# 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 $$1\ \mathrm{atm}$$ until it doubles in volume. Given initial temperature of $$25\ \mathrm{^\circ C}$$ and initial pressure of $$5\ \mathrm{atm}$$, calculate final $$T$$ (etc.).*

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

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 $$V_2=2V_1$$, $$T_2=274\ \mathrm K$$, $$T_1=298\ \mathrm K$$,

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

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.