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

In thermodynamics, the isothermal bulk modulus $B$ of an ideal gas is equal to its pressure $P$ at that state.
Given that $B = 3 \times 10^5 \text{ N/m}^2$, we can take the initial pressure $P_0$ to be $3 \times 10^5 \text{ Pa}$.

Step 1: Identify the process and formula.
The process is isothermal compression. For an isothermal process of an ideal gas, the work done on the gas is given by:
$$W = nRT \ln\left(\frac{V_i}{V_f}\right)$$
Since $PV = nRT$, we can substitute $nRT$ with $P_0 V_0$. Thus:
$$W = P_0 V_0 \ln\left(\frac{V_0}{V_f}\right)$$

Step 2: Substitute the known values.
Initial volume $V_0 = 3 \text{ litre} = 3 \times 10^{-3} \text{ m}^3$.
Final volume $V_f = \frac{1}{3} V_0 = 1 \text{ litre} = 1 \times 10^{-3} \text{ m}^3$.
Bulk modulus (Pressure) $P_0 = 3 \times 10^5 \text{ N/m}^2$.

Step 3: Calculate the work.
$$W = (3 \times 10^5) \times (3 \times 10^{-3}) \times \ln\left(\frac{3}{1}\right)$$
$$W = 900 \times \ln(3)$$
Using the value $\ln(3) \approx 1.0986$:
$$W = 900 \times 1.0986 = 988.74 \text{ J}$$
The magnitude of work done rounded to the nearest integer is $989 \text{ J}$.