Question

For an organic monoprotic acid solution of concentration $C_o$ $mol$ ${litre}^{-1}$, if $K_a$ has a value comparable to $K_w$, show that the hydronium ion concentration is given by:
$\left[H^{+}\right]==[\frac{K_w}{\left[H^{+}\right]}+\frac{K_a\cdot Co}{[K_a+H^{+}]}]$
If $\left[H^{+}\right]={10}^{-3}M$ and $C_o={10}^{-1}M$ in a solution of some organic monoportic acid. According to the above equation, the order of magnitude of $K_a$ must be ${10}^{-X}$, the value of $X$ should be:

Answer 1

$\left[H^{+}\right]=[\frac{K_w}{\left[H^{+}\right]}+\frac{K_a\cdot Co}{[K_a+H^{+}]}]$
Substitute values in the above expression.
${10}^{-3}=\frac{{10}^{-14}}{{10}^{-3}}+\frac{K_a\times {10}^{-1}}{K_a+{10}^{-3}}$
$\frac{K_a}{K_a+{10}^{-3}}={10}^{-4}$
$1+\frac{{10}^{-3}}{K_a}={10}^4$
$K_a={10}^{-7}={10}^{-X}$
$⟹X=7$