Question

Show that the slope of an adiabatic curve at any point is γ times the slope of an isothermal curve at the corresponding point.

Answer 1

For an isothermal change PV = K

Differentiating both the sides, we get

P.dV + V.dP = 0 or V.dP = -PdV

∴ Slope of an isothermal curve,

\((\frac{dP}{dV})_{iso} = -\frac{P}{V}\)

For an adiabatic change, PV^γ = K'

Differentiating both sides, we get

P.^γV^γ-1.dV + V^γ.dP = 0

or γPdV + VdP = 0 or VdP = -γPdV

Slope of an adiabatic curve,

\((\frac{dP}{dV})_{adia} = -\frac{\gamma P}{V}\)

Clearly, slope of an adiabatic curve = γ x slope of an isothermal curve

As, γ > 1, so an adiabatic P - V curve is steeper than the corresponding isothermal P - V curve.