Page 116 - Mechanical Engineers' Handbook (Volume 4)

6 Relations among Thermodynamic Properties 105

Note that this formula holds for an infinitesimal change of state along any path (because dU
is path-independent); however, TdS matches Q and PdV matches W only if the path is
reversible. In general, Q TdS and W PdV. The formula derived above for dU can
be written for a unit mass: Tds Pdv du. Additional identities implied by this relation
are
T P
u
u
s v v s

u T P
2
s v v s
s v
where the subscript indicates which variable is held constant during partial differentiation.
Similar relations and partial derivative identities exist in conjunction with other derived
functions such as enthalpy, Gibbs free energy, and Helmholtz free energy:

• Enthalpy (defined as h u Pv)
dh Tds v dP
T v
h
h
s P P s

h T v
2
s P P s
s P
• Gibbs free energy (defined as g h Ts)
dg sdT v dP
s v
g
g
T P P T

g s v
2
T P P T
T P
• Helmholtz free energy (defined as ƒ u Ts)
df sdT Pdv
s P
ƒ
ƒ
T v v T

2
ƒ s P
T v v T
T v
In addition to the (P, v, T) surface, which can be determined based on measure-
1
ments (Fig. 3), the following partial derivatives are furnished by special experiments :
• The specific heat at constant volume, c ( u/ T) , follows directly from the constant
v
v
volume ( W 0) heating of a unit mass of pure substance.
• The specific heat at constant pressure, c ( h/ T) , is determined during the
P
P
constant-pressure heating of a unit mass of pure substance.