Question

A given ideal gas with $\gamma = \frac{C_{p}}{C_{v}} = 1.5$ at a temperature $T$. If the gas is compressed adiabatically to one-fourth of its initial volume, the final temperature will be

• A. $2\sqrt{2}T$
• B. $4 T$
• C. $2 T$
• D. $8 T$

Answer 1

Answer: (C)

$TV^{\gamma-1} =$ constant
$T_{1}V_{1}^{\gamma-1} = T_{2}V_{2}^{\gamma-1}$
$\Rightarrow T \left(V\right)^{1/2} = T_{2} \left(\frac{V}{4}\right)^{1/2}$
$\left[\because \gamma = 1.5, T_{1}^{ }=T, V_{1} = V \,and \, V_{2} = \frac{V}{4}\right]$
$\therefore \quad T_{2} = \left(\frac{4V}{V}\right)^{1/2} T = 2T$