= Solution
Assume the added gas mixes throughout the convective column and retain constant <mass absorption coefficient> for the original gas. The original absorber column <mass> is unchanged, so its total effective <optical depth> $D=\tau_\infty$ is unchanged. Under <transparent-gas addition at fixed absorber mass>, the absorber <mass> fraction becomes $q=p_s/p_a$, and consequently $t=Dp/p_a$. The new radiating <pressure> is $p_{{\rm rad},a}\simeq p_a/D$, even though the ratio of surface to radiating <pressure> stays $D$.
The mixture's <mass>-specific constants are
$$
R_a=\frac{p_sR_g+(p_a-p_s)R_t}{p_a},\qquad
c_{pa}=\frac{p_sc_{pg}+(p_a-p_s)c_{pt}}{p_a}.
$$
These <pressure> weights represent column <mass> fractions, not mole fractions. Repeating the adiabatic argument gives
$$
\boxed{T_{s,a}=T_eD^{R_a/c_{pa}}\simeq T_eD^{R_t/c_{pt}}\quad(p_a\gg p_s).}
$$
At fixed $D>1$, this is warmer than the original surface precisely if
$$
\boxed{\frac{R_a}{c_{pa}}>\frac{R_g}{c_{pg}}
\quad\Longleftrightarrow\quad\frac{R_t}{c_{pt}}>\frac{R_g}{c_{pg}}.}
$$
The equivalence follows by cross multiplication of the positive heat capacities; it holds for any positive added <mass>. If the two ratios agree, adding a transparent gas leaves this model's surface <temperature> unchanged. Increased total <pressure> alone is not a new opacity in a model that fixes absorber <mass> and <mass absorption coefficient>. <Pressure broadening> or <collision-induced absorption> could change the answer in a more realistic atmosphere, but neither is part of the stated constant-opacity model.
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