Reversible adiabat

The reversible adiabat is the transformation with the constraint $Q = 0$: the gas has no thermal contact with the outside. Since there are no thermostats, there is no entropy exchanged with the environment either, $\Delta S_\text{term} = 0$, and the reversibility condition $\Delta S_\text{tot} = 0$ falls entirely on the gas:

$\Delta S_\text{gas} = nC_v\,\ln\frac{T_B}{T_A} + nR\,\ln\frac{V_B}{V_A} = 0$

This is no longer just a value to calculate: it is an equation linking temperature and volume throughout the whole process. Balancing the two terms and exponentiating, the combination $T\,V^{\,R/C_v}$ must remain constant; introducing the adiabatic exponent $\gamma = C_p/C_v$ the relation takes its celebrated form, which in $P$-$V$ variables becomes $P\,V^\gamma = \text{const}$.

Key formula

$\ev{T\,V^{\gamma-1} = \text{const} \qquad P\,V^\gamma = \text{const}}$ where $\gamma = \dfrac{C_p}{C_v} = \dfrac{f+2}{f}$.

The coefficient $\gamma$ depends only on the degrees of freedom $f$ of the molecule: for a monatomic gas $f=3$ and $\gamma = 5/3$, for a diatomic gas $f=5$ and $\gamma = 7/5$.

On the energy side, with $Q = 0$ the first law says that all the work done by the gas is drawn from its internal energy. An adiabatic expansion therefore cools the gas, while a compression heats it:

$\Delta E_\text{int} = -L_\text{gas} = nC_v(T_B - T_A)$

Principle — Summary of the reversible adiabat

$\begin{aligned} Q &= 0 \\ \Delta S_\text{gas} &= 0 \\ \Delta E_\text{int} &= -L_\text{gas} = nC_v(T_B - T_A) \end{aligned}$

Historical note

The formula $P\,V^\gamma = \text{const}$ was first found by Siméon Poisson in 1823, even before the concept of entropy was clearly defined. Only decades later was it understood to be a direct consequence of the condition $\Delta S_\text{gas} = 0$.

Links

Topics: Thermodynamics Concepts: Thermodynamic transformations · Entropy · First law of thermodynamics · Internal energy · Second law of thermodynamics Skills: Entropy balance · Solving a thermodynamic cycle Objects: Ideal gas

Related exercises: Worked exercise — reversible vs irreversible isothermal expansion · Worked exercise — free expansion is not reversible · Gas in reversible adiabatic expansion