Equations of state

   We have already remarked that the state of any sample of substance can be specified by giving the values of the following properties:

V, the volume the sample occupies

p, the pressure of the sample

T, the temperature of the sample

n, the amount of substance in the sample

  However, an astonishing experimental fact is that these four quantities are not independent of one another. For instance, we cannot arbitrarily choose to have a sample of 0.555 mol H₂O in a volume of 100 cm³ at 100 kPa and 500 K: it is found experimentally that that state simply does not exist. If we select the amount, the volume, and the temperature, then we find that we have to accept a particular pressure (in this case, close to 230 kPa). The same is true of all substances, but the pressure in general will be different for each one. This experimental generalization is summarized by saying the substance obeys an equation of state, an equation of the form

p = f(n,V,T)        .1

  This expression tells us that the pressure is some function of amount, volume, and temperature and that if we know those three variables, then the pressure can have only one value. The equations of state of most substances are not known, so in general we cannot write down an explicit expression for the pressure in terms of the other variables.

   However, certain equations of state are known. In particular, the equation of state of a low-pressure gas is known and proves to be very simple and very useful. This equation is used to describe the behavior of gases taking part in reactions, the behavior of the atmosphere, as a starting point for problems in chemical engineering, and even in the description of the structures of stars. We now pay some attention to gases because they are the simplest form of matter and give insight, in a reasonably uncomplicated way, into the time scale of events on a molecular scale. The equation of state of a low-pressure gas was among the first results to be established in physical chemistry. The original experiments were carried out by  Robert Boyle in the seventeenth century, and there was a resurgence in interest later in the century when people began to fly in balloons. This technological progress demanded more knowledge about the response of gases to changes of pressure and temperature and, like technological advances in other fields today, that interest stimulated a lot of experiments. The experiments of Boyle and his successors led to the formulation of the following perfect gas equation of state:

pV = nRT           .2

In this equation (which has the form of eqn.1 when we rearrange it into p = nRT/V), the gas constant, R, is an experimentally determined quantity that turns out to have the same value for all gases. It may be determined by evaluating  R= pV/nRT as the pressure is allowed to approach zero or by measuring the speed of sound (which depends on R). Values of R in different units are given in Table .1.

In SI units the gas constant has the value R= 8.314 47 J K^-1 mol^-1

The perfect gas equation of state—more briefly, the “perfect gas law”—is so called because it is an idealization of the equations of state that gases actually obey.

   Specifically, it is found that all gases obey the equation ever more closely as the pressure is reduced toward zero. That is, eqn.2 is an example of a limiting law, a law that becomes increasingly valid as the pressure is reduced and is obeyed exactly at the limit of zero pressure. A hypothetical substance that obeys eqn .2 at all pressures is called a perfect gas.

   From what has just been said, an actual gas, which is termed a real gas, behaves more and more like a perfect gas as its pressure is reduced toward zero. In practice, normal atmospheric pressure at sea level (p = 100 kPa) is already low enough for most real gases to behave almost perfectly, and unless stated otherwise, we shall always assume in this text that the gases we encounter behave like a perfect gas.