Summary
531
SUMMARY
Acids and bases are important in our everyday lives. They contribute to food tastes,
have properties that are helpful for cleaning, and participate in many chemical processes. The Brønsted-Lowry theory is most widely used to describe the behavior of
acids and bases. According to this theory, an acid is a substance that donates an H+
ion to another substance; a base is an H+ acceptor. Acids and bases can be ranked
according to their strength, or how readily they donate or accept H+ ions. When
strong acids dissolve in water, they completely ionize, so essentially no acid molecules remain unreacted. When a weak acid dissolves in water, only a small percentage of the acid molecules ionize. Most remain as molecules. Equilibrium constants
called acid ionization constants, Ka, can be used to compare the strengths of different
weak acids. The stronger the acid, the larger the Ka value.
The relative concentrations of H3O+ and OH– ions in an aqueous solution at
25°C are determined by the ion-product constant for water, Kw:
2H2O(l)
H3O+(aq) + OH–(aq) Kw = 1.0 × 10–14
[H3O+][OH–] = 1.0 × 10–14
Neutral solutions have equal concentrations of H3O+ and OH–. Acidic solutions
contain a greater concentration of H3O+ than OH–. Basic solutions contain a greater
concentration of OH– than H3O+. The acidity of a solution is commonly expressed
in terms of pH, the negative logarithm of the H3O+ concentration:
pH = –log[H3O+]
At 25°C, acidic solutions have a pH less than 7, basic solutions have a pH greater
than 7, and neutral solutions have a pH equal to 7.
Buffered solutions contain a weak acid and its conjugate base (or a weak base
and its conjugate acid) in similar concentrations. Buffers help to prevent large
changes in pH by reacting with small amounts of added acid or base.
Math Toolbox 13.1 | Log and Inverse Log Functions
Log Functions
1
Let’s consider several substances that vary in their H3O+(aq)
concentrations:
HCl (1 M)
HCl (0.1 M)
HCl (0.01 M)
lemon juice
vinegar
beer
urine
pure water
[H3O ]
1M
0.1 M (or 10–1 M)
0.01 M (or 10–2 M)
0.001 M (or 10–3 M)
0.0001 M (or 10–4 M)
0.00001 M (or 10–5 M)
0.000001 M (or 10–6 M)
0.0000001 M (or 10–7 M)
Except for pure water, these substances are all acids. Suppose we
wanted to compare their concentrations graphically. In a bar graph,
notice what happens to those with the lower concentrations:
0.8
0.7
[H3O+], (M)
Substance
0.9
+
0.6
0.5
0.4
0.3
0.2
0.1
ce
Vi
ne
ga
r
Be
er
Ur
Pu ine
re
wa
ter
)
ju
i
M
on
Le
0.
l(
HC
m
M
01
1
0.
l(
HC
HC
l(
1
M
)
)
0
Substance
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532
Chapter 13
Acids and Bases
Math Toolbox 13.1 (continued )
Can you distinguish lemon juice, which we can consume, from
stomach acid, which causes damage to the esophagus? No. Logarithms are very useful in situations like this because logarithms
make it easier to compare numbers that are inconvenient to work
with and differ by orders of magnitude.
The log function on your calculator stands for logarithm to
the base 10. When you take the log of a number, you are asking,
“10 to what power will give the value after the word log?” For
example, the log of 1000 is 3, because 10 must be raised to the
third power to get 1000:
log(1000) = 3.0
Likewise, the log of 0.001 is –3 because 10 must be raised to the
–3 power to get 0.001:
log(0.001) = –3.0
To use the log function on most calculators, you first press the
log key and then enter the number. (On some calculators, these
two steps may be reversed.) Try taking the log of 0.0010 on your
calculator. You should get –3.0.
Step 1: Press the log key.
Step 2: Enter 0.001 (or 1 × 10–3), and then the ENTER or
= key.
You should get –3.0. If this does not work, try reordering the key
strokes.
Numerical values with large positive or negative exponents,
such as concentrations of H3O+, are inconvenient to work with. A
more convenient way to express such values is to use a logarithmic
scale. The pH scale is an example of a logarithmic scale. By
convention, the “p” of something is its negative logarithm:
pX = –log X
In chemistry, the most common examples are the pH scale and
the pOH scale:
pH = –log[H+] = –log[H3O+]
pOH = –log[OH–]
To obtain the pH or pOH of a solution, you must first take the log
of the concentration and then multiply by –1 (change the sign).
Try using your calculator to determine the pH of a solution that
has an H3O+ concentration of 5.0 × 10–4 M.
EXAMPLE 13.14 Log Functions
Use your calculator to find the pH of the following solutions.
(a) [H3O+] = 1.0 × 10–8 M
(b) [H3O+] = 6.2 × 10–1 M
Solution:
To calculate the pH from the H3O+ concentration, use the following
relationship:
pH = –log[H3O+]
Using your calculator, first enter a negative sign, followed by
the log key and the H3O+ concentration. Then hit the ENTER
or = key.
(a) pH = –log(1.0 × 10–8) = 8.00
(b) pH = –log(6.2 × 10–1) = 0.21
Practice Problem 13.14
Use your calculator to find the pH of the following solutions.
(a) [H3O+] = 7.5 × 10–3 M
(b) [H3O+] = 5.6 × 10–8 M
Further Practice: Questions 13.3 and 13.4 at the end of the
chapter
If we take the log of our original data and then change the
sign, we get the following values:
HCl (1 M)
HCl (0.1 M)
HCl (0.01 M)
lemon juice
vinegar
beer
urine
pure water
pH = –log(5.0 × 10–4) = 3.30
If you did not get the right answer, your calculator may require
you to reorder the key strokes.
–log[H3O+]
1M
0.1 M (or 10–1 M)
0.01 M (or 10–2 M)
0.001 M (or 10–3 M)
0.0001 M (or 10–4 M)
0.00001 M (or 10–5 M)
0.000001 M (or 10–6 M)
0.0000001 M (or 10–7 M)
0
1
2
3
4
5
6
7
A plot of the log values now allows us to distinguish the H3O+
content in each of these substances:
pH = –log[H3O+]
8
7
6
5
pH
Step 1: Press the +/– (change of sign) key.
Step 2: Press the log key.
Step 3: Enter the H3O+ concentration, and then the ENTER
or = key.
[H3O+]
Substance
4
3
2
1
r
wa
te
e
Pu
re
Ur
in
r
Be
e
ice
ju
Vi
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ga
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)
M
m
on
1
Le
(0
.0
(0
.1
M
HC
l
HC
l
HC
l(
1
M
)
)
0
Substance
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Math Toolbox 13.1
Log and Inverse Log Functions
533
Math Toolbox 13.1 (continued )
The pH scale is more convenient to use. However, we have to keep
in mind that an integer difference in pH is a 10-fold difference in
H3O+ concentration. For example, the pH of beer is around 5 and
that of vinegar is around 4. Vinegar is 10 times as acidic as beer
(10–4 for vinegar versus 10–5 for beer).
Did you get [H3O+] = 0.0010 or 1.0 × 10–3? If not, your calculator
may require you to reorder the key strokes. The steps for
calculating [OH–] from pOH are the same as calculating [H3O+]
from pH:
[H3O+] = 10–pH
x
[OH–] = 10–pOH
Inverse Log (10 ) Functions
When you have an equation that involves a log function, you
sometimes need to solve for the quantity that follows the log.
For example, if you are given pH, you might want to solve for
[H3O+]:
pH = –log[H3O+]
To solve for [H3O+], we must take the inverse log (antilog) of both
sides of the equation. To rearrange the equation so that [H3O+] is
on one side by itself, first multiply both sides by –1:
–pH = log[H3O+]
+
To get rid of the log expression in front of [H3O ], take the inverse
log of both sides:
inverse log (–pH) = inverse log (log[H3O+])
The value of [H3O+], then, equals the inverse log of the negative pH:
+
inverse log (–pH) = [H3O ]
Taking the inverse log of –pH is the same as 10 raised to the
–pH power. The inverse log function on your calculator is the
10x function, which is often found above the log function as an
alternate function for the same key. Another way to describe the
relationship between [H3O+] and pH is to say [H3O+] equals 10
raised to the –pH power:
[H3O+] = 10–pH
To use the inverse log (10x) function for this calculation on most
calculators, you first press the INV , SHIFT , or 2nd button
followed by the log key. Then enter the negative of the pH value.
The inverse log function, or 10x function, should appear on your
calculator above the log function as the alternative function.
By pressing INV , SHIFT , or 2nd button first, you tell the
calculator to use the alternative function. (On some calculators,
you have to enter the –pH value first.) Try taking the inverse log
where pH is 3.00.
The general calculation steps for calculating [H3O+] or [OH–] are
Step 1: Press the INV , SHIFT , or 2nd button.
Step 2: Press the log button.
Step 3: Enter –pH (or –pOH), and then the ENTER
or = key.
(On some calculators you may have to reorder the key strokes.)
EXAMPLE 13.15 Inverse Log Functions
Calculate the H3O+ concentrations for the solutions with the following pH values.
(a) pH = 9.40
(b) pH = 3.60
Solution:
To calculate the H3O+ concentration from the pH, use the following relationship:
[H3O+] = 10–pH
Using your calculator, first enter the INV , SHIFT , or 2nd button. Then press the log key to give you the 10x function. (On some
calculators the 10x button is its own key.) Finally, press the +/–
(change sign) key, enter the pH, and then the ENTER or = key.
(a) [H3O+] = 10–9.40 = 4.0 × 10–10
(b) [H3O+] = 10–3.60 = 2.5 × 10–4
Practice Problem 13.15
Calculate the OH– concentration from the following pOH values.
(a) pOH = 8.00
(b) pOH = 2.60
Further Practice: Questions 13.5 and 13.6 at the end of the
chapter
[H3O+] = 10–pH
Step 1: Press the INV , SHIFT , or 2nd button.
Step 2: Press the log button to use the 10x function.
Step 3: Enter –3.00, and then the ENTER or = key.
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