Question

if gamma denotes the ratio of the two specfic of a gas then the ratio of the slope of adiabetic and isothermal p v curve at their point of intersection is
​

Answer 1

In Isothermal process,
PV = constant
Differentiating both sides,

P dv + V dP = 0
So,
dP/dV = - P/V
This will be the slope of an Isothermal process

Now,
In adiabatic process, by Poisson equation

PV^gamma = constant
So,
P*gamma*V^gamma-1 dV + V^gamma dP = 0
So,
dP/dV = -gamma*P/V
So,
ratio of the slopes of adiabatic to Isothermal process
= -gamma(P/V)/(-P/V) = gamma

Hope this helps you !