humans need to feel a sense of belonging and acceptance, whether it comes from larger community affiliations or simply a few close friends. Without these connections, many people become susceptible to loneliness, social anxiety, and clinical depression. This need for belonging can sometimes overcome physiological and security needs. For example, an anorexic may ignore the need to eat and the security of health for a feeling of control and belonging.
Esteem
Esteem represents the normal human desire to be accepted and valued by others. People need to engage themselves to gain recognition and have an activity or activities that give the person a sense of contribution, to feel self-valued, be it in a profession or hobby. Imbalances at this level can result in low self-esteem or an inferiority complex. Many people with low self-esteem will not be able to improve their view of themselves simply by receiving fame, respect, and glory externally, but must first accept themselves internally. Psychological imbalances, such as depression, can prevent one from obtaining self-esteem on both levels.
Self-actualization
This level of need refers to what a person’s full potential is and the realization of that potential. Maslow describes this level as the desire to accomplish everything that one can, to become the most that one can be. Individuals may perceive or focus on this need very specifically. For example, one individual may have the strong desire to become an ideal parent. In another, the desire may be expressed athletically. For others, it may be expressed in paintings, pictures, or inventions. Maslow believed that to acquire this level of need, the person must adequately achieve the previous needs.
Freud’s Defense Mechanisms
Defense mechanisms are psychological mechanisms aimed at reducing anxiety. They were first discussed by Sigmund Freud as part of his psychoanalytic theory and further developed by his daughter, Anna Freud. Often unconscious, defense mechanisms are used to protect an individual from psychological pain or anxiety.
While such mechanisms may seem to be helpful in the short term, they can easily become a substitute for addressing the underlying cause and lead to additional problems. The solution, therefore, is to address the underlying causes of the pain these mechanisms are used to alleviate.
Here are a few examples:
| Defense Mechanism | Description | Example |
|---|---|---|
| Repression | Unknowingly placing an unpleasant memory or thought in the conscious | Not remembering a traumatic event such as being sexually abused as a child. |
| Regression | Reverting back to an immature behavior from an earlier stage of development | Throwing temper tantrums as an adult when you don’t get your way |
| Displacement | Redirecting feelings or actions from the intended source to a safer, substitute target | Taking your anger towards your boss out on family members by yelling at them in place of your boss. |
| Sublimation | Replacing socially unacceptable impulses with socially acceptable behavior | Channeling aggressiveness into playing football |
| Reaction formation | Overacting in the opposite way to one’s true feelings. | Being overly protective of an unwanted child. |
| Projection | Attributing one’s own unacceptable feelings and thoughts to others and not yourself | Accusing your boy/girlfriend of cheating on you because you have thoughts about cheating on him/her |
| Rationalization | Justifying actions, thoughts, or unwanted outcomes with excuses or faulty logic | Blaming the teaching style of a professor for why you failed an exam. |
Contributors and Attributions
CC Licensed Content, Shared Previously
- Maslow's Hierarchy of Needs. Authored by: Boundless. Located at: www.boundless.com/business/textbooks/boundless-business-textbook/motivation-theories-and-applications-11/theories-of-motivation-75/maslow-s-hierarchy-of-needs-356-3204/. License: CC BY-SA: Attribution-ShareAlike
- Defense Mechanism. Authored by: New World Encyclopedia. Located at: http://www.newworldencyclopedia.org/entry/Defense_mechanism . License: CC BY-SA: Attribution-ShareAlike


15.3.5: Raising Long-Term Financing
What are the Key Differences Between Debt and Equity, and What are the Major Types and Features of Long-term Debt?
A basic principle of finance is to match the term of the financing to the period over which benefits are expected to be received from the associated outlay. Short-term items should be financed with short-term funds, and long-term items should be financed with long-term funds. Long-term financing sources include both debt (borrowing) and equity (ownership). Equity financing comes either from selling new ownership interests or from retaining earnings. Financial managers try to select the mix of long-term debt and equity that results in the best balance between cost and risk.
Debt Versus Equity Financing
Say that the Boeing Company plans to spend $2 billion over the next four years to build and equip new factories to make jet aircraft. Boeing’s top management will assess the pros and cons of both debt and equity and then consider several possible sources of the desired form of long-term financing.
The major advantage of debt financing is the deductibility of interest expense for income tax purposes, which lowers its overall cost. In addition, there is no loss of ownership. The major drawback is financial risk : the chance that the firm will be unable to make scheduled interest and principal payments. The lender can force a borrower that fails to make scheduled debt payments into bankruptcy. Most loan agreements have restrictions to ensure that the borrower operates efficiently.
Equity, on the other hand, is a form of permanent financing that places few restrictions on the firm. The firm is not required to pay dividends or repay the investment. However, equity financing gives common stockholders voting rights that provide them with a voice in management. Equity is more costly than debt. Unlike the interest on debt, dividends to owners are not tax-deductible expenses. Table <IP_ADDRESS> summarizes the major differences between debt and equity financing.
Debt Financing
Long-term debt is used to finance long-term (capital) expenditures. The initial maturities of long-term debt typically range between 5 and 20 years. Three important forms of long-term debt are term loans, bonds, and mortgage loans.
Table \(\PageIndex{1}\): Major Differences between Debt and Equity Financing
| Debt Financing | Equity Financing | |
|---|---|---|
| Have a say in management | Creditors typically have none, unless the borrower defaults on payments. Creditors may be able to place restraints on management in event of default. | Common stockholders have voting rights. |
| Have a right to income and assets | Debt holders rank ahead of equity holders. Payment of interest and principal is a contractual obligation of the firm. | Equity owners have a residual claim on income (dividends are paid only after paying interest and any scheduled principal) and no obligation to pay dividends. |
| Maturity (date when debt needs to be paid back) | Debt has a stated maturity and requires repayment of principal by a specified date. | The company is not required to repay equity, which has no maturity date. |
| Tax treatment | Interest is a tax-deductible expense. | Dividends are not tax-deductible and are paid from after-tax income. |
A term loan is a business loan with a maturity of more than one year. Term loans generally have maturities of 5 to 12 years and can be unsecured or secured. They are available from commercial banks, insurance companies, pension funds, commercial finance companies, and manufacturers’ financing subsidiaries. A contract between the borrower and the lender spells out the amount and maturity of the loan, the interest rate, payment dates, the purpose of the loan, and other provisions such as operating and financial restrictions on the borrower to control the risk of default. The payments include both interest and principal, so the loan balance declines over time. Borrowers try to arrange a repayment schedule that matches the forecast cash flow from the project being financed.
Bonds are long-term debt obligations (liabilities) of corporations and governments. A bond certificate is issued as proof of the obligation. The issuer of a bond must pay the buyer a fixed amount of money—called interest , stated as the coupon rate —on a regular schedule, typically every six months. The issuer must also pay the bondholder the amount borrowed—called the principal , or par value —at the bond’s maturity date (due date). Bonds are usually issued in units of $1,000—for instance, $1,000, $5,000, or $10,000—and have initial maturities of 10 to 30 years. They may be secured or unsecured, include special provisions for early retirement, or be convertible to common stock.
A mortgage loan is a long-term loan made against real estate as collateral. The lender takes a mortgage on the property, which lets the lender seize the property, sell it, and use the proceeds to pay off the loan if the borrower fails to make the scheduled payments. Long-term mortgage loans are often used to finance office buildings, factories, and warehouses. Life insurance companies are an important source of these loans. They make billions of dollars’ worth of mortgage loans to businesses each year.
CONCEPT CHECK
- Distinguish between debt and equity.
- Identify the major types and features of long term debt.


6.5: Culture
In this section
- You will learn about the high school and college admission in the Arab World and particularly Egypt.
Watch this video and discuss:
What is the setting?
What are they talking about? What do all agree on?
Culture Notes
- In this lesson, you learned about the types of school in Arab countries which include المدرسة الإبتدائية followed by المدرسة الإعدادية then المدرسة الثانوية
- The last year of high school is referred to as الثانوية العامة which entails the country-wide examinations that student sit for at the end of that year.
- In Egypt, for example, as you see in the video, this is a high stakes exam that cause stress for students as they race for places in colleges. To survive in the race, families have taken it for granted that their kids take private tutoring دروس خصوصية which entails financial burdens as well.
- The process of college admission in public universities is based on the total percentage in all exams. This creates tense competition as students may miss their chance to study at top colleges by a few points or even half a point.
- Based on prestigious national ranking, some colleges are referred to as top colleges كليات القمة . They admit students with above a 95% and some only admit students with a score higher than 99%. Examples of top colleges are the college of medicine, college of pharmacology, and college of political sciences. Average colleges include college of Arts, college of Law and college of Business.
Discuss with your classmates the differences between the college systems in both the Arab World and your country in terms of the following:
- High school
- Cumulative total
- Top colleges
- Grading
- Private tutoring
- College admission
Key Takeaways
- You can search the differences in high school and college admission in different Arab countries.