(6.52) |

This result is known as the

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 changes. Let us calculate the relationship between the
pressure and volume of the gas during adiabatic expansion.
According to the first law of thermodynamics,

(6.53) |

in an adiabatic process (in which no heat is absorbed). [See Equation (6.35).] The ideal gas equation of state, (6.10), can be differentiated, yielding

(6.54) |

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

(6.55) |

which reduces to

(6.56) |

Dividing through by yields

where

(6.58) |

It turns out that is a slowly-varying function of temperature in most gases. Consequently, it is usually a good approximation to treat the ratio of specific heats, , as a constant, at least over a limited temperature range. If is constant then we can integrate Equation (6.57) to give

(6.59) |

or

This result is known as the

(6.61) |

and

Equations (6.60)-(6.62) are all completely equivalent.