Two different adiabatic paths for the same gas intersect two isothermal curves as shown P - V in diagram. ​

Question

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in \(\mathrm{P}-\mathrm{V}\) diagram. The relation between the ratio \(\frac{V_{a}}{V_{d}}\) and the ratio \(\frac{\mathrm{V}_{\mathrm{b}}}{\mathrm{V}_{\mathrm{c}}}\) is :  

(1) \(\frac{V_{a}}{V_{d}}=\left(\frac{V_{b}}{V_{c}}\right)^{-1}\)

(2) \(\frac{V_{a}}{V_{d}} \neq \frac{V_{b}}{V_{c}}\)

(3) \(\frac{V_{a}}{V_{d}}=\frac{V_{b}}{V_{c}}\)

(4) \(\frac{V_{a}}{V_{d}}=\left(\frac{V_{b}}{V_{c}}\right)^{2}\)  

Answer 1

Correct option is : (3) \(\frac{V_{a}}{V_{d}}=\frac{V_{b}}{V_{c}}\)  

For adiabatic process

\(\mathrm{TV}^{\gamma-1}=\) constant

\(\mathrm{T}_{\mathrm{a}} \cdot \mathrm{V}_{\mathrm{a}}^{\gamma-1}=\mathrm{T}_{\mathrm{d}} \cdot \mathrm{V}_{\mathrm{d}}^{\gamma-1}\)

\(\left(\frac{V_{a}}{V_{d}}\right)^{\gamma-1}=\frac{T_{d}}{T_{a}}\)

\(\mathrm{T}_{\mathrm{b}} \cdot \mathrm{V}_{\mathrm{b}}^{\gamma-1}=\mathrm{T}_{\mathrm{c}} \cdot \mathrm{V}_{\mathrm{c}}^{\gamma-1}\)

\(\left(\frac{\mathrm{V}_{\mathrm{b}}}{\mathrm{V}_{\mathrm{c}}}\right)^{\gamma-1}=\frac{\mathrm{T}_{\mathrm{c}}}{\mathrm{T}_{\mathrm{b}}}\) 

\( \frac{\mathrm{V}_{\mathrm{a}}}{\mathrm{V}_{\mathrm{d}}}=\frac{\mathrm{V}_{\mathrm{b}}}{\mathrm{V}_{\mathrm{c}}}\)        

\(\left(\begin{array}{r}\because \mathrm{T}_{\mathrm{d}}=\mathrm{T}_{\mathrm{c}} \\ \mathrm{T}_{\mathrm{a}}=\mathrm{T}_{\mathrm{b}}\end{array}\right)\)