In an adiabatic process, the First Law gives:  dE = dW.

Use this result to determine adiabatic paths (adiabats) in the P-V diagram.

dE = dW  →   nC_V dT = - P dV = - n RT dV/V.

→   C_V dT/T = - R dV/V but R = (C_P-C_V).

dT/T = - (dV/V) [(C_P-C_V)/ C_V].

→   d ln T = - (γ - 1) d ln V   or   d [ln T V ^{(γ - 1)}] = 0.

→   T V ^{(γ - 1)} = constant.

Equivalently use T ∝ PV to give P V ^γ = constant.

Note: adiabats are steeper than isotherms!

• Adiabat:   P V ^γ = const.   or     T V ^{(γ - 1)} = const.
• Isotherm: P V = const.