For eq 3b and c, respectively, using the definitions
In the absence of work, the isothermal equilibrium
of Gibbs free energy, enthalpy, and Helmholtz free
condition is dA = 0; furthermore, a spontaneous pro-
energy, differentiating and substituting eq 3b or 3c for
cess produces negative Helmholtz free energy. In other
dU results in
words, a constant temperature system minimizes Helm-
holtz free energy.
dG = SdT + VdP
(4a)
A similar derivation can be done for constant pres-
sure and temperature processes to show that the spon-
or
taneity condition is
dG = -SdT + VdP + ∑ i dηi
(4b)
i
dG ≥ δW.
(11)
dH = TdS + VdP
(5a)
Thus, at constant temperature and pressure and in the
or
absence of work, the equilibrium condition is dG = 0;
and, a spontaneous process produces negative Gibbs
dH = -TdS + VdP + ∑ i dηi
(5b)
free energy.
i
dA = SdT PdV
(6a)
Thermal equilibrium
Substituting TdS ≥ δQ into the first law results in
or
dA = -SdT - PdV + ∑ i dηi .
dU δW + TdS ≥ 0.
(12)
(6b)
i
For an isolated system, dU = δW = δQ = 0; thus, eq
Equations 3 through 6 are known as the four funda-
12 applied to an isolated system is
mental equations of thermodynamics.
dS ≥ 0.
(13)
THERMODYNAMIC EQUILIBRIUM
Since dS = (δQrev / T ) , if a positive quantity of heat
Equilibrium thermodynamic relations are often used
passes from region a to b within an isolated system,
in soil freezing and frost heave models (analytical and
then
numerical); therefore, their derivations are now present-
ed. For a system undergoing cyclical changes in state,
1
1
the process is reversible if, at the end of each cycle, the
dS = dSa + dSb = - δQrev
(14)
Tb Ta
surroundings are restored to their original state. At every
stage of this process, the system departs from equilib-
and for a spontaneous process, dS > 0; therefore, Ta >
rium only infinitesimally. Thus, the condition for revers-
ibility is an equilibrium condition (e.g., Castellan 1983):
in thermal equilibrium has the same temperature in all
TdS = δQrev
(7a)
regions, and when it is not in equilibrium, heat flows
from regions of high temperature to low temperature.
and for irreversible (natural) processes
Mechanical equilibrium
TdS > δQrev.
(7b)
For a constant-volume, constant-temperature system
divided into regions a and b, if region a expands revers-
The composite functions are used to describe equi-
ibly by dVa then region b contracts by dVb = dVa.
librium and spontaneous transformation conditions of
According to eq 6a, ( A/ V)T = P, or dA = PdV, and
systems under the constraints for which they were
dA = dAa+ dAb. Therefore, dA = (Pb Pa) dVa. Since
defined. For example, at constant temperature, d(TS) =
δW = 0 (for a constant volume), from eq 10 and the
TdS, and applying eq 7 (TdS ≥ δQ) together with the
second law, dA ≤ 0 and, therefore, Pa > Pb. In other
first law and the substitution that δQ = TdS = d(TS)
words, for a spontaneous expansion of region a into b,
results in
dU + d(TS) ≥ δW
Pb. This is a lot of work to get an obvious result, but
(8)
this type of analysis is helpful when less intuitive pro-
d(U TS) ≥ δW
cesses are described as below.
(9)
For a constant-volume, constant-entropy system
consisting of two phases a and b with an interface, ψ,
dA ≥ δW.
or
(10)
4
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