42
2 Macroscopic Thermodynamics
Along the same line of reasoning that we have employed to define the enthalpy
thermodynamic state function for use in the description of adiabatic and isobaric
processes, it is convenient to define similarly a new thermodynamic state function
to be employed in the descriptions of isothermal and isochoric processes and
having temperature and volume as its natural-independent variables. This may be
accomplished via a Legendre transform of the internal energy U that replaces
the extensive thermodynamic natural variable S by the corresponding intensive
thermodynamic variable T , namely,
A ≡ U − T S ,
(2.3.5a)
whose total differential, dA, can be obtained as
dA = −SdT − P dV .
(2.3.5b)
From Eq. (2.3.5b) we see that the natural-independent thermodynamic variables for
A, which is known as the Helmholtz energy, are T and V . Note that this also requires
that we treat the entropy and pressure as functions of T and V .
It will now also be clear that thermodynamic analyses of processes that occur
either under isothermal or under isobaric conditions will be aided by employing
another new thermodynamic state function, G, obtained using the Legendre transform
G = H − T S
(2.3.6a)
of the enthalpy H or via the Legendre transform
G = A + P V
(2.3.6b)
of the Helmholtz energy A or even via the double Legendre transform
G = U + P V − T S
(2.3.6c)
of the internal energy U . This thermodynamic state function is known as the Gibbs
energy, and has the total differential
dG = −SdT + V dP ,
(2.3.6d)
from which we see that its natural-independent thermodynamic variables are indeed
T and P .
We see from expression (2.3.6d) that for an isothermal process, integration over
pressure from an initial pressure P i to a final pressure P f gives
2 Macroscopic Thermodynamics
Along the same line of reasoning that we have employed to define the enthalpy
thermodynamic state function for use in the description of adiabatic and isobaric
processes, it is convenient to define similarly a new thermodynamic state function
to be employed in the descriptions of isothermal and isochoric processes and
having temperature and volume as its natural-independent variables. This may be
accomplished via a Legendre transform of the internal energy U that replaces
the extensive thermodynamic natural variable S by the corresponding intensive
thermodynamic variable T , namely,
A ≡ U − T S ,
(2.3.5a)
whose total differential, dA, can be obtained as
dA = −SdT − P dV .
(2.3.5b)
From Eq. (2.3.5b) we see that the natural-independent thermodynamic variables for
A, which is known as the Helmholtz energy, are T and V . Note that this also requires
that we treat the entropy and pressure as functions of T and V .
It will now also be clear that thermodynamic analyses of processes that occur
either under isothermal or under isobaric conditions will be aided by employing
another new thermodynamic state function, G, obtained using the Legendre transform
G = H − T S
(2.3.6a)
of the enthalpy H or via the Legendre transform
G = A + P V
(2.3.6b)
of the Helmholtz energy A or even via the double Legendre transform
G = U + P V − T S
(2.3.6c)
of the internal energy U . This thermodynamic state function is known as the Gibbs
energy, and has the total differential
dG = −SdT + V dP ,
(2.3.6d)
from which we see that its natural-independent thermodynamic variables are indeed
T and P .
We see from expression (2.3.6d) that for an isothermal process, integration over
pressure from an initial pressure P i to a final pressure P f gives