Isothermal and Adiabatic Expansion

Suppose that the temperature of an ideal gas is held constant by keeping the gas in thermal contact with a heat reservoir. If the gas is allowed to expand quasi-statically under these so-called isothermal conditions then the ideal gas law, (5.97), tells us that

$$p\,V = {\rm constant}.$$ (5.114)

This result is known as the isothermal gas law.

Suppose, now, that the gas is thermally isolated from its surroundings. If the gas is allowed to expand quasi-statically under these so-called adiabatic conditions then it does work on its environment, and, hence, its internal energy is reduced, and its temperature decreases. Let us calculate the relationship between the pressure and volume of the gas during adiabatic expansion. According to Equation (5.109),

$$dQ = \nu \,c_V \,dT +p\,dV = 0,$$ (5.115)

in an adiabatic process (in which no heat is absorbed). The ideal gas law, (5.97), can be differentiated, yielding

$$p\,dV + V\,dp = \nu\, R \,dT.$$ (5.116)

The temperature increment, $dT$, can be eliminated between the previous two expressions to give

$$0 = \frac{c_V}{R} \,(p\,dV + V\,dp) + p \,dV = \left(\frac{c_V}{R} + 1\right) p\, dV +\frac{c_V}{R} \,V\,dp,$$ (5.117)

which reduces to

$$(c_V +R)\,p\,dV + c_V \,V\, dp = 0.$$ (5.118)

Dividing through by $c_V\, p\, V$ yields

$$\gamma\,\frac{dV}{V} + \frac{dp}{p}=0,$$ (5.119)

where

$$\gamma \equiv \frac{c_p}{c_V} = \frac{c_V + R}{c_V}$$ (5.120)

is termed the ratio of specific heats. [See Equation (5.113).] Given that $c_V$ is a constant in an ideal gas, the ratio of specific heats, $\gamma$, is also a constant. In fact, Equations (5.106), (5.107), and the previous equation, imply that

$$\gamma = \frac{5}{3}$$ (5.121)

for a monatomic gas, and

$$\gamma = \frac{7}{5}$$ (5.122)

for a diatomic gas.

Because $\gamma$ is a constant for an ideal gas, we can integrate Equation (5.119) to give

$$\gamma \ln V + \ln p = {\rm constant},$$ (5.123)

or

$$p \,V^{\gamma} = {\rm constant}.$$ (5.124)

This result is known as the adiabatic gas law. It is straightforward to obtain analogous relationships between $V$ and $T$, and between $p$ and $T$, during adiabatic expansion or contraction. In fact, because $p = \nu\, R\,T/V$, the previous formula also implies that

$$T \,V^{\gamma-1} = {\rm constant},$$ (5.125)

and

$$p^{1-\gamma} \,T^{\gamma} = {\rm constant}.$$ (5.126)

Equations (5.124)–(5.126) are all completely equivalent.