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Q.

The pressure of one mole of an ideal gas varies according to the equation ${p=p}_{0} {-aV}^{2}$ , where $p_{0}$ and a are positive constants. The maximum temperature attained by the gas during the process is

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a

$\frac{{3p}_{0}}{2R} {(\frac{p_{0}}{3a})}^{\frac{1}{2}}$

b

$\frac{p_{0}}{R} {(\frac{p_{0}}{3a})}^{\frac{1}{2}}$

c

$\frac{{2p}_{0}}{R} {(\frac{p_{0}}{3a})}^{\frac{1}{2}}$

d

$\frac{{2p}_{0}}{3R} {(\frac{p_{0}}{3a})}^{\frac{1}{2}}$

answer is D.

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Detailed Solution

$P V = n R T$
$(P_{0} - a V^{2}) V = n R T$
On differentiation, we get
$(P_{0} - 3 a V^{2}) = n R \frac{d T}{d V}$
$\frac{d T}{d V} = 0 & \frac{d^{2} T}{d V^{2}} = - v e$
$0 = \frac{d T}{d V} \Rightarrow 3 a V^{2} = P_{0}$ or $V = \sqrt{\frac{P_{0}}{3 a}}$
$∴ T_{\max} = \frac{2 P_{0}}{3 R} \sqrt{\frac{P_{0}}{3 a}}$

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