Question

An ideal gas at ${27}^{o}C$ is compressed adiabatically and reversibly to $\frac{8}{27}$ of its original volume. If $\gamma =\frac{5}{3},$ then the rise in temperature is

• A. $450K$
• B. $375K$
• C. $225K$
• D. $405K$

Answer 1

The correct option is B $375K$

For an adiabatic process
$\frac{T_2}{T_1}={\left(\frac{V_1}{V_2}\right)}^{\gamma -1}$
where, $T_1=$ initial temperature
$T_2=$ final temperature
$V_1=$ initial volume
$V_2=$ final volume $=\frac{8}{27}{V}_1$
$\gamma =\frac{5}{3}$
$\Rightarrow T_2=300{\left(\frac{27}{8}\right)}^{\frac{5}{3}-1}=300{\left(\frac{27}{8}\right)}^{\frac{2}{3}}$
$=300{\left(\frac{3}{2}\right)}^2=300\left(\frac{9}{4}\right)=675K$
therefore rise in tempearture is :
$\Rightarrow △T=675-300=375K$