The temperature T of one mole of an ideal gas varies with its volume V as T - - V 3 - - V 2- where - and - are positive constants- The maximum pressure of gas during this process is

The correct option is D β^ 2 R 4α T= α V^3+β V^2 PVR= α #10;V^2+β V^2 P=RV( α V+β) #10; #10;For maximum pressure #10; #10; d ...

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Standard XII

Physics

Ideal Gas Equation

The temperatu...

Question

The temperature (T) of one mole of an ideal gas varies with its volume (V) as $T = - α V^{3} + β V^{2}$ , where $α$ and $β$ are positive constants. The maximum pressure of gas during this process is

\frac{α β}{2 R}

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\frac{β^{2} R}{4 α}

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\frac{(α + β) R}{{2 β}^{2}}

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\frac{α^{2} R}{2 β}

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Solution

The correct option is D $\frac{β^{2} R}{4 α}$
$T = - α V^{3} + β V^{2} \frac{P V}{R} = - α V^{2} + β V^{2} P = R V (- α V + β)$

For maximum pressure

$\frac{d P}{d V} = 0 \Rightarrow - 2 α V + β = 0 \Rightarrow V = \frac{β}{2 α} P = R [- α (\frac{β}{2 α})^{2} + \frac{β}{2 α} \times β] P = R [\frac{- β^{2}}{4 α} + \frac{β^{2}}{2 α}] P = \frac{R β^{2}}{4 α}$

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