Blood Buffers

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In this question [$A$] means the concentration of the chemical $A$ at equilibrium.

For a balanced chemical equation, where $A, B, C$ and $D$ are chemicals in aqueous solution and $a, b, c, d$ are whole numbers,
$$ aA + bB \rightleftharpoons cC + dD+ eH_2O $$
the law of mass action tells us that for a fixed temperature, there is a constant $K$ (called the equilibrium constant) such that
$$
\frac{[C]^c[D]^d}{[A]^a[B]^b} = K
$$
(note the absence of the solvent concentration $[H_2O]$ ).
In the blood, the carbonic-acid-bicarbonate buffer prevents large changes in the pH of the blood. Chemically, it consists of two reactions which are simultaneously in equilibrium


$$H^+ +HCO^-_3+H_2O\rightleftharpoons^{K_1} H_2CO_3+H_2O \rightleftharpoons^{K_2} 2H_2O+CO_2$$
Show that
$$
pH =pK -\log \left(\frac{[CO_2]}{[HCO^-_3]}\right)\quad \mbox{where }K=\frac{1}{K_1K_2}
$$

Think about this equation. It shows that the pH of the blood is dependent on the ratio of the concentrations of $CO_2$ and $HCO^-_3$. These are large in the blood, so small changes in the relative concentrations leads to very small changes in the pH of the blood. They act as a 'buffer' against pH change.


Now, make a new variable $x$ to be the fraction of the buffer in the form of $HCO^-_3$. Thus,
$$
x = \frac{[HCO^-_3]}{[HCO^-_3]+[H_2CO_3]+[CO_2]}
$$
Show that
$$
pH = pK-\log\left(\frac{1}{x}-1 -K_1[H^+]\right)
$$
By taking the value $pK=6.1$ and treating $K_1[H^+]$ as very small, reproduce the titration curve




Extension:
1. Why is it numerically valid to ignore the $[H^+]$ term in the equation giving rise to the graph?
2. With the assumption that $K_1[H^+]=0$, use calculus to show that the second derivative of the pH is zero when $x=0.5$. Graphically, what does this correspond to?