Page 41 - Physical Chemistry

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2
2
Chapter 1 Provided
z/(
x
y) and
z/(
y
x) are continuous, as is generally true in physical
Thermodynamics
applications, one can show that they are equal (see any calculus text):
2
2
0 z 0 z
(1.36)*
0x 0y 0y 0x
The order of partial differentiation is immaterial.
Fractions are sometimes written with a slant line. The convention is that
a
a>bc d d
bc

1.7 EQUATIONS OF STATE
Experiment generally shows the thermodynamic state of a homogeneous system with a
fixed composition to be specified when the two variables P and T are specified. If the
thermodynamic state is specified, this means the volume V of the system is specified.
Given values of P and T of a fixed-composition system, the value of V is determined.
But this is exactly what is meant by the statement that V is a function of P and T.
Therefore, V u(P, T), where u is a function that depends on the nature of the system.
If the restriction of fixed composition is dropped, the state of the system will depend
on its composition as well as on P and T. We then have

V f 1P, T, n , n , . . .2 (1.37)
2
1
where n , n , . . . are the numbers of moles of substances 1, 2, . . . in the homogeneous
1
2
system and f is some function. This relation between P, T, n , n , . . . , and V is called
1
2
a volumetric equation of state, or, more simply, an equation of state. If the system
is heterogeneous, each phase will have its own equation of state.
For a one-phase system composed of n moles of a single pure substance, the equa-
tion of state (1.37) becomes V f(P, T, n), where the function f depends on the nature
of the system; f for liquid water differs from f for ice and from f for liquid benzene. Of
course, we can solve the equation of state for P or for T to get the alternative form P
g(V, T, n) or T h(P, V, n), where g and h are certain functions. The laws of thermo-
dynamics are general and cannot be used to deduce equations of state for particular
systems. Equations of state must be determined experimentally. One can also use
statistical mechanics to deduce an approximate equation of state starting from some
assumed form for the intermolecular interactions in the system.
An example of an equation of state is PV nRT, the equation of state of an ideal
gas. In reality, no gas obeys this equation of state.
The volume of a one-phase, one-component system is clearly proportional to the
number of moles n present at any given T and P. Therefore the equation of state for
any pure one-phase system can be written in the form

V nk1T, P2
where the function k depends on what substance is being considered. Since we usually
deal with closed systems (n fixed), it is convenient to eliminate n and write the equa-
tion of state using only intensive variables. To this end, we define the molar volume
V of any pure, one-phase system as the volume per mole:
m
V V>n (1.38)*
m
V is a function of T and P; V k(T, P). For an ideal gas, V RT/P. The m sub-
m m m
script in V is sometimes omitted when it is clear that a molar volume is meant. (A
m
commonly used alternative symbol for V is .)V
m