42 Second Law of Thermodynamics (M17Q3)

The Second Law of Thermodynamics

In the quest to identify a property that may reliably predict the spontaneity of a process, we have identified a very promising candidate: entropy. Processes that involve an increase in entropy of the system (ΔS > 0) are very often spontaneous; however, examples to the contrary are plentiful. By expanding consideration of entropy changes to include the surroundings, we may reach a significant conclusion regarding the relation between this property and spontaneity. In thermodynamic models, the system and surroundings comprise everything, that is, the universe, and so the following is true:

To illustrate this relationship, consider again the process of heat flow between two objects, one identified as the system and the other as the surroundings. There are three reasonable possibilities for such a process:

1. The objects are at different temperatures, and heat flows from the hotter to the cooler object. This always occurs spontaneously. Designating the hotter object as the system and invoking the definition of entropy yields the following:
   ΔS_sys = [latex]\frac{-q_{rev}}{T_{sys}}[/latex]    and    ΔS_surr = [latex]\frac{q_{rev}}{T_{surr}}[/latex]

   The arithmetic signs of q_rev denote the loss of heat by the system and the gain of heat by the surroundings. Since T_sys > T_surr in this scenario, the magnitude of the entropy change for the surroundings will be greater than that for the system, and so the sum of ΔS_sys and ΔS_surr will yield a positive value for ΔS_univ. This process involves an increase in the entropy of the universe.

2. The objects are at different temperatures, and heat flows from the cooler to the hotter object. This is never observed to occur spontaneously. Again designating the hotter object as the system and invoking the definition of entropy yields the following:
   ΔS_sys = [latex]\frac{q_{rev}}{T_{sys}}[/latex]    and    ΔS_surr = [latex]\frac{-q_{rev}}{T_{surr}}[/latex]

   The arithmetic signs of q_rev denote the gain of heat by the system and the loss of heat by the surroundings. The magnitude of the entropy change for the surroundings will again be greater than that for the system, but in this case, the signs of the heat changes will yield a negative value for ΔS_univ. This process involves a decrease in the entropy of the universe.

3. The temperature difference between the objects is infinitesimally small, T_sys ≈ T_surr, and so the heat flow is thermodynamically reversible. In this case, the system and surroundings experience entropy changes that are equal in magnitude and therefore sum to yield a value of zero for ΔS_univ. This process involves no change in the entropy of the universe.

These results lead to a profound statement regarding the relation between entropy and spontaneity known as the second law of thermodynamics: all spontaneous changes cause an increase in the entropy of the universe. A summary of these three relations is provided in Table 1.

For many realistic applications, the surroundings are vast in comparison to the system. In such cases, the heat gained or lost by the surroundings  represents a very small, nearly infinitesimal, fraction of its total thermal energy. For example, combustion of a fuel in air involves transfer of heat from a system (the fuel and oxygen molecules reactants) to surroundings that are infinitely more massive (the earth’s atmosphere). As a result, q_surr is a good approximation of q_rev, and the second law may be stated as the following:

We may use this equation to predict the spontaneity of a process as illustrated in Example 1.

We can understand the relationship between entropy and enthalpy by recalling the equation just before Example 1:

The first law requires that q_surr = −q_sys, and at constant pressure q_sys = ΔH, and so this expression may be rewritten as the following:

The Third Law of Thermodynamics

The previous section described the various contributions of matter and energy dispersal that contribute to the entropy of a system. With these contributions in mind, consider the entropy of a pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K). This system may be described by a single microstate, as its purity, perfect crystallinity and complete lack of motion means there is only one possible location for each atom or molecule comprising the crystal, giving the system a single microstate (W = 1). According to the Boltzmann equation, the entropy of this system is zero.

This limiting condition for a system’s entropy represents the third law of thermodynamics: the entropy of a pure, perfect crystalline substance at 0 K is zero.

Standard Entropies

We can make careful calorimetric measurements to determine the temperature dependence of a substance’s entropy and to derive absolute entropy values under specific conditions. Standard entropies are given the label for values determined for one mole of substance at a pressure of 1 bar and a temperature of 298 K. The standard entropy change (ΔS°) for any process may be computed from the standard entropies of its reactant and product species like the following:

For example, ΔS° for the following reaction at room temperature:

is computed as the following:

Table 2 lists some standard entropies at 298.15 K. You can find additional standard entropies in Appendix F.

Glossary

second law of thermodynamics
all spontaneous changes cause an increase in the entropy of the universe

standard entropies
values determined for one mole of a substance at a pressure of 1 bar and a temperature of 298 K (Sº₂₉₈)

third law of thermodynamics
the entropy of a pure, perfect crystalline substance at 0 K is zero