Question

Prove that the ratio of slope of adiabatic curve to slope of isothermal is $\gamma$ .

Answer 1

For veriable isothermal process

$pv=$constant

differentiating

$pdv+vdp=0\Rightarrow \frac{dp}{dv}=\frac{p}{v}$ (=slope of isothermal curve)

for adiabatic process,

$p{v}^r=$ constant

$\Rightarrow rp{v}^{r-1}dv+v^{r}dp=0\Rightarrow \frac{dp}{dv}=-r\frac{p}{v}$ (=slope of adiabatic curve)

so ratio of the slopes:-

$\frac{m_1}{m_2}=\frac{-p/v}{-Y\left(p/v\right)}\Rightarrow \frac{m_2}{m_1}=r$