Statistical Mechanical Interpretation of Entropy P. G. Nelson University of Hull, Hull HU6 7RX, England
Most advanced programs in physical chemistry include a course in statistical mechanics ( I ) . One of the aims of such a course is to provide an understanding of the thermodynamic quantity, entropy, S. This is usually achieved by making the identification S~klnW
The change in this quantity as a result of the thermal contact between Aand B is given by 6S, = a , + a, (3) From the definition of entropy and rewriting,2we get
(1)
where W is the weight of the most probable configuration of a system, and k is the Boltzmann constant. However, a deeper understanding of entropy can be obtained as described below (2). The treatment also provides an explanation of temperature T .
If TA< TB,q~ is positive, and if TA> Tg, q~ is negative. If T A= TB,then (11T~- 1ITg)is Hence for spontaneous changes,
Entropy of Thermodynamically Isolated Systems Consider the system, shown in the figure, comprising two subsystems, A and B, which are thermally insulated from each other and from their surroundings. Each subsystem has rigid and impermeable walls, so its volume and overall chemical composition are fixed. A small device C enables thermal contact to be made between the two, without any other interaction taking place. What happens when wntact is made for a short period, and a small quantity of energy flows from one subsystem to the other?
In t e r n of entropy, therefore, the condition for Aand B to be in equilibrium when they are brought into contact is &St = 0 or 6SB= -=A. At constant volume and composition, qx = 6Ux. Thus, eq 4 can also be written as
StatisticalMechanical Treatment To derive the thermodynamic properties of an isolated system from statistical mechanics, it is necessary to assume that such a system is not perfectly isolated (otherwise it would remain in a single micmscopic state). Rather it is subject to small perturbations that cause it to spend time in all possible microscopic states, consistent with the macroscopic state, having ansnergy E within a small range (%El of its mean value (E = U). This has been fully discussed by Denbigh and Denbigh (3).The magnitude of 66E is not critical (2). Suppose therefore that, before contact is made between the two systems in the figure, the microscopic motion of system Aranges over RA distinct states and that of B over RB.Consider the quantity
A thermodynamically isolated system comprising two subsystems (A and B), with a device (C)for enabling thermal contact to be made
between them.
Suppose that, before contact is made, each subsystem is in an equilibrium state with internal energy Ux (X = Aor B), entropy Sx,and temperature Tx. Suppose further that, aRer contact has been made, each subsystem settles again into an equilibrium state. Let the heat absorbed by subsystem Aduring contact be q,, and by B q ~where , q~ =
