THE STATISTICAL INTERPRETATION OF ENTROPY
4.8 CONFIGURATIONAL ENTROPY AND THERMAL ENTROPY In the preceding discussion entropy was considered in terms of the number of ways in
which energy can be distributed among identical particles, and the given example of a mixing process involved the redistribution of thermal energy among the particles of two closed systems when placed in thermal contact. The change in entropy accompanying this redistribution is a change in thermal entropy. Entropy can also be considered in terms of the number of ways in which particles themselves can be distributed in space, and this consideration gives rise to the concept of configurational entropy.
Consider two crystals at the same temperature and pressure, one containing atoms of the element A and the other containing atoms of the element B. When the two crystals are placed in physical contact with one another the spontaneous process which occurs is the diffusion of A atoms into the crystal of B and the diffusion of B atoms into the crystal of A. As this process is spontaneous, entropy is produced, and intuitively it might be predicted that equilibrium will be reached (i.e., the entropy of the system will reach a maximum value) when the diffusion processes have occurred to the extent that all concentration gradients in the system have been eliminated. This is the mass transport analog of the heat transfer case in which heat flows irreversibly between two bodies until the temperature gradients have been eliminated.
Consider a crystal containing four atoms of A placed in contact with a crystal containing four atoms of B. The initial state of this system, in which all four atoms of A lie to the left of XY and all four of the B atoms lie to the right of XY, is shown in Fig. 4.5.
The number of distinguishable ways in which this arrangement can be realized is unity, as interchange among the identical A atoms on the left of XY and/or
90 Introduction to the Thermodynamics of Materials
Figure 4.5 Representation of a crystal of A in contact with a crystal of B.
interchange among the identical B atoms on the right of XY does not produce a different configuration. Thus
(in which the notation indicates four atoms of A on the left of XY and none on the right.) When one A atom is interchanged with one B atom across XY, the B atom can be located on any of four sites, and hence the left side of XY can be realized in four different ways. Similarly the exchanged A atom can be located on any of four sites, and hence the right side can be realized in four different ways. As any of the four former arrangements can be combined with any of the four latter arrangements, the total number of distinguishable configurations of the arrangement 3:1 is 44=16, i.e.,
When a second A atom is exchanged with a second B atom across XY, the first B atom on the left of XY can be located in any of four positions, and the second B atom can be located in any of the three remaining positions, giving, thus, 43=12 configurations.
However, these 12 configurations include those which occur as a result of interchange of the two B atoms themselves, which, being indistinguishable, must be discounted. The number of distinguishable configurations on the left of XY is thus (43)/2!=6. Similarly six distinguishable arrangements occur on the right of XY, and hence the total number of distinguishable configurations in the arrangement 2:2 is 66=36, i.e.,
When a third A atom is exchanged with a third B atom across XY, the first B atom can be located on any of four sites, the second on any of the three remaining sites and the third on either of the two remaining sites. Factoring out the number of indistinguishable configurations caused by interchange of the three B atoms among themselves gives the number of distinguishable configurations on the left of XY as (432)/3!=4. Similarly four distinguishable configurations occur on the right of XY, and hence,
Interchange of the final A and B atoms across XY gives
Thus the total number of spatial configurations available to the system is 1+16+
36+16+1=70, which is the number of distinguishable way s in which four particles of one kind and four particles of another kind can be arranged on eight sites, i.e.,
If, as before, it is assumed that each of these configurations is equally probable, then the probability of finding the system in the arrangement 4:0 or 0:4 is 1/70, the probability of arrangement 3:1 or 1:3 is 16/70, and the probability of finding the system in the arrangement 2:2 is 36/70. Arrangement 2:2 is thus the most probable and thus corresponds to the equilibrium state, in which the concentration gradients have been eliminated. Again, as
it is seen that maximization of maximizes the entropy. In this case the increase in entropy occurs as a result of the increase in the number of spatial configurations which become available to the system when the crystals of A and B are placed in contact with one another. The increase in the entropy of the system arises from an increase in its configurational entropy, Sconf. The mixing process can be expressed as
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Thus
(4.18)
The total entropy of a system consists of its thermal entropy, Sth, which arises from the number of ways in which the energy of the system can be shared among the particles, and its configurational entropy, Sconf, which arises from the number of distinguishable ways in which the particles can fill the space available to them. Thus
The number of spatial configurations available to two closed systems placed in thermal contact, or to two open chemically identical systems placed in thermal contact, is unity.
Thus in the case of heat flow down a temperature gradient between two such systems, as only th changes, the increase in the entropy arising from the heat transfer which takes the system from state 1 to state 2 is
Similarly in the mixing of particles of A with particles of B, Stotal only equals Sconf if the mixing process does not cause a redistribution of the particles among the energy levels, i.e., if th(1)=th(2). This condition corresponds to “ideal mixing” of the particles and requires that the quantization of energy be the same in crystals A and B. Ideal mixing is the exception rather than the rule, and, generally, when two or more components are mixed at constant U, V, and n, th(2) does not have the same value as th(1); thus completely random mixing of the particles does not occur. In such cases either clustering of like particles (indicating difficulty in mixing) or ordering (indicating a tendency toward compound formation) occurs. In all cases, however, the equilibrium state of the system is that which, at constant U, V, and n, maximizes the product thconf.
1. A single macrostate of a system, which is determined when the independent variables of the system are fixed, contains a very large number of microstates, each of which is characterized by the manner in which the thermal energy of the system is distributed among the particles and the manner in which the particles are distributed in the space available to them.
2. Although the occurrence of a system in any one of its microstates is equally probable, greatly differing numbers of microstates occur in differing distributions. The distribution which contains the largest number of microstates is the most probable distribution, and in real systems the number of microstates in the most probable distribution is significantly larger than the sum of all of the other microstates occurring in all of the other distributions. This most probable distribution is the equilibrium thermodynamic state of the system.
3. The relationship between the number of microstates available to the system, , and the entropy of the system is given by Boltzmann’s equation as S=k ln , in which k is Boltzmann’s constant. Thus, if a situation arises which allows an increase in the number of microstates available to the system, then spontaneous redistribution of the energy among the particles (or particles over the available space) occurs until the newly available most probable distribution occurs. The Boltzmann equation shows that an increase in the number of microstates made available to the system causes an increase in the entropy of the system.
4. The total entropy of a system is the sum of the thermal entropy, Sth, and the configurational entropy, Sconf. The former arises from the number of ways in which the thermal energy available to the system can be shared among the constituent particles, th, and the latter arises from the number of ways in which the particles can be distributed over the space available to them, conf. As any of the thermal distributions can be combined with any of the configurational distributions, the total number of microstates available to the system is the product thconf, and hence, from the logarithmic form of Boltzmann’s equation, the total entropy of the system is the sum of Sth and Sconf.
4.10 NUMERICAL EXAMPLES
Example 1
Spectroscopic observation of molecular N2 in an electrical discharge shows that the relative numbers of molecules in excited vibrational states with energies given by
(4.19) 4.9 SUMMARY
94 Introduction to the Thermodynamics of Materials
i 0 1 2 3
1.00 0.250 0.062 0.016
Show that the gas is in thermodynamic equilibrium with respect to the distribution of vibrational energy, and calculate the temperature of the gas. In Eq. (4.19), i is an integer which has values in the range zero to infinity, h is Planck’s constant of action (=6.62521034 joules·s), and the vibration frequency, v, is 7.00 1013
From Eqs. (4.13), (4.14), and (4.19),
Observation shows that
Thus,
which gives
Then, from Eq. (4.13), are
and
Normalizing gives
which shows that the gas is in equilibrium with respect to the distribution of vibrational energy. The temperature of the gas is obtained as
Example 2 The isotopic composition of lead in atomic percent is
atomic weight atomic percent
204 1.5
206 23.6
207 22.6
208 52.3
Calculate the molar configurational entropy of lead. The configurational entropy is obtained from Boltzmann's equation
(4.17)
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Stirling’s theorem gives
Therefore, the molar configurational entropy is
PROBLEMS