We have seen that the \((P, V)\)-relationship during a reversible adiabatic process in an ideal gas is governed by the exponent \(\gamma\), such that

\[
P V^{\gamma}=\text { const. }
\]

Consider a mixture of two ideal gases, with mole fractions \(f_{1}\) and \(f_{2}\) and respective exponents \(\gamma_{1}\) and \(\gamma_{2}\). Show that the effective exponent \(\gamma\) for the mixture is given by

\[
\frac{1}{\gamma-1}=\frac{f_{1}}{\gamma_{1}-1}+\frac{f_{2}}{\gamma_{2}-1}
\]