Adiabatic process

In an adiabatic process the system is insulated, so no heat enters or leaves: ΔQ = 0. The first law then gives ΔW = −ΔU: work done by the gas comes out of its internal energy, so an ideal gas cools as it expands adiabatically.

For an ideal gas undergoing a quasi-static adiabatic change, PV^γ = constant, where γ = C_p/C_v (ratio of the specific heats, ordinary or molar). The result is quoted without proof.

So between two states: P₁V₁^γ = P₂V₂^γ. Since γ > 1, an adiabat is steeper than an isotherm through the same point: for the same compression the pressure rises more.

Fig. 11.8: two adiabats connect two isotherms on a P–V diagram.

Work done by the gas from (P₁, V₁, T₁) to (P₂, V₂, T₂): W = ∫P dV with P = constant/V^γ, giving W = [P₁V₁ − P₂V₂]/(γ − 1) = μR(T₁ − T₂)/(γ − 1).

If the gas does work (W > 0), T₂ < T₁: it cools. If work is done on it (W < 0), T₂ > T₁: it warms up.

Example: a gas with γ = 1.4 (such as hydrogen) compressed adiabatically to half its volume: P₂/P₁ = 2^1.4 ≈ 2.64, while an isothermal halving only doubles the pressure.

For the same case, TV^(γ−1) = constant gives T₂ = T₁ × 2^0.4 ≈ 1.32 T₁: 300 K becomes about 396 K.