4.1 Criteria for Spontaneous Processes and for Equilibrium: The Gibbs and Helmholtz Energies
153
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 that dV 0:
dU − TdS ≤ 0 (simple system, T and V constant)
(4.1-7)
The Helmholtz energy is denoted by A and is defined by
A U − 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
dA dU − TdS − SdT
(4.1-9)
so that if T is constant
dA dU − TdS (constant T )
(4.1-10)
Equation (4.1-7) is the same as
dA ≤ 0 (simple system, T and V 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 to P ext and to P (transmitted). We refer to this case simply as
“constant pressure.” In this case,
dU + PdV − TdS ≤ 0
(simple system,
T and P constant)
(4.1-12)
The Gibbs energy is defined by
G U + 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
G H − TS A + 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 symbol F has been used in the past for both the
Helmholtz energy and the Gibbs energy. To avoid confusion, the symbol F should not
be used for either of these functions and the term “free energy” should not be used.
The differential of the Gibbs energy is
dG dU + PdV + VdP − TdS − SdT
(4.1-15)
153
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 that dV 0:
dU − TdS ≤ 0 (simple system, T and V constant)
(4.1-7)
The Helmholtz energy is denoted by A and is defined by
A U − 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
dA dU − TdS − SdT
(4.1-9)
so that if T is constant
dA dU − TdS (constant T )
(4.1-10)
Equation (4.1-7) is the same as
dA ≤ 0 (simple system, T and V 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 to P ext and to P (transmitted). We refer to this case simply as
“constant pressure.” In this case,
dU + PdV − TdS ≤ 0
(simple system,
T and P constant)
(4.1-12)
The Gibbs energy is defined by
G U + 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
G H − TS A + 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 symbol F has been used in the past for both the
Helmholtz energy and the Gibbs energy. To avoid confusion, the symbol F should not
be used for either of these functions and the term “free energy” should not be used.
The differential of the Gibbs energy is
dG dU + PdV + VdP − TdS − SdT
(4.1-15)