Eliminate
seems to work well. Stealing @Wolfgang's expressions:
eqs = DeltaQ == ((p1 Cp)/R) (V2 - V1) + ((V2 CV)/R) (p2 - p1)
given = {p1 V1^(gamma) == p2 V2^(gamma), gamma == Cp/CV}
If you want to eliminate p2
Eliminate[{eqs}~Join~ given, {p2}] // Solve[#, DeltaQ] & // FullSimplify
{{DeltaQ$\to \frac{\text{p1} \left(\text{Cp} (\text{V2}-\text{V1})+\text{CV} \text{V2} \left(\text{V1}^{\text{gamma}} \text{V2}^{-\text{gamma}}-1\right)\right)}{R}$}}
And if you don't want a expressions on your answers, you can tell Mathematica that they are a variable. Removing gamma from the above expression:
Eliminate[{eqs}~Join~ given, {p2}] // Solve[#, {DeltaQ, gamma}] & // FullSimplify
{{DeltaQ $\to \frac{\text{p1} \left(\text{CV} \text{V2} \left(\text{V1}^{\text{Cp}/\text{CV}} \text{V2}^{-\frac{\text{Cp}}{\text{CV}}}-1\right)+\text{Cp} (\text{V2}-\text{V1})\right)}{R}$}}
In general, if you Eliminate
variables you get simpler answers.
Eliminate[{eqs}~Join~ given, {p2}];
Eliminate[{%}~Join~ given, {gamma}];
% // Solve[#, DeltaQ] &;
% // FullSimplify;
{{Delta$\to \frac{\text{Cp} \text{p1} (\text{V2}-\text{V1})+\text{CV} \text{V2} (\text{p2}-\text{p1})}{R}$}}
Disclaimer: I couldn't make Mathematica prove that those last two equations were equivalent, so there could be an error there!
Stealing again from @Wolfgang's answer, if you want to eliminate V2
:
Eliminate[{eqs}~Join~given, {V2}] // Solve[#, {DeltaQ}] & // FullSimplify
{{DeltaQ$\to \frac{\text{CV} ((\text{gamma}-1) \text{p1}+\text{p2}) \left(\frac{\text{p1} \text{V1}^{\text{gamma}}}{\text{p2}}\right)^{1/\text{gamma}}-\text{CV} \text{gamma} \text{p1} \text{V1}}{R}$}}
Again, do not use those blindly. If you have some numbers to plug and check the equivalence, please use them!