Page 260 - Theory and Problems of BEGINNING CHEMISTRY

CHAP. 17] ACID-BASE THEORY 249

The value of the equilibrium constant is given by
−3
−3
[C 2 H 3 O 2 ][H 3 O ] (1.64 × 10 )(1.64 × 10 )
+
−
K a = = = 1.82 × 10 −5
[HC 2 H 3 O 2 ] 0.148
EXAMPLE 17.7. Calculate the hydronium ion concentration of a 0.250 M solution of acetic acid, using the equilibrium
constant of Example 17.6.
Ans. In this example, we will use x for the unknown concentration of hydronium ions. We will solve in terms of x.
HC 2 H 3 O 2 + H 2 O ←− C 2 H 3 O 2 + H 3 O +
−→
−
Initial 0.250 0 0
Change −x x x
Equilibrium 0.250 − x x x

[C 2 H 3 O 2 ][H 3 O ] (x)(x) −5
+
−
K a = = = 1.82 × 10
[HC 2 H 3 O 2 ] 0.250 − x
Since the acid is weak, we can solve this equation most easily by assuming that x is small enough to neglect when
added to or subtracted from a greater concentration. Therefore,
(x)(x) −5
K a = = 1.82 × 10
0.250
2
x = 4.55 × 10 −6
x = 2.13 × 10 −3 = [H 3 O ]
+
In cases where x is too large to neglect from the concentration from which it is subtracted, a more exact method is
required. The quadratic formula is used (Appendix).
(x)(x)
K a = = 1.82 × 10 −5
0.250 − x
−5
−5
2
x = 1.82 × 10 (0.250 − x) = 4.55 × 10 −6 − 1.82 × 10 x
−5
2
x + 1.82 × 10 x − 4.55 × 10 −6 = 0
−5 2
−6
−1.82 × 10 −5 + (1.82 × 10 ) − 4(−4.55 × 10 )
x =
2
x = 2.12 × 10 −3
In this case, the approximate solution gives almost the same answer as the exact solution. In general, you should
use the approximate method and check your answer to see that it is reasonable; only if it is not should you use the
quadratic equation.

17.4. AUTOIONIZATION OF WATER
Since water is defined as both an acid and a base (Sec. 17.2), it is not surprising to find that water can react
with itself, even though only to a very limited extent, in a reaction called autoionization:
−→
+
H 2 O + H 2 O ←− H 3 O + OH −
acid base acid base
An equilibrium constant for this reaction, called K w , does not have terms for the concentration of water; otherwise
it is like the other equilibrium constants considered so far.
K w = [H 3 O ][OH ]
+
−
The value for this constant in dilute aqueous solution at 25 Cis1.0 × 10 −14 . Thus, water ionizes very little
◦
when it is pure, and even less in acidic or basic solution. You must remember this value.
The equation for K w indicates that there is always some H 3 O and always some OH in any aqueous
−
+
solution. Their concentrations are inversely proportional. A solution is acidic if the H 3 O concentration exceeds
+