# Liquefaction of gases by reversible adiabatic expansion

In Zemansky and Dittman's Heat and Thermodynamics (7th ed.), the authors while discussing the liquefaction of gases by Joule-Thomson expansion, state the following (p. 284): "An approximately reversible adiabatic expansion against a piston or a turbine blade always produces a decrease in temperature, no matter what the original temperature...But, this method has the disadvantage that the temperature drop on adiabatic expansion decreases as the temperature decreases".

What is the quantitative expression (based on the First Law of Thermodynamics or changes in enthalpy) which qualifies the above statements ?

Expansion against a piston or a turbine blade (or any resistance) constitutes $$P$$$$V$$ work done by the system, where $$P$$ is the pressure and $$V$$ is the volume. Thus, incoming work $$W<0$$. In addition, adiabatic conditions ensure that heating $$Q=0$$. Thus, $$\Delta U=W+Q<0$$ by the First Law, where $$U$$ is the internal energy. For single-component systems near equilibrium, we always find that $$U$$ decreases monotonically with decreasing temperature (i.e., the heat capacity is positive), which provides the first statement.
• It is unclear. $C_v=\left(\frac{\partial U}{\partial T}\right)_V$ if $dV=0$, but over here $P$-$V$ work is non-zero. Commented Feb 22, 2021 at 4:40
• I don't make any assumption of constant volume. For reversible adiabatic processes, $C_V=\left(T\alpha K/P\right)\left(\partial U/\partial T\right)_S$, where $\alpha$ is the thermal expansion coefficient and $K$ is the bulk modulus. These parameters are all positive for gases. Commented Feb 22, 2021 at 5:15