98
2 Macroscopic Thermodynamics
molecular sizes and intermolecular forces vary from molecule to molecule, the
equation of state (often expressed in terms of the system volume as a function
of temperature and pressure) will be specific to the substance constituting the
thermodynamic system.
If we therefore consider the volume, V , to be a function of T and P , then we
may express the total differential, dV , for the volume as
dV =
∂V
∂T
P
dT +
∂V
∂P
T
dP
=
1
V
∂V
∂T
P
V dT +
1
V
∂V
∂P
T
V dP
or, upon introducing the thermal expansivity, α, and the isothermal compressibility,
κ T , defined in Eq. (2.4.2), as
dV = αV dT − κ T V dP .
(2.8.7)
Division on both sides of this equation by V , followed by integration, allows us to
obtain
V (T , P ) = V (T 0 , P 0 )e
α(T −T 0 )−κ T (P −P 0 )
(2.8.8a)
as the equation of state for a liquid or solid substance.
Typical values of α for liquids are found to lie between 10 −3 and 10 −4 K −1 ,
while values of α for solids are typically between one and two orders of magnitude
smaller. Similarly, typical values of κ T for liquids are of order 10 −5 bar
−1 and for
solids of order 10 −7 bar
−1 . Given the small magnitudes of α and κ T , a reasonable
approximate equation of state for a liquid or solid thermodynamic system is thus
V (T , P ) V (T 0 , P 0 )[1 + α(T − T 0 ) − κ T (P − P 0 )] .
(2.8.8b)
2.8.1 The Van der Waals Model
The most widely-known model for a real thermodynamic system is that proposed
for a fluid (that is, applying to both gaseous and liquid fluid phases) by Van der
Waals in 1875. The equation of state for a Van der Waals fluid can be written either
in the form