Molecular Degree Freedom
Figure 2.4 The number of degrees of freedom of molecular hydrogen as a function of temperature.

For some estimates, it is sufficient to assume that these quantities are constant. One can assume v = 5 for H2 and O2, and v = 7 for the more complicated molecule H2O, all at ambient temperature. More precise calculations require looking up these values in tables. See, for instance, a listing of observed values of cp and of 7 for H2, O2, and H2O in Table 9.5.

Some general trends should be remembered: More complex molecules or higher temperatures lead to a larger number of degrees of freedom and consequently larger specific heats and smaller 7.

2.14 Entropy

When one considers different forms of energy, one can intuitively rank them in order of their "nobility." Electric energy must be quite "noble"—it can easily be transformed into any other kind of energy. The same is true of mechanical energy because it can (theoretically) be transformed into electricity and vice versa without losses. Heat, however, must be "degraded" energy. It is well known that it cannot be entirely transformed into either electric or mechanical energy (unless it is working against a heat sink at absolute zero). It turns out that chemical energy has a degree of "nobility" lower than that of electricity but higher than that of heat.

Still, intuitively, one can feel that the higher the temperature, the higher the corresponding "nobility" of the heat—that is, the more efficiently it can be transformed into some other form of energy.

Let us try to put these loose concepts on a more quantitative basis.

Consider two large adiabatic reservoirs of heat: one (which we shall call the source) at a temperature, TH, and one (the sink) at a lower temperature, TC. The reservoirs are interconnected by a slender metal rod forming a thermally conducting path between them. We shall assume that, for the duration of the experiment, the heat transferred from source to sink is much smaller than the energy stored in the reservoirs. Under such circumstances, the temperatures will remain unaltered.

Assume also that the rod makes thermal contact with the reservoirs but not with the environment. The amount of heat that leaves the source must then be exactly the same as that which arrives at the sink. Nevertheless, the heat loses part of its "nobility" because its arrival temperature is lower than that at the departure. "Nobility" is lost in the conduction process.

Form the Q/T ratio at both the heat source and the heat sink. Q is the amount of heat transferred. Clearly, Q/TH < Q/TC. We could use this ratio as a measure of "ignobility" (lack of "nobility"), or, alternately, the ratio —Q/T as a measure of "nobility." Loosely, entropy is what we called "ignobility":

It is important to realize that in the above experiment, energy was conserved but "nobility" was lost. It did not disappear from the experimental system to emerge in some other part of the universe—it was lost to the universe as a whole. There is no law of conservation of "ignobility" or entropy. In any closed system, at best, the entropy will not change, but if it does, it always increases.^ This is a statement of the second law of thermodynamics.

Since there is no heat associated with electric or mechanical energy, these forms have zero entropy.

2.14.1 Changes in Entropy

Returning to the question of functions of state, it is important to know that entropy is such a function. To determine the change in entropy in any process, it is sufficient to determine the entropies of the final and the initial states and to form the difference.

Some processes can drive a system through a full cycle of changes (pressure, volume, and temperature) in such a way that, when the cycle is complete, the system is returned to the initial state. Such processes are reversible. To be reversible, the net heat and the net work exchanged t However, a given system does not always tend toward maximum entropy. Systems may spontaneously create complicated structures such as life forms emerging from some primeval soup—a reduction of entropy.

with the environment must be zero. In any reversible process, the change in entropy of a substance owing to a change from State 1 to State 2 is

Here, S is the entropy and AS is the change in entropy. In an adiabatic processes, dQ = 0. Hence

In an isothermal process, AS = Q/T because T is constant. Notice, however, that according to the first law of thermodynamics, AU = Q — W, but, in an isothermal change, AU = 0, hence W = Q and, since in such a change, W = poVo lnpi/p2,

In an isobaric process,

Cp T

and if cp is constant,

In an isometric process,

and if cv is constant,

The change in entropy of | kilomoles of a substance owing to an isobaric change of phase is

where QL is the latent heat of phase change (per kilomole) and T is the temperature at which the change takes place. We have collected all these results in Table 2.3.

Table 2.3 Changes in Entropy


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