Question

If $\gamma$ denotes the ratio of the two specific heats of a gas, the ratio of the slopes of adiabatic and isothermal $p-V$ curves at their point of intersection is

• A. $1/\gamma$
• B. $\gamma$
• C. $\gamma -1$
• D. $\gamma +1$

Answer 1

Answer: (B)

$\gamma$

Isothermal process : $PV=$ constant

Differentiating, we get $PdV+VdP=0$

$⟹$ Slope of isothermal curve $(\frac{dP}{dV}{)}_{iso}=\frac{-P}{V}$ ...........(1)

Adiabtic process : $P{V}^{\gamma }=$ constant

Differentiating, we get $P\gamma V^{\gamma -1}dV+V^{\gamma }dP=0$

$⟹$ Slope of adiabatic curve $(\frac{dP}{dV}{)}_{adi}=\frac{-\gamma P}{V}$ ...........(2)

$⟹$ Ratio of slopes $\frac{(\frac{dP}{dV}{)}_{adi}}{(\frac{dP}{dV}{)}_{iso}}=\frac{(-\gamma P/V)}{-P/V}=\gamma$