General Interaction Between Macrosystems

(5.42) |

Thus, the energy of system is determined once the energy of system is given, and vice versa. In fact, could be regarded as a function of . Furthermore, if the two systems can interact mechanically then, in general, the parameters are some function of the parameters . As a simple example, if the two systems are separated by a movable partition, in an enclosure of fixed volume , then

(5.43) |

where and are the volumes of systems and , respectively.

The total number of microstates accessible to is clearly a function of , and the parameters (where runs from 1 to ), so . We have already demonstrated (in Section 5.2) that exhibits a very pronounced maximum at one particular value of the energy, , when is varied but the external parameters are held constant. This behavior comes about because of the very strong,

(5.44) |

increase in the number of accessible microstates of (or ) with energy. However, according to Section 3.8, the number of accessible microstates exhibits a similar strong increase with the volume, which is a typical external parameter, so that

(5.45) |

It follows that the variation of with a typical parameter, , when all the other parameters and the energy are held constant, also exhibits a very sharp maximum at some particular value, . The equilibrium situation corresponds to the configuration of maximum probability, in which virtually all systems in the ensemble have values of and very close to and , respectively. The mean values of these quantities are thus given by and .

Consider a quasi-static process in which the system is brought from an equilibrium state described by and to an infinitesimally different equilibrium state described by and . Let us calculate the resultant change in the number of microstates accessible to . Because , the change in follows from standard mathematics:

However, we have previously demonstrated that

(5.47) |

[in Equations (5.30) and (5.41), respectively], so Equation (5.46) can be written

Note that the temperature parameter, , and the mean conjugate forces, , are only well defined for equilibrium states. This is why we are only considering quasi-static changes in which the two systems are always arbitrarily close to equilibrium.

Let us rewrite Equation (5.48) in terms of the thermodynamic temperature, , using the relation . We obtain

where

Equation (5.49) is a differential relation that enables us to calculate the quantity as a function of the mean energy, , and the mean external parameters, , assuming that we can calculate the temperature, , and mean conjugate forces, , for each equilibrium state. The function is termed the

According to Equation (4.16), the net amount of work performed during a quasi-static change is given by

(5.51) |

It follows from Equation (5.49) that

(5.52) |

Thus, the thermodynamic temperature, , is the integrating factor for the first law of thermodynamics,

(5.53) |

which converts the inexact differential into the exact differential . (See Section 4.5.) It follows that the entropy difference between any two macrostates and can be written

(5.54) |

where the integral is evaluated for any process through which the system is brought

All of the concepts that we have encountered up to now in this course, such as temperature, heat, energy, volume, pressure, et cetera, have been fairly familiar to us from other branches of physics. However, entropy, which turns out to be of crucial importance in thermodynamics, is something quite new. Let us consider the following questions. What does the entropy of a thermodynamic system actually signify? What use is the concept of entropy?