The general criterion for spontaneous processes is that the entropy of the universe must increase. We now express this criterion in terms of the properties of a closed system that can exchange heat and work with its surroundings. We make the simplest possible assumption about the surroundings: that the surroundings have a very large thermal conductivity and a very large heat capacity so that all processes in the surround- ings are reversible. Equation (3.2-27) asserts that under these conditions all processes must obey
dS≥ dq
Tsurr (dU−dw) Tsurr
(4.1-1) whereTsurris the temperature of the surroundings and where we have used the first law to write the second equality. SinceTsurrmust be positive, we can multiply byTsurr
without changing the direction of the inequality, obtaining
dU−dw−TsurrdS≤0 (4.1-2)
We now consider several special cases.
The first case is that of anisolated system, which is a closed system that cannot exchange either heat or work with its surroundings. In this casedq0 anddw0 so thatdU 0 and
dS≥0 (isolated system) (4.1-3)
An isolated system is anadiabatic systemwith the additional requirement thatdw0, so Eq. (4.1-3) is a special case of Eq. (3.2-22) for an adiabatic system.
The second case is thatdS 0 anddw0,
dU ≤0 (Sconstant,dw0) (4.1-4)
This is a case that is not likely to be encountered in thermodynamics since there is no convenient way to keep the entropy of a system constant while a process occurs.
The third case is that of anisothermal system (constant-temperature system). If the system is maintained at the constant temperature of the surroundings (T Tsurr), Eq. (4.1-2) becomes
dU−TdS−dw≤0 (T Tsurrconstant) (4.1-5)
If our system is simple,dw −P(transmitted)dV, and
dU−TdS+P(transmitted)dV ≤0 (simple system,T constant) (4.1-6) There are two important cases for isothermal closed simple systems. The first case is that of a constant volume, so thatdV 0:
dU−TdS≤0 (simple system, T andV constant) (4.1-7) TheHelmholtz energyis denoted byAand is defined by
AU−TS (definition) (4.1-8)
The Helmholtz energy has been known to physicists as the “free energy” and as the
“Helmholtz function.” It has been known to chemists as the “work function” and as the “Helmholtz free energy.” The differential of the Helmholtz energy is
dAdU−TdS−SdT (4.1-9)
so that ifT is constant
dAdU−TdS (constantT) (4.1-10)
Equation (4.1-7) is the same as
dA≤0 (simple system,T andV constant) (4.1-11) The Helmholtz energy is named for
Hermann Ludwig von Helmholtz, 1821–1894, already mentioned in Chapter 2 as the first person to announce the first law of thermodynamics.
The second important isothermal case is the case that the pressure of the system is constant and equal toPext and toP (transmitted). We refer to this case simply as
“constant pressure.” In this case,
dU+PdV−TdS≤0 (simple system,
T andP constant) (4.1-12) TheGibbs energyis defined by
GU+PV−TS (definition) (4.1-13)
The Gibbs energy is named for Josiah Willard Gibbs, 1839 –1903, an American physicist who made fundamental contributions to thermodynamics and statistical mechanics and who was the first American scientist after Benjamin Franklin to gain an international scientific reputation.
The Gibbs energy is related to the enthalpy and the Helmholtz energy by the relations
GH−TSA+PV (4.1-14)
The Gibbs energy has been called the “free energy,” the “Gibbs free energy,” the “Gibbs function,” and the “free enthalpy.” The symbolF has been used in the past for both the Helmholtz energy and the Gibbs energy. To avoid confusion, the symbolFshould not be used for either of these functions and the term “free energy” should not be used.
The differential of the Gibbs energy is
dGdU +PdV +VdP−TdS−SdT (4.1-15)
IfT andP are constant, this equation becomes
dGdU +PdV −TdS (T andP constant) (4.1-16) The relation in Eq. (4.1-12) is the same as
dG≤0 (simple system,T andP constant) (4.1-17) The criteria for finite processes are completely analogous to those for infinitesimal processes. For example, for a simple system at constant pressure and temperature, a finite spontaneous process must obey
∆G≤0 (simple system,T andPconstant) (4.1-18) Constant temperature and pressure are the most common circumstances in the lab- oratory, so Eq. (4.1-18) is the most useful criterion for the spontaneity of chemical reactions. Thermodynamics does not distinguish between chemical and physical pro- cesses, and Eq. (4.1-17) is valid for physical processes such as phase transitions as well as for chemical reactions.
In the 19th century, Berthelot incorrectly maintained that all spontaneous chemical reactions must beexothermic(∆H <0). The incorrectness of Berthelot’s conjecture was shown by Duhem, who established Eq. (4.1-18), which can be written in the form
∆H−T∆S≤0 (simple system,
T andPconstant) (4.1-19)
The∆H term dominates at sufficiently low temperature, but theT∆Sterm becomes important and can dominate at sufficiently high temperature. In many chemical reac- tions near room temperature theT∆Sterm is numerically less important than the∆H term, and most spontaneous chemical reactions are exothermic, a fact that presumably led Bertholet to his thermodynamically incorrect assertion.
Pierre Eugene Marcelin Berthelot, 1827–1907, was a French chemist who synthesized many useful compounds, but who argued against Dalton’s atomic theory of matter.
Pierre-Maurice-Marie Duhem, 1861–1916, was a French physicist whose doctoral dissertation showing Berthelot’s conjecture to be false was initially rejected because of Berthelot’s objection.
We can illustrate the interplay of∆Hand∆Swith the vaporization of a liquid, for which∆Hand∆Sare both positive. At a given pressure there is some temperature at which the vaporization is a reversible process so that∆G0 and∆HT∆S. When T is smaller than this equilibrium temperature, the∆Hterm dominates and∆G >0 for the vaporization process. That is, condensation is spontaneous and the equilibrium phase is the liquid phase. At a higher temperature theT∆Sterm dominates and∆G <0 for the vaporization process, which is spontaneous. The gas phase is the equilibrium phase.
Some people say that there are two tendencies: (1) that of the enthalpy or energy of a system tends to decrease, and (2) that the entropy of the system tends to increase.
In fact, there is only one fundamental tendency, that of the entropy of the universe to increase. We can separately focus on the two terms in Eq. (4.1-19), but you should remember that the lowering of the enthalpy corresponds to an increase in the entropy of the surroundings.
We summarize our results for the cases considered:
1. If a system is isolated, its entropy cannot decrease.
2. If Sof a closed system is fixed and no work is done,Ucannot increase.
3. If a simple closed system is at constantTandV, Acannot increase.
4. If a simple closed system is at constantTandP, Gcannot increase.