7.5 Buffer Solutions
333
We let m eq (OH − )/m ◦ y and m eq (H + )/m ◦ x.
K a
γ 2 x
m s /m ◦ + x − y
m a /m ◦ − x + y
When y is replaced by K w /γ 2 x, we obtain the equation
γ 2 x 3 + (γ 2 m s /m ◦ + K)x 2 − (K w + Km a /m ◦ ) x − K a K w /γ 2 0
(7.5-6)
The ionic strength of the solution is equal to m s + x, so that the activity coefficient γ
can be estimated by setting I m s in the Davies equation. The cubic equation is solved
numerically and the result is that x 2.13 × 10 −5 , γ 0.740, a(H + ) 1.576 × 10 −5 ,
and pH 4.802. The pH value differs from that in part a by 0.130. If the activity coefficient
is included but x and y are neglected compared with m s and m a the resulting pH is 4.800.
The most significant source of error in the Henderson–Hasselbalch equation is the neglect
of the activity coefficient, not the neglect of ionization and hydrolysis.
Exercise 7.13
a. Verify Eq. (7.5-6).
b. Substitute the numerical result of part b of Example 7.16 into the equilibrium expression,
Eq. (7.5-3), to verify its correctness.
c. Carry out the solution of Example 7.16 including the activity coefficient but neglecting x and
y compared with m a and m s .
Biological Buffering
The principal buffering system in mammalian blood is that of carbonic acid (H 2 CO 3 ).
We have the following equilibria:
CO 2 (g) CO 2 (aq)
(7.5-7a)
CO 2 (aq) + H 2 O H 2 CO 3 (aq)
(7.5-7b)
H 2 CO 3 (aq) H
+
+ HCO
−
3
(7.5-7c)
HCO
−
3 H
+
+ CO
2−
3
(7.5-7d)
E X A M P L E 7.17
The Henry’s law constant for CO 2 in water at 25 ◦ C is equal to 1.24 × 10 6 torr. The equilibrium constant for reaction (7.5-7b) is equal to 2.58 × 10 −3 , that for reaction (7.5-7c)
is equal to 1.70 × 10 −4 , and that for reaction in Eq. (7.5-7d) is equal to 4.69 × 10 −11 . It
is not possible by chemical analysis to distinguish CO 2 (aq) from H 2 CO 3 , but it has been
determined that the molality of CO 2 (aq) is larger than that of H 2 CO 3 (aq). Find the value
of the first ionization constant of carbonic acid, using the combined molalities of CO 2 (aq)
and H 2 CO 3 (aq) in the equilibrium expression instead of the molality of H 2 CO 3 (aq). Explain
why reaction in Eq. (7.5-7c) can act as an effective buffering reaction near the normal pH of
blood, around pH 7.4.
333
We let m eq (OH − )/m ◦ y and m eq (H + )/m ◦ x.
K a
γ 2 x
m s /m ◦ + x − y
m a /m ◦ − x + y
When y is replaced by K w /γ 2 x, we obtain the equation
γ 2 x 3 + (γ 2 m s /m ◦ + K)x 2 − (K w + Km a /m ◦ ) x − K a K w /γ 2 0
(7.5-6)
The ionic strength of the solution is equal to m s + x, so that the activity coefficient γ
can be estimated by setting I m s in the Davies equation. The cubic equation is solved
numerically and the result is that x 2.13 × 10 −5 , γ 0.740, a(H + ) 1.576 × 10 −5 ,
and pH 4.802. The pH value differs from that in part a by 0.130. If the activity coefficient
is included but x and y are neglected compared with m s and m a the resulting pH is 4.800.
The most significant source of error in the Henderson–Hasselbalch equation is the neglect
of the activity coefficient, not the neglect of ionization and hydrolysis.
Exercise 7.13
a. Verify Eq. (7.5-6).
b. Substitute the numerical result of part b of Example 7.16 into the equilibrium expression,
Eq. (7.5-3), to verify its correctness.
c. Carry out the solution of Example 7.16 including the activity coefficient but neglecting x and
y compared with m a and m s .
Biological Buffering
The principal buffering system in mammalian blood is that of carbonic acid (H 2 CO 3 ).
We have the following equilibria:
CO 2 (g) CO 2 (aq)
(7.5-7a)
CO 2 (aq) + H 2 O H 2 CO 3 (aq)
(7.5-7b)
H 2 CO 3 (aq) H
+
+ HCO
−
3
(7.5-7c)
HCO
−
3 H
+
+ CO
2−
3
(7.5-7d)
E X A M P L E 7.17
The Henry’s law constant for CO 2 in water at 25 ◦ C is equal to 1.24 × 10 6 torr. The equilibrium constant for reaction (7.5-7b) is equal to 2.58 × 10 −3 , that for reaction (7.5-7c)
is equal to 1.70 × 10 −4 , and that for reaction in Eq. (7.5-7d) is equal to 4.69 × 10 −11 . It
is not possible by chemical analysis to distinguish CO 2 (aq) from H 2 CO 3 , but it has been
determined that the molality of CO 2 (aq) is larger than that of H 2 CO 3 (aq). Find the value
of the first ionization constant of carbonic acid, using the combined molalities of CO 2 (aq)
and H 2 CO 3 (aq) in the equilibrium expression instead of the molality of H 2 CO 3 (aq). Explain
why reaction in Eq. (7.5-7c) can act as an effective buffering reaction near the normal pH of
blood, around pH 7.4.