Question

One mole of an ideal monoatomic gas is taken round the cyclic process ABCA as shown in figure. Calculate the maximum temperature attained by the gas during the cycle.

Answer 1

The maximum temperature attained in cycle is along BC curve

Explanation of BC

$\Rightarrow P=-\left[\frac{2P_0}{V_0}\right]V+5P_0⟶\left(1\right)$

Where $\frac{2P_0}{V_0}$ id slope of BC $5P_0$ is intercept $y-axis$

Putting $P=\frac{RT}{V}$ in equation $\left(1\right)$

$\Rightarrow RT=-\frac{2P_0}{V_0}{V}^2+5P_{0}V⟶\left(2\right)$

For maximum $\frac{dt}{dv}=0$

$\Rightarrow 0=-\frac{2P_0}{V_0}\times 2V+5P_0$

$\Rightarrow V=\frac{5P_0}{4}⟶\left(3\right)$

Putting equation $\left(3\right)$ in equation $\left(2\right)$

$\Rightarrow R{T}_{max}=-\frac{2P_0}{V_0}+{\left[\frac{5V_0}{4}\right]}^2+5P_0\left[\frac{5V_0}{4}\right]$

$=-2P_0{V}_0\times \frac{25}{16}+\frac{25P_0{V}_0}{4}=\frac{25P_0{V}_0}{8}$

$\Rightarrow T_{max}=\frac{25}{8}\frac{P_0{V}_0}{R}$

Hence, the answer is $\frac{25}{8}\frac{P_0{V}_0}{R}.$