# Finding final temperature for adiabatic expansion

An ideal gas, molar heat capacity $C_v=\frac{5}{2}R,$ is expanded adiabatically against a constant pressure of 1 atm until it doubles in volume. Given initial temperature of 25$^o$C and initial pressure of 5 atm, calculate final T (etc.).*

The author uses the First Law to calculate $T_2$ as 274K.

My question is, why can't we use the relation $T_1V_1^{\gamma-1}=T_2V_2^{\gamma-1}?$

It appears we cannot, since $\gamma =C_p/C_v$ and so, since $V_2=2V_1,T_2=274,T_1=298,$

$$log_2\frac{T_1}{T_2}+1 =\gamma = \frac{C_p}{5/2}\rightarrow C_p\approx 2.8,$$

so assuming $C_p - C_v = R,$ giving $C_p = \frac{7}{2}R,$ does not seem to work.

Can someone explain why the highlighted equation doesn't apply?

*This is adapted from an example in Castellan, Physical Chemistry.