Question

An ideal monoatomic gas at 27${}^{\circ }$C is compressed adiabatically to $\frac{8}{27}$ times the present volume. The increase in temperature of the gas is

• A. 175${}^{\circ }$ C
• B. 475${}^{\circ }$ C
• C. 375${}^{\circ }$C
• D. 402${}^{\circ }$ C

Answer 1

Answer: (C)

Let the initial volume of the gas be $V$ i.e $V_1=V$
Thus final volume of the gas $V_2=\frac{8}{27}V$
Given: $T_1=27^{\circ }C=300K$
For monoatomic gas, $\gamma =\frac{5}{3}$
Using $T{V}^{\gamma }=$ constant
$<br/>\Rightarrow \frac{T_2}{T_1}={\left(\frac{V_1}{V-2}\right)}^{\gamma -1}<br/>$
$<br/>\therefore \frac{T_2}{300}={\left(\frac{27}{8}\right)}^{\frac{5}{3}-1}<br/>$
$<br/>\begin{array}{l}<br/>\text{ OR }\frac{T_2}{300}={\left(\frac{27}{8}\right)}^{\frac{2}{3}}={\left(\frac{3}{2}\right)}^2\Rightarrow T_2=300\times 2.25=675K\\ <br/>⟹T_2=675-273=402^{\circ }C<br/>\end{array}<br/>$
Thus increase in temperature $\Delta T=T_2-T_1=402-27=375^{\circ }C$