Question

A certain gas is isothermally compressed to $(\frac{1}{3})^{rd}$ of its initial volume ($V_0 = 3$ litre) by applying required pressure. If the bulk modulus of the gas is $3 \times 10^5 \text{ N/m}^2$, the magnitude of work done on the gas is ____ J.

Hint:

Use the fact that for an isothermal process in gases, Bulk Modulus is equal to Pressure ($B=P$). Apply the work done formula for isothermal compression: $W = P_i V_i \ln(V_i/V_f)$.

Answer 1

Correct Answer: 989

The bulk modulus $B$ is defined as $B = -V \frac{dP}{dV}$. For an isothermal process of an ideal gas, $PV = \text{constant}$.
Differentiating $PV = C$ with respect to $V$, we get $P + V \frac{dP}{dV} = 0$, which means $P = -V \frac{dP}{dV}$.
Hence, for an isothermal process, the bulk modulus $B$ equals the pressure $P$.

1. Given bulk modulus $B = 3 \times 10^5 \text{ N/m}^2$, the pressure at the start of compression is $P = 3 \times 10^5 \text{ Pa}$.
2. Work done in an isothermal process is $W = \int_{V_i}^{V_f} P dV = \int_{V_i}^{V_f} \frac{nRT}{V} dV = nRT \ln\left(\frac{V_f}{V_i}\right)$.
3. The magnitude of work done on the gas is $|W| = P_i V_i \ln\left(\frac{V_i}{V_f}\right)$.
4. Plugging in the values:
$P_i = 3 \times 10^5 \text{ Pa}$
$V_i = 3 \text{ L} = 3 \times 10^{-3} \text{ m}^3$
$V_f = 1 \text{ L} = 1 \times 10^{-3} \text{ m}^3$
$|W| = (3 \times 10^5) (3 \times 10^{-3}) \ln\left(\frac{3}{1}\right) = 900 \ln 3$.
5. Calculating the final value:
$|W| = 900 \times 1.0986 = 988.74 \approx 989 \text{ J}$.