JEE Main 2026 — Thermodynamics Question with Solution

For an isothermal process, the isothermal bulk modulus equals the gas pressure:
$B_T=-V\frac{dP}{dV}$

Since $PV=\text{constant}$, differentiating gives $PdV+VdP=0$, so $\frac{dP}{dV}=-\frac{P}{V}$, which yields $B_T=P$.

Therefore, the initial pressure is:
$P_o=3\times 10^5{\text{ N/m}}^2$

Volumes:
$V_o=3\text{ L}=3\times 10^{-3}{\text{ m}}^3$
$V_f=\frac{1}{3}{V}_o=10^{-3}{\text{ m}}^3$

Work done on the gas during an isothermal compression:
$W_{\text{on}}=-W_{\text{by}}=-P_o{V}_o\ln \left(\frac{V_f}{V_o}\right)=P_o{V}_o\ln \left(\frac{V_o}{V_f}\right)$

$W_{\text{on}}=(3\times 10^5)(3\times 10^{-3})\ln 3$

$W_{\text{on}}=900\ln 3\text{ J}$

Using $\ln 3\approx 1.0986$:
$W_{\text{on}}\approx 900\times 1.0986\approx 988.74\text{ J}$

Rounded to the nearest integer:
$W_{\text{on}}\approx 989\text{ J}$

Hence, the work done on the gas is approximately $989\text{ J}$.