In a certain thermodynamic process, the pressure of a gas depends on its volume as $k V^{3}$ . The work done when the temperature changes from $100 ° C$ to $300 ° C$ will be $________n R$ , where $n$ denotes number of moles of a gas.

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Solution

Step 1: Given
Pressure of gas: $P = k V^{3}$
Change in temperature: $∆ T = 300 ° - 100 ° = 200 °$
Step 2: Formula Used
$P V = n R T$
$W o r k = \int P d V$
Step 3: Calculate the work done
Find an expression for $P d V$ by substituting $P = k V^{3}$ in the formula $P V = n R T$ and differentiating it.
$(k V^{3}) V = n R T \Rightarrow \frac{d}{d V} (k V^{4}) = \frac{d}{d V} (n R T) \Rightarrow 4 k V^{3} d V = n R d T \Rightarrow k V^{3} d V = \frac{n R d T}{4} \Rightarrow P d V = \frac{n R d T}{4}$
Calculate the work done by integrating the expression obtained.
$\begin{array}{rcl} W & = & \int P d V \\ = & \int \frac{n R d T}{4} \\ = & \frac{n R}{4} ∆ T \\ = & \frac{n R}{4} \times 200 \\ = & 50 n R \end{array}$
Hence, the work done is $\begin{array}{rcl} 50 n R \end{array}$ .

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