I see answers to similar questions like this and this that confuse me. I feel like they ignore how the expansion of the gases occurs.
Consider two chambers of gases at $ n_{1i}, P_{1i} $ and $ n_{2i}, P_{2i} $ each with constant initial temperature $ T_i $. Put them in adiabatic contact such that the total volume $ V = V_1 + V_2 $ is constant, no heat exchange $ dT_i = 0 $, and the chambers are allowed to expand against each other based on their pressures. Final pressure will be equilibrium $ P_f $
As they expand, they should be subject to Adiabatic relation for P-T,
$$ P_{1i}^{1-\gamma} T_i^\gamma = P_f^{1-\gamma} T_{1f}^\gamma \\ P_{2i}^{1-\gamma} T_i^\gamma = P_f^{1-\gamma} T_{2f}^\gamma \\ $$
Dividing those, you can get a relation, $$ T_{1f} = \left(\frac{P_{1i}}{P_{2i}}\right)^{(1 - \gamma)/\gamma} T_{2f} $$
Only now do gas mixing according to the formulas in the linked posts,
$$\begin{align} T_{mixed} &= \frac{n_1 \left(\frac{P_{1i}}{P_{2i}}\right)^{(1 - \gamma)/\gamma} + n_2}{n_1 + n_2} T_{2f} \\ &= \frac{n_1 \left(\frac{P_{1i}}{P_f}\right)^{(1 - \gamma)/\gamma} + n_2 \left(\frac{P_{2i}}{P_f}\right)^{(1 - \gamma)/\gamma}}{n_1 + n_2} T_{i} \\ \end{align}$$
$ \gamma = 7/5 $ for diatomic, so $ (1 - \gamma)/\gamma < 0 $ implying that we would expect cooling $ T_{mixed} < T_i $ if combining a low pressure gas with a high pressure gas ($ P_{2i} > P_{1i} $) in equal volumes (implying $ n_2 > n_1 $) which intuitively makes sense.
Am I wrong? If I am correct, then this would imply that this is either incorrect or incomplete because they predict no temperature change.
