What is the reaction equation for a mixture of trichloroethanoic acid and dimethylpropanoic acid?
TL;DR
Contrary to what the answers/comments have suggested, I would say that no reaction happens here. The acid–base reaction between the two given species is thermodynamically unfeasible, with an equilibrium constant $K \sim 10^{-9}$.


If you want an acid-base reaction to occur between A and B, it's no use comparing the $\mathrm pK_\mathrm a$ values of A and B. That tells you how acidic A and B are, but you're not interested in both of them acting as acids. You're interested in one of them being an acid, and the other being a base. So the relevant quantities are the $\mathrm pK_\mathrm a$ of A and the $\mathrm pK_\mathrm b$ of B!
In this context, let's keep trichloroacetic acid $\ce{Cl3CCOOH}$ as the acid; we already know that it has $\mathrm pK_\mathrm a = 0.65$. The question tells us that for the &quot;base&quot; $\ce{Me3CCO2H}$, its $\mathrm pK_\mathrm a$ is $5.05$. But we're not interested in that value; we're more interested in its $\mathrm pK_\mathrm b$.
To find that out, we need to look for data beyond the question text itself. The veritable Evans tables states that the $\mathrm pK_\mathrm a$ of $\ce{PhC(OH)+OH}$ is $-7.8$. That species is the conjugate acid of $\ce{PhCOOH}$, so the $\mathrm pK_\mathrm b$ of $\ce{PhCOOH}$ is $14 - (-7.8) = 21.8$. Recall that a larger $\mathrm pK_\mathrm b$ means that a species is a weaker base. This gigantic number should already tip you off to the fact that $\ce{PhCOOH}$ is a terrible base. The species in the question, $\ce{Me3CCOOH}$, will likely have a value that's broadly similar.
How do we use these values to find whether an acid-base reaction is plausible or not? Returning to the reaction
$$\ce{AH + B &lt;=&gt; A- + BH+,}$$
we have the equilibrium constant
$$\begin{align}
K &amp;= \frac{[\ce{A-}][\ce{BH+}]}{[\ce{AH}][\ce{B}]} \\
&amp;= \frac{[\ce{A-}][\ce{H3O+}]}{[\ce{AH}]} \cdot \frac{[\ce{BH+}][\ce{OH-}]}{[\ce{B}]} \cdot \frac{1}{[\ce{H3O+}][\ce{OH-}]} \\
&amp;= \frac{K_\mathrm a(\ce{HA}) \cdot K_\mathrm b(\ce{B})}{K_\mathrm w}
\end{align}$$
For our molecules, we know that the relevant quantities are $\mathrm pK_\mathrm a(\ce{HA}) = 0.65$ and $\mathrm pK_\mathrm b(\ce{B}) \approx 21.8$. Putting in $K_\mathrm a = 10^{-0.65}$, $K_\mathrm b = 10^{-21.8}$, and $K_\mathrm w = 10^{-14}$, we get an equilibrium constant of
$$K = \frac{10^{-0.65} \cdot 10^{-21.8}}{10^{-14}} = 3.548 \times 10^{-9}$$
which is so small that it makes very little sense to claim that this reaction &quot;happens&quot;.