Question

One mole of an ideal gas $[C_{v,m}=\frac{5}{2}R]$ at $300$ K and $5$ atm is expanded adiabatically to a final pressure of $2$ atm against a constant pressure of $2$ atm. The final temperature (in Kelvin) of the gas is:

• A. $270.2$
• B. $273.0$
• C. $248.5$
• D. $200.5$

Answer 1

Answer: (C)

$248.5$

Since the process is an adiabatic processs, $q=0$ and hence, $\Delta U=w$

$n{C}_{v,m}(T_2-T_1)=-P_{ext}[\frac{nR{T}_2}{P_2}-\frac{nR{T}_1}{P_1}]$

For one mole, we have $C_{v,m}(T_2-T_1)=P_{ext}R[\frac{T_1}{P_1}-\frac{T_2}{P_2}]$

$\frac{5}{2}R(T_2-300)=2\times R[\frac{300}{5}-\frac{T_2}{2}]$ $⟹T_2\approx 248.5K$

Hence, the final temperature of the gas is $248.5K$.