QJEEM24Phbf47: JEE Main 2024 Physics - Class 11 JEEM24Phbf47 Solution: The pressure and volume of an ideal gas are related as…

To find the work done by the gas when it goes from state A to state B, we can look at the definition of work done on or by a gas in a thermodynamic process. For a quasi-static process, the work done $W$ is given by the integral of the pressure $P$ with respect to the volume $V$ :

$W = \int_{V_1}^{V_2} P dV$

Given the relationship $PV^{\frac{3}{2}} = K$ , we can solve for $P$ :

$P = \frac{K}{V^{\frac{3}{2}}}$

Now, substitute $P$ into the work integral and evaluate it:

$W = \int_{V_1}^{V_2} \frac{K}{V^{\frac{3}{2}}} dV = K \int_{V_1}^{V_2} V^{-\frac{3}{2}} dV$

To integrate this, we'll use the power rule for integration. The integral of $V^{-\frac{3}{2}}$ is $-2V^{-\frac{1}{2}}$ , so the work done is:

$W = K \left[-2V^{-\frac{1}{2}}\right]_{V_1}^{V_2} = K \left(-2V_2^{-\frac{1}{2}} + 2V_1^{-\frac{1}{2}} \right)$

We can rewrite $V^{-\frac{1}{2}}$ as $\frac{1}{\sqrt{V}}$ :

$W = K \left(-2\frac{1}{\sqrt{V_2}} + 2\frac{1}{\sqrt{V_1}} \right)$

Since $P_2V_2^{\frac{3}{2}} = K$ and $P_1V_1^{\frac{3}{2}} = K$ , we can express $K$ in terms of $P_1$ and $V_1$ (or $P_2$ and $V_2$ , but we’ll use $P_1$ and $V_1$ for now):

$W = P_1V_1^{\frac{3}{2}} \left(-2\frac{1}{\sqrt{V_2}} + 2\frac{1}{\sqrt{V_1}} \right)$

$W = -2P_1V_1\sqrt{V_1}\frac{1}{\sqrt{V_2}} + 2P_1V_1\sqrt{V_1}\frac{1}{\sqrt{V_1}}$

$W = -2P_1V_1\frac{\sqrt{V_1}}{\sqrt{V_2}} + 2P_1V_1$

Since $\frac{\sqrt{V_1}}{\sqrt{V_2}} = \sqrt{\frac{V_1}{V_2}}$ , and recalling $P_2V_2^{\frac{3}{2}} = K$ once more:

$W = -2P_1V_1\sqrt{\frac{V_1}{V_2}} + 2P_1V_1$

$W = -2 \frac{P_1V_1^{\frac{3}{2}}}{\sqrt{V_2}} + 2P_1V_1$

Putting $P_1V_1^{\frac{3}{2}}$ back as $K$ :

$W = -2\frac{K}{\sqrt{V_2}} + 2P_1V_1$

$W = -2P_2V_2 \sqrt{V_2}\frac{1}{\sqrt{V_2}} + 2P_1V_1$

$W = -2P_2V_2 + 2P_1V_1$

So the work done is:

$W = 2P_1V_1 - 2P_2V_2$

Therefore, the correct answer matching the given options is:

Option D: $2\left(\mathrm{P}_1 \mathrm{~V}_1-\mathrm{P}_2 \mathrm{~V}_2\right)$

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