Question

One mole of an ideal gas at temperature $T_1$ expands according to the law $\left(\frac{P}{V}\right)=\text{constant}$. Find the work done when the final temperature becomes $T_2$.

• A. $R(T_2-T_1)$
• B. $\frac{R}{2}(T_2-T_1)$
• C. $\left(\frac{R}{4}\right)(T_2-T_1)$
• D. $PV(T_2-T_1)$

Answer 1

The correct option is B $\frac{R}{2}(T_2-T_1)$

Work done by gas is given as,
$W={\int }_{V_1}^{V_2}PdV={\int }_{V_1}^{V_2}kVdV$
$(\because \frac{P}{V}=k=\text{constant})$
$\Rightarrow W=\frac{1}{2}k(V_2^2-V_1^2)...\left(i\right)$
Applying ideal gas equation for $1\text{mol}$,
$PV=RT$
$\&P=kV$
$\Rightarrow k{V}^2=RT...\left(ii\right)$
From Eq.$\left(i\right)\&\left(ii\right)$ we can write,
$\Rightarrow k(V_2^2-V_1^2)=R{T}_2-R{T}_1=R(T_2-T_1)$
Substituting this value in equation $\left(i\right)$,
$\Rightarrow W=\frac{R}{2}(T_2-T_1)$