Page 30 - Physical chemistry eng

1.4 EQUATIONS OF STATE AND THE IDEAL GAS LAW 7

all species, equilibrium can exist with respect to P, but not with respect to concentra-
tion. Because N 2 and O 2 cannot diffuse through the (idealized) bubble, the system and
surroundings are in equilibrium with respect to P, but not to concentration. Equilibrium
with respect to temperature is a special case that is discussed next.
Two systems that have the same temperature are in thermal equilibrium. We use
the concepts of temperature and thermal equilibrium to characterize the walls
between a system and its surroundings. Consider the two systems with rigid walls
shown in Figure 1.7a. Each system has the same molar density and is equipped with a
pressure gauge. If we bring the two systems into direct contact, two limiting behav-
(a)
iors are observed. If neither pressure gauge changes, as in Figure 1.7b, we refer to the
walls as being adiabatic. Because P Z P 2 , the systems are not in thermal equilib-
1
rium and, therefore, have different temperatures. An example of a system surrounded
2
by adiabatic walls is coffee in a Styrofoam cup with a Styrofoam lid. Experience
shows that it is not possible to bring two systems enclosed by adiabatic walls into
thermal equilibrium by bringing them into contact, because adiabatic walls insulate
against the transfer of “heat.” If we push a Styrofoam cup containing hot coffee
against one containing ice water, they will not reach the same temperature. Rely on
experience at this point regarding the meaning of heat; a thermodynamic definition
will be given in Chapter 2.
The second limiting case is shown in Figure 1.7c. In bringing the systems into inti- (b)
mate contact, both pressures change and reach the same value after some time. We
= T , and say that they are
conclude that the systems have the same temperature, T 1 2
in thermal equilibrium. We refer to the walls as being diathermal. Two systems in
contact separated by diathermal walls reach thermal equilibrium because diathermal
walls conduct heat. Hot coffee stored in a copper cup is an example of a system sur-
rounded by diathermal walls. Because the walls are diathermal, the coffee will quickly
reach room temperature.
The zeroth law of thermodynamics generalizes the experiment illustrated in
Figure 1.7 and asserts the existence of an objective temperature that can be used to define
the condition of thermal equilibrium. The formal statement of this law is as follows:
(c)
Two systems that are separately in thermal equilibrium with a third system are FIGURE 1.7
also in thermal equilibrium with one another. (a) Two separated systems with rigid
walls and the same molar density have
The unfortunate name assigned to the “zeroth” law is due to the fact that it was formu- different temperatures. (b) Two systems
lated after the first law of thermodynamics, but logically precedes it. The zeroth law tells us are brought together so that their adiabatic
walls are in intimate contact. The pressure
that we can determine if two systems are in thermal equilibrium without bringing them into
in each system will not change unless heat
contact. Imagine the third system to be a thermometer, which is defined more precisely in
transfer is possible. (c) As in part (b), two
the next section. The third system can be used to compare the temperatures of the other two
systems are brought together so that their
systems; if they have the same temperature, they will be in thermal equilibrium if placed diathermal walls are in intimate contact.
in contact. The pressures become equal.

1.4 Equations of State and the Ideal Gas Law

Macroscopic models in which the system is described by a set of variables are based on
experience. It is particularly useful to formulate an equation of state, which relates the
state variables. A dilute gas can be modeled as consisting of point masses that do not
interact with one another; we call this an ideal gas. The equation of state for an ideal
gas was first determined from experiments by the English chemist Robert Boyle. If the
pressure of He is measured as a function of the volume for different values of tempera-
ture, the set of nonintersecting hyperbolas as shown in Figure 1.8 is obtained. The
curves in this figure can be quantitatively fit by the functional form
PV = aT (1.17)

2
In this discussion, Styrofoam is assumed to be a perfect insulator.