The heat capacity at constant volume is given by

Likewise, the heat capacity at constant pressure is written

Experimentally, the parameters that are most easily controlled are the temperature, , and the pressure, . Let us consider these as the independent variables. Thus, , which implies that

(6.122) |

in an infinitesimal quasi-static process in which an amount of heat is absorbed. It follows from Equation (6.121) that

(6.123) |

Suppose that . The previous equation can be written

(6.124) |

At constant volume, . Hence, Equation (6.120) gives

This is the general relationship between and . Unfortunately, it contains quantities on the right-hand side that are not readily measurable.

Consider . According to the Maxwell relation (6.119),

(6.126) |

Now, the quantity

which is known as the

(6.128) |

in Equation (6.125).

Consider the quantity . Writing , we obtain

(6.129) |

At constant volume, , so we obtain

(6.130) |

The (usually positive) quantity

which is known as the

in Equation (6.125). It follows that

(6.133) |

and

where is the

As an example, consider an ideal gas, for which

(6.135) |

At constant , we have

(6.136) |

Hence,

(6.137) |

and the expansion coefficient defined in Equation (6.127) becomes

At constant , we have

(6.139) |

Hence,

(6.140) |

and the compressibility defined in Equation (6.131) becomes

Finally, the molar volume of an ideal gas is

Hence, Equations (6.134), (6.138), (6.141), and (6.142) yield

(6.143) |

which is identical to Equation (6.39).