Thermodynamics and Chemistry

CHAPTER 3 THE FIRST LAW

3.5 APPLICATIONS OFEXPANSIONWORK 74

the phase remains uniform. Consider the surface elementof the boundary, with area
As;, indicated in the figure by a short thick curve. Because the phase is isotropic, the
forceFsysDpAs;exerted by the system pressure on the surroundings is perpendicular
to this surface element; that is, there is no shearing force. The forceFsurexerted by the
surroundings on the system is equal in magnitude toFsysand is directed in the opposite
direction. The volume change for an infinitesimal displacement dsthat reduces the volume

is dVDAs;ds, so that the work at this surface element (from Eq.3.4.1with (^) D 0 ) is
∂wDpdV.
By summing the work over the entire boundary, we find the total reversible expan-
sion work is given by the same expression as for a gas in a piston-and-cylinder device:
∂wDpdV. This expression can be used for deformation caused by reversible displace-
ments of a confining wall, or for a volume change caused by slow temperature changes
at constant pressure. It is valid if the system is an isotropic fluid phase in which other
phases are immersed, provided the fluid phase contacts all parts of the system boundary.
The expression is not necessarily valid for ananisotropicfluid or solid, because the angle
(^) appearing in Eq.3.4.1might not be zero.

3.4.4 Generalities

The expression∂wD pdV for reversible expansion work of an isotropic phase is the
product of a work coefficient,p, and the infinitesimal change of a work coordinate,V.
In the reversible limit, in which all states along the path of the process are equilibrium
states, the system has two independent variables, e.g.,pandV orTandV. The number of
independent variables is one greater than the number of work coordinates. This will turn out
to be a general rule:The number of independent variables needed to describe equilibrium
states of a closed system is one greater than the number of independent work coordinates
for reversible work.
Another way to state the rule is as follows: The number of independent variables is one
greater than the number of differentkindsof reversible work, where each kindiis given by
an expression of the form∂wiDYidXi.

3.5 Applications of Expansion Work

This book usesexpansion workas a general term that includes the work of both expansion
and compression of an isotropic phase.

3.5.1 The internal energy of an ideal gas

The model of an ideal gas is used in many places in the development of thermodynamics.
For examples to follow, the following definition is needed: An ideal gas is a gas
1.whose equation of state is the ideal gas equation,pVDnRT; and
2.whose internal energy in a closed system is a function only of temperature.^11

(^11) A gas with this second property is sometimes called a “perfect gas.” In Sec.7.2it will be shown that if a gas
has the first property, it must also have the second.