Question

Figure illustrates a cycle conducted with $n$ moles of an ideal gas. In the states $a$ and $b$ the gas temperatures are $T_a$ and $T_b$ respectively. Temperature of the gas in the state $c$ is

• A. $\sqrt{T_a{T}_b}$
• B. $T_a+T_b$
• C. $\frac{T_a{T}_b}{T_a+T_b}$
• D. $\frac{(T_a+T_b)}{2}$

Answer 1

The correct option is A $\sqrt{T_a{T}_b}$

From ideal gas equation, we can write,

$PV=nRT$

$\Rightarrow$$T=\frac{PV}{nR}$

For the points $\text{a}$ and $\text{b}$, as shown in figure

$T_a=\frac{P_0{V}_0}{nR},T_b=\frac{x^2{P}_0{V}_0}{nR}$

$x=\sqrt{\frac{T_b}{T_a}}$

Now, temprature at $C$,

$T_c=\frac{x{P}_0{V}_0}{nR}=\left(\sqrt{\frac{T_b}{T_a}}\right)T_a=\sqrt{T_a{T}_b}$

$\begin{array}{c}\text{Why this question}?\\ \text{This question tests the application}\\ \text{of thermodynamic laws on a P-V graph and}\\ \text{interpretation of cyclic process.}\end{array}$