The slope of isothermal and adiabatic curves are related as

isothermal curve slope= adiabatic curve slope

isothermal curve slope = γ × adiabatic curve slope

adiabatic curve slope = γ× isothermal curve slope

adiabatic curve slope = $\frac{1}{2}$ × isothermal curve slope

Solution

adiabatic curve slope = γ× isothermal curve slope

For isothermal process pV = constant

⇒ $(\frac{dp}{dV}) = - \frac{p}{V}$ = slope of isothermal curve

For adiabatic process, pV^γ = constant

⇒ $\frac{dp}{dV} = - \frac{γ p}{V}$ = slope of adiabatic curve

Clearly

${(\frac{dp}{dV})}_{adiabatic} = γ {(\frac{dp}{dV})}_{isothermal}$

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