MATH 1010 PROJECT
ACIDS, BASES, AND LOGARITHMS
Water is a chemical compound formed by combination of two hydrogen (H) atoms and
one oxygen (O) atom. Thus water is often chemically designated as H2O. In general, water is
regarded as a stable substance. However, individual water molecules are in constant motion,
and through collisions with one another they sometimes collide with so much force that they ionize
(break apart into electrically charged ions):
(–)
(+)
W
+
water
hydroxide ion
H2O
OH–
hydrogen ion
H+
The double arrow indicates that the reaction proceeds in both directions. Thus, water molecules
constantly gain, lose, and swap hydrogen atoms: molecules dissociate to form ions; the ions
recombine to reform water molecules. At any given time about two of every billion water
molecules are ionized.
Pure water has equal concentrations of hydrogen ions (H+) and hydroxide ions (OH–), but in
many solutions the concentrations of H+ and OH– ions are not the same. If the concentration of H+
ions exceeds the concentration of OH– ions, the solution is acidic; if the concentration of OH– ions
is greater, the solution is basic. The exact amounts of hydrogen and hydroxide present are
controlled by the product:
Kw = [H+]*[OH–].
For pure water, the concentrations of H+ and OH– must be equal (see reaction above).
Experimentally, they have been determined to be 1 x 10–7 each. Therefore, Kw has a value of 1 x
10–14. This is a constant, so whenever one concentration is changed, the other must change in
response to keep their product equal to 1 x 10–14. Although pure water contains both hydrogen
ions (acid) and hydroxide ions (base), it is considered to be neutral overall because these ions
are present in equal amounts.
Any substance which, when added to water, increases the concentration of hydrogen ions
is called an acid. For example, when hydrogen chloride (HCl) is added to water, virtually all the
HCl molecules separate into hydrogen (H+) and chloride (Cl–) ions.
HCl
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H+ + Cl–
The water now has an excess of hydrogen ions, and is considered to be acidic. This does not
mean that there are no hydroxide ions remaining, just that there are relatively few of them
remaining. Any such addition of hydrogen (H+) ions to water will always result in a corresponding
decline in hydroxide ions (OH–). For example, if [H+] were increased to 1 x 10–6, then [OH–] must
decrease to 1 x 10–8 so that their product remains constant at 1 x 10–14.
Any substance which, when added to water, increases the concentration of hydroxide ions
is called a base. For example, when potassium hydroxide (KOH) is dissolved in water, it
separates to form K+ and OH– ions.
KOH
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K+ + OH–
The water now contains an excess of hydroxide ions, and is considered basic.
As shown above, the concentrations of H+ and OH– in water are usually very small, and
therefore are often written in exponential notation. However, a more convenient method of
expressing the concentration of hydrogen ions is by pH. The pH of a solution is defined to be the
negative log of its hydrogen ion concentration. Concentrations of most acid or base
solutions are generally much less than one; therefore the concentration, when expressed
exponentially, carries a negative exponent (e.g. 1 x 10–7). The letter "p" is used as an
abbreviation for "–log", and is meant to express the concentration as a power of ten. Thus the pH
of pure water would be:
[H+] = 1 x 10–7
pH = –log[H+] = –log(1 x 10–7)
pH = –(–7) Y pH = 7
As the concentration of H+ increases, e.g. [H+] = 1 x 10–2, the pH decreases.
pH = –log[H+] Y pH = –log[1 x 10–2]
pH = –(–2) Y pH = 2
The pH of an acidic solution will always be less than 7.
As the concentration of OH– increases, e.g. [OH–] = 1 x 10–3, the hydrogen ion concentration must
decrease. How much?
Kw = [H+]*[OH–]
1 x 10–14 = [H+]*[1 x 10–3]
1 x 10–11 = [H+]
pH = –log(1 x 10–11)
pH = –(–11) Y pH = 11
The pH of a basic solution will always be greater than 7.
A solution that has a pH between 7 and 9 is slightly basic, one with a pH between 9 and 11 is
moderately basic, and one with a pH between 12 and 14 is strongly basic.
REFERENCE TABLES
In any water solution, the hydrogen ion and hydroxide ion concentrations are inversely
proportional. The concentrations of these ions are regulated by the expression:
Kw = [H+]*[OH–], where Kw = 1x 10–14.
pH is an expression of the hydrogen ion concentration with regard to its power of ten:
pH = –log[H+]
A neutral solution contains equal amounts of H+ and OH–. At 25EC, they are both
1 x 10–7.
[H+] = [OH–] = 1 x 10–7 Y pH = 7
All acidic solutions contain a relative excess of H+ ions.
[H+] > 1 x 10–7 Y [OH–] < 1 x 10–7
Therefore, the pH of an acidic solution must be below 7.
All basic solutions contain a relative excess of OH– ions.
[H+] < 1 x 10–7 Y [OH–] > 1 x 10–7
Therefore, the pH of a basic solution must be above 7.
<------------------increasing acidity-----neutral-----increasing basicity------------------>
[H+]
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6.4 7.0 7.4 7.8 8.1
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11.5
13.0
The pH scale is logarithmic: each unit of change in pH represents a tenfold change in the
concentration of acid or base!
EXAMPLES
Example 1: Calculate the concentration of H+ ions in wine.
pH = 3.3 = –(–3.3) = –log(1 x 10–3.3) | H+ = 1 x 10–3.3
Example 2: Which is more acidic, wine or seawater?
Wine is more acidic than seawater, since it is to the left of seawater on the pH table.
Seawater is more basic than wine, since it is to the right of wine on the pH table.
Example 3: How many times more acidic is wine than acid rain?
Concentration of H+ ions in wine = 1 x 10 – 3.3 and
concentration of H+ ions in acid rain = 1 x 10 – 4.2
1 x 10 – 3.3
1 x 10
– 4.2
= 1 x 10 0.9 . 7.9 times as acidic
Example 4: How many times more basic is pure water than vinegar?
Since the pH value increases by 4 units, the basicity increases by
1 x 10 4 = 10,000 times as basic.
Example 5: Calculate the concentration of OH– ions in household ammonia.
pH = 11.5 | concentration of H+ = 1 x 10 – 11.5
Since Kw = [H+] ( [OH–], where Kw = 1 x 10 – 14, and we know [H+] = 1 x 10 – 11.5,
1 x 10 – 14
= 1 x 10 – 2.5 .
then 1 x 10 – 14 = 1 x 10 – 11.5 ( [OH–] | [OH–] =
– 11.5
1 x 10
In one of the exercises you will be asked to prove that 14 = pH + pOH, a commonly
used shortcut. Since the pH of household ammonia is given as 11.5, its pOH must
be 2.5, which implies that [OH–] = 1 x 10 – 2.5.
Example 6: Find the pH of a substance which has a hydrogen ion concentration of 4.1 x 10 – 2.
pH = – log[H+] = – log(4.1 x 10 – 2) = –(log 4.1 + log 10 – 2) . 1.4
PROJECT
1. Calculate the concentration of H+ ions in stomach acid.
2. What is the concentration of H+ ions in acid rain?
3. What is the concentration of H+ ions in oven cleaner?
4. If the pH of a substance is known to be n, what will be the concentration of H+ ions in the
substance?
5. How many times more acidic is acid rain than normal rain?
6. How many times more acidic is battery acid than orange juice?
7. Which substance is 100 times as acidic as cola:
8. What would be the pH of a substance that is 1000 times less acidic than lemon juice?
9. Calculate the concentration of OH– ions in an egg.
10. Find the concentration of OH– ions in seawater.
11. How many times more basic is seawater than normal rain?
12. Which substance is 10,000 times as basic as baking soda?
13. What would be the pH of substance that is 10 times less basic than beer?
14. Use logarithms and the identity Kw = [H+]*[OH–], where Kw = 1x 10–14, to derive the expression
14 = pH + pOH.
15. Use the above shortcut to complete this table.
[ H+ ]
16.
[ OH– ]
[ H+ ]
milk
oven cleaner
normal rain
cola
blood
stomach acid
A typical tomato has a hydrogen ion concentration of 3.2 x 10 – 4.
Is this acidic or basic?
What is its pH?
What is the concentration of hydroxide ions present?
Judy Ahrens/Jerry Burns
Pellissippi State Technical Community College
September 1997
[ OH– ]