The State of a System

We specify the state of a system - say, a sample of material - by specifying the values of all the variables describing the system. If the system is a sample of a pure substance this would mean specifying the values of the temperature, *T*, the pressure, *p*, the volume, *V*, and the number of moles of the substance, *n*.

(We must assume that the system is at equilibrium. That is, none of the variables is changing in time and they have the values they would have if we let time go to infinity. We will give a thermodynamic definition of equilibrium later, but this one will suffice for now.)

(If the system is a mixture you also have to specify the composition of the mixture as well as *T*, *p*, and *V*. This could be done by specifying the number of moles of each component, n_{1}, n_{2}, n_{3}, . . . , or by specifying the total number of moles of all the substances in the mixture and the mole fraction of each component, X_{1}, X_{2}, X_{3}, . . . . We will not deal with mixtures on this page.)

Equations of State

Let's consider a sample of a pure substance, say *n* moles of the substance. It is an experimental fact that the variables, *T*, *p*, *V*, and *n* are not independent of each other. That is, if we change one variable one (or more) of the other variables will change too. This means that there must be an equation connecting the variables. In other words, there is an equation that relates the variables to each other. This equation is called the "equation of state." The most general form for an equation of state is,

. (1)

This equation is not very useful because it does not tell us the detailed form of the function, *f*. However, it does tell us that we should be able to solve the equation of state for any one of the variables in terms of the other three. For example, we can, in principle, find

(2)

or

(3)

and so on. (These last two equations should be read as, "*p* is a function of *V*, *T*, and *n*" and "*V* is a function of *p*, *T*, and *n*." )

If we do some more experiments we will notice that when we hold *p* and *T* constant we can't change *n* without changing *V* and vice versa. In fact, *V* is proportional to *n*. That is, if we double *n,* the volume, *V*, will also double, and so on. Because *V* is proportional to *n* these two variables must always appear in the equation of state as *V/n* (or *n/V*). This means that the most general form of the equation of state is simpler than that shown above. The most general form of the equation of state really has the form,

(4)

which can be solved for *p*, *V/n*, or *T* in terms of the other two. For example,

(5)

and so on.

All isotropic^{1} substances have, in principle, an equation of state, but we do not know the equation of state for any real substance. All we have is some approximate equations of state which are useful over a limited range of temperatures and pressures. Some of the approximate equations of state are pretty good and some are not so good. Our best equations of state are for gases. There are no general equations of state for liquids and solids, isotropic or otherwise. On another page we will show you how to obtain an approximate equation of state for isotropic liquids and solids which is acceptable for a limited range of temperatures around 25^{o}C and for a limited range of pressures near one atmosphere.

The Ideal Gas Equation of State

The best known equation of state for a gas is the "ideal gas equation of state." It is usually written in the form,

(6)

This equation contains a constant, *R*, called the gas constant or, sometimes, the universal gas constant^{2}. We can write this equation in the forms shown above if we wish. For example, the analog of Equation (1) is,

(7)

The analogs of Equations (2) and (3) are.

(8)

and

(9)

respectively, and so on.

No real gas obeys the ideal gas equation of state for all temperatures and pressures. However, all gases obey the ideal gas equation of state in the limit as pressure goes to zero (except possibly at very low temperatures). Another way to say this is to say that all gases become ideal in the limit of zero pressure. We will make use of this fact later on in these pages (see "fugacity" for example).

The ideal gas equation of state is the consequence of a model in which the molecules are point masses - that is, they have no size - and in which there are no attractive forces between the molecules.

The van der Waals Equation of State

The van der Waals equation of state is,

. (10)

Notice that the van der Waals equation of state differs from the ideal gas by the addition of two adjustable parameters, *a*, and *b* (among other things). These parameters are intended to correct for the omission of molecular size and intermolecular attractive forces in the ideal gas equation of state. The parameter *b* corrects for the finite size of the molecules and the parameter, *a*, corrects for the attractive forces between the molecules.

The argument goes something like this: Assume that an Avogadro's number of molecules (i.e., a mole of the molecules) takes up a volume of space - just by their physical size - of *b* Liters. Then any individual molecule doesn't have the whole (measured) volume, *V*, available to move around in. The space available to any one molecule is just the measured volume less the volume taken up by the molecules themselves, *nb*. So the "effective" volume, which we shall call *V*_{eff}, is *V - nb*. The effective pressure, *p*_{eff}, is a little bit trickier. Consider a gas where the molecules attract each other. The molecules at the edge of the gas (near the container wall) are attracted to the interior molecules. The number of "edge" molecules is proportional to *n/V* and the number of interior molecules is proportional to *n/V* also. The number of pairs of interacting molecules is thus proportional to *n*^{2}/*V*^{2} so that the forces attracting the edge molecules to the interior are proportional to *n*^{2}/*V*^{2}. These forces give an additional contribution to the pressure on the gas proportional to *n*^{2}/*V*^{2}. We will call the proportionality constant *a* so that the effective pressure becomes, .

We now guess that the gas would obey the ideal gas equation of state if only we used the effective volume and pressure instead of the measured volume and pressure. That is,

. (11)

Inserting our forms for the effective pressure and volume we get,

(12)

which **is** the van der Waals equation of state.

The van der Waals constants, *a* and *b*, for various gases must be obtained from experiment or from some more detailed theory. They are tabulated in handbooks and in most physical chemistry textbooks.

- Isotropic means that the properties of the material are independent of direction within the material. All gases and most liquids are isotropic, but crystals are not. The properties of the crystal may depend on which direction you are looking with respect to the crystal lattice. As we said above, most liquids are isotropic, but liquid crystals are not. That's why they are called liquid crystals. Amorphous solids and polycrystalline solids are usually isotropic.
- The value of the gas constant,
*R*, depends on the units being used.

*R* = 8.314472 J/K mol = 0.08205746 L atm/K mol = 1.987207 cal/K mol = 0.08314472 L bar/K mol.

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