Magnitude of Work Done in Isothermal Gas Compression

To find the magnitude of work done on the gas during the isothermal compression, we analyze the given parameters and relationships involving the bulk modulus and volume change.

1. Given Data:

• Initial volume of the gas, $V_0 = 3\text{ litres} = 3 \times 10^{-3}\text{ m}^3$
• Final volume of the gas, $V_f = \frac{1}{3}V_0 = 1\text{ litre} = 1 \times 10^{-3}\text{ m}^3$
• Bulk modulus of the gas, $B = 3 \times 10^5\text{ N/m}^2$

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2. Calculation of Volume Change:

The change in volume ($\Delta V$) during the compression process is: $\Delta V = V_0 - V_f = V_0 - \frac{1}{3}V_0 = \frac{2}{3}V_0$

Substituting the value of $V_0$: $\Delta V = \frac{2}{3} \times (3 \times 10^{-3}\text{ m}^3) = 2 \times 10^{-3}\text{ m}^3$

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3. Work Done Calculation:

The bulk modulus $B$ of a substance represents the ratio of volumetric stress to volumetric strain. The magnitude of work done ($W$) in compressing the gas by a volume change $\Delta V$ under the bulk modulus $B$ is given by: $W = B \cdot \Delta V$

Substituting the given values into the formula: $W = (3 \times 10^5\text{ N/m}^2) \times (2 \times 10^{-3}\text{ m}^3)$

$W = 600\text{ J}$

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Final Answer:

The magnitude of work done on the gas is 600 J.