Question

In the following $P-V$ diagram of an ideal gas, $BC\&DA$ are adiabatic curves,while $AB$ and $DC$ are isotherms at $T_1\&T_2$ respectively.
The value of $V_B/V_C$ is

• A. $=V_A/V_D$
• B. $<V_A/V_D$
• C. $>V_A/V_D$
• D. Cannot say

Answer 1

The correct option is A $=V_A/V_D$

For adiabatic process -
$P{V}^{\gamma }=\text{constant}.......\left(i\right)$
From the ideal gas equation,
$PV=nRT$
$P=\frac{nRT}{V}$
Put the value of $P$ in equation (i)
$\left[\frac{nRT}{V}\right]V^{\gamma }=\text{constant}$
$T{V}^{\gamma -1}=\text{constant}$
For $BC$
$\left[\frac{T_2}{T_1}\right]={\left[\frac{V_B}{V_C}\right]}^{\gamma -1}$
For $AD$
$\left[\frac{T_2}{T_1}\right]={\left[\frac{V_A}{V_D}\right]}^{\gamma -1}$

$\Rightarrow \frac{V_B}{V_C}=\frac{V_A}{V_D}$