Question

Ratio of slope of isothermal curve to adiabatic curve is 
1. 1/γ 
2. γ 
3. 1
4. -1

Answer 1

Correct Answer - Option 1 : 1/γ 

Concept:

For Isothermal process, PV = C

Differentiating on both sides,

⇒ P dV + V dP = 0

\( \Rightarrow slop{e_{isothermal}} = \frac{{dP}}{{dV}} = - \frac{P}{V}\;\;\;\;\; - - - \left( 1 \right)\)

For adiabatic process, PV^γ = C

Differentiating on both sides,

⇒ P γ V^{γ -1} dV + V^γ dP = 0

\( \Rightarrow slop{e_{adiabatic}} = \frac{{dP}}{{dV}} = \frac{{ - P\gamma {V^{\gamma - 1}}}}{{{V^\gamma }}} = \gamma \left( {\frac{{ - P}}{V}} \right)\;\;\;\;\; - - - \left( 2 \right)\)

Ratio of slope of isothermal to adiabatic curve is 

\(\frac{{slop{e_{isothermal}}}}{{slop{e_{adiabatic}}}} = \frac{1}{\gamma }\)