Does the acidity constant depend on the solvent?

Question

Today I came across $pK_a$ values that depend on the solvent. For the hypothetical reaction $HA \rightarrow H^+ + A^-$, I had in mind that the acidity constant is defined simply as $K_a = \frac{\{H^+\}\{A^-\}}{\{HA\}}$. Therefore, no reference to the solvent seems to appear.

Of course, you wouldn't find protons by themselves, but would it still be a "pure" acidity constant if you used $H_3O^+$ and $H_2O$? In this case you would then have $HA + H_2O\rightarrow H_3O^+ + A^-$. As water is a liquid $\{H_2O\} = 1$. Thus, for the constants to be equal $\{H^+\} = \{H_3O^+\}$. But that should not hold for all solvents since different values come up, right?

Answer 1

The hypothetical reaction you're referencing is an over-simplified version of the actual reaction that is taking place. In strict terms, the hydronium ion does no exist as "$[H^+]$", but rather as "$[H_3O^+]$". The complete reaction (if the solvent is water) would be:

$$\ce{HA(aq) + H_2O(l) <=> H_3O^+(aq) + A^-(aq)}$$

And the equilibrium constant for this reaction would be:

$$K_{eq}=\frac{[H_3O^+][A^-]}{[HA][H_2O]}$$

However, because the solvent (in this case, water) is in excess, its concentration is considered a constant value. This means $[H_2O]$ can be grouped up with $K_{eq}$ (another constant value):

$$[H_2O]*K_{eq}=\frac{[H_3O^+][A^-]}{[HA]}$$

This product is defined as:

$$K_a=[H_2O]*K_{eq}$$

So we have:

$$K_a=\frac{[H_3O^+][A^-]}{[HA]}$$

Even though using $[H_2O]=1$ leads to $K_a = K_{eq}$, that's not the case because $K_a ≠ K_{eq}$

The actual value of $[H_2O]$ at 25°C is:

$$[H_2O]=\frac{\rho_{H_2O}}{M_{H2O}}=\frac{1000 g/L}{18 g/mol}=55.56 mol/L$$

So, if we repeat the reaction using a different solvent "$S$":

$$\ce{HA(aq) + S(l) <=> H_3O^+(aq) + A^-(aq)}$$

$$K_{eq}=\frac{[H_3O^+][A^-]}{[HA][S]}$$

$$[S]*K_{eq}=\frac{[H_3O^+][A^-]}{[HA]}$$

$$K_a=[S]*K_{eq}$$

$$[S]=\frac{\rho_{S}}{M_{S}}$$

In other words, $K_a$ is a function of $[S]$

Since "S" in this case isn't water, it follows that:

$$[S]≠[H_2O]$$

In conclusion:

(1) For the same reaction at the same temperature, $K_a$ calculated with water as a solvent is different than $K_a$ calculated with a different solvent.

(2) $K_a$ depends on the solvent used.