In the following P-V diagram, two adiabatics cut two isothermals at temperatures ${\mathrm{T}}_1 \mathrm{and} {\mathrm{T}}_2$ (fig.). The value of  $\frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{d}}}$ Will be

For adiabatic process  $T_1{V}_b^{\gamma -1}=$ Constant
For bc curve  or   $T_1{V}_b^{\gamma -1}=T_2{V}_c^{\gamma -1}\text{or}\frac{T_2}{T_1}={\left(\frac{V_b}{V_c}\right)}^{\gamma -1}$          …..(i)
For ad curve   or  ${\mathrm{T}}_1{\mathrm{V}}_{\mathrm{a}}^{\mathrm{\gamma }-1}={\mathrm{T}}_2{\mathrm{V}}_{\mathrm{d}}^{\mathrm{\gamma }-1}\text{or}\frac{{\mathrm{T}}_2}{{\mathrm{T}}_1}={\left(\frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{d}}}\right)}^{\mathrm{\gamma }-1}$          …..(ii)
From equation (i) and (ii)  $\frac{V_b}{V_c}=\frac{V_a}{V_d}$