Physical Chemistry Third Edition

6 1 The Behavior of Gases and Liquids

several variables, you can manipulate the equation symbolically to turn any one of

them into the dependent variable.

The ideal gas law might not be accurate enough for some gases under some condi-

tions. If so, we can find some other function that will give the value of the pressure to

greater accuracy. It is an experimental fact that the pressure of a gas or liquid of one

substance at equilibrium is given by a function that depends on only three independent

variables. We represent such a function by

PP(T,V,n) (1.1-4)

A mathematician would writePf(T,V,n) for the functional relation in Eq. (1.1-4),

using the letterPfor the variable and the letterffor the function. Chemists have

too many variables to use two letters for each variable, so we use the same letter for

the variable and the function. A functional relation that relatesP,V,T, andnfor a

gas or a liquid at equilibrium is called anequation of stateand is said to represent

thevolumetric behaviorof the gas or liquid. We will introduce several equations of

state later in this chapter.

EXAMPLE 1.1

Assume that the volume of a liquid is a linearly decreasing function ofP, is a linearly

increasing function ofT, and is proportional ton. Write a formula expressing this functional

relationship.

Solution

LetV 0 represent the volume at some reference temperatureT 0 , some reference pressureP 0 ,

and some reference amount of substancen 0.

VV 0

n

n 0

[ 1 −k(P−P 0 )+a(T−T 0 )]nVm0[ 1 −k(P−P 0 )+a(T−T 0 )]

wherekandaare constants and whereVmrepresents the molar volume, equal toV/n, and

Vm0representsV 0 /n 0.

A two-dimensional graph can represent a function of one independent variable.

You plot the value of the independent variable on the horizontal axis and represent

the value of the dependent variable by the height of a curve in the graph. To make a

two-dimensional graph that represents the ideal gas law, we must keep two of the three

independent variables fixed. Figure 1.1a shows a set of graphical curves that represent

the dependence ofPonVfor an ideal gas forn1.000 mol and for several fixed

values ofT.

A three-dimensional graph can represent a function of two independent variables.

Figure 1.1b shows a perspective view of a graphical surface in three dimensions that

represents the dependence ofPonVandTfor an ideal gas with a fixed value ofn

(1.000 mol). Just as the height of a curve in Figure 1.1a gives the value ofPfor a

particular value ofV, the height of the surface in Figure 1.1b gives the value ofPfor

a particular value ofTand a particular value ofV. Such graphs are not very useful for

numerical purposes, but help in visualizing the general behavior of a function of two

independent variables.