Question

Pressure of one mole of an ideal gas varies with volume as per equation $P=b–aV$, where $b$ and $a$ are positive constants and $V<\frac{b}{a}$ always. Find the maximum temperature achievable by the gas during the process $(R$ is the molar gas constant)

• A. $\frac{b^2}{4aR}$
• B. $\frac{b^2}{2aR}$
• C. $\frac{b^2}{aR}$
• D. $\frac{a}{2b^{2}R}$

Answer 1

The correct option is A $\frac{b^2}{4aR}$

Given, $P=b-aV$

$\Rightarrow PV=bV-a{V}^2$

$\Rightarrow RT=bV-a{V}^2[\because PV=nRT;n=1]$

$\Rightarrow T=\frac{bV-a{V}^2}{R}$

For, $T=T_{max}$

$\Rightarrow \frac{dT}{dV}=0$

$\Rightarrow \left(\frac{b-2aV}{R}\right)=0$

$\Rightarrow V=\frac{b}{2a}$

At, $V=\frac{b}{2a}$

$T=\frac{b\times \frac{b}{2a}-a{\left(\frac{b}{2a}\right)}^2}{R}$

$\Rightarrow$ $T=\frac{b^2}{4aR}$

Hence, option $\left(A\right)$ is the correct answer.