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 conditions. 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
P P(T , V , n)
(1.1-4)
A mathematician would write P f (T , V , n) for the functional relation in Eq. (1.1-4),
using the letter P for the variable and the letter f for 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 relates P, V , T , and n for a
gas or a liquid at equilibrium is called an equation of state and is said to represent
the volumetric behavior of the gas or liquid. We will introduce several equations of
state later in this chapter.
E X A M P L E 1.1
Assume that the volume of a liquid is a linearly decreasing function of P, is a linearly
increasing function of T , and is proportional to n. Write a formula expressing this functional
relationship.
Solution
Let V 0 represent the volume at some reference temperature T 0 , some reference pressure P 0 ,
and some reference amount of substance n 0 .
V V 0
n
n 0
[1 − k(P − P 0 ) + a(T − T 0 )] nV m0 [1 − k(P − P 0 ) + a(T − T 0 )]
where k and a are constants and where V m represents the molar volume, equal to V /n, and
V m0 represents V 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 of P on V for an ideal gas for n 1.000 mol and for several fixed
values of T .
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 of P on V and T for an ideal gas with a fixed value of n
(1.000 mol). Just as the height of a curve in Figure 1.1a gives the value of P for a
particular value of V , the height of the surface in Figure 1.1b gives the value of P for
a particular value of T and a particular value of V . Such graphs are not very useful for
numerical purposes, but help in visualizing the general behavior of a function of two
independent variables.
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 conditions. 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
P P(T , V , n)
(1.1-4)
A mathematician would write P f (T , V , n) for the functional relation in Eq. (1.1-4),
using the letter P for the variable and the letter f for 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 relates P, V , T , and n for a
gas or a liquid at equilibrium is called an equation of state and is said to represent
the volumetric behavior of the gas or liquid. We will introduce several equations of
state later in this chapter.
E X A M P L E 1.1
Assume that the volume of a liquid is a linearly decreasing function of P, is a linearly
increasing function of T , and is proportional to n. Write a formula expressing this functional
relationship.
Solution
Let V 0 represent the volume at some reference temperature T 0 , some reference pressure P 0 ,
and some reference amount of substance n 0 .
V V 0
n
n 0
[1 − k(P − P 0 ) + a(T − T 0 )] nV m0 [1 − k(P − P 0 ) + a(T − T 0 )]
where k and a are constants and where V m represents the molar volume, equal to V /n, and
V m0 represents V 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 of P on V for an ideal gas for n 1.000 mol and for several fixed
values of T .
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 of P on V and T for an ideal gas with a fixed value of n
(1.000 mol). Just as the height of a curve in Figure 1.1a gives the value of P for a
particular value of V , the height of the surface in Figure 1.1b gives the value of P for
a particular value of T and a particular value of V . Such graphs are not very useful for
numerical purposes, but help in visualizing the general behavior of a function of two
independent variables.