Equations of State

Thermodynamics is concerned only with the macroscopic properties of a body and not with its atomic properties, such as the distance between the atoms in a molecule. These macroscopic properties form a large class and include the pressure, surface, volume, tension, viscosity, temperature etc. They may be divided into two groups as follows:

The extensive properties, such as volume and mass, depends on the amount of substance in the sample. They are additive, in the sense that the value of the property for the whole of a body is the sum of the values for all of its constituent parts.

The intnsive properties, such as pressure, density, etc., is independent of the amount of substance in the sample. They can be specified at each point in a system and which may vary from point to point, when there is an absence of equilibrium. Such properties are not additive and do not require any specification of the quantity of the sample to which they refer.

Specifying the state of a system means describing the condition of the system by giving the values of a sufficient set of numerical variables. Consider a closed system which consists of a single phase which is in a state of equilibrium, and is not significantly affected by external fields. For such a system it is usually found that the specification of any two of the intensive variables will determine the values of the rest. For example, for any fixed amount of a pure gas, two state variables are pressure, P, and volume, V. Each can be controlled independently of each other. The pressure can be varied while the volume is kept constant, or vice versa (either implies changing T). Temperature, T, is another state variable that can be changed independently from P and V.

For example, if I1, I2, ... , Ij, ... ,In,., are the intensive properties then the fixing of, say, I1 and I2 will give the values of all the others. Thus

Ij = f(I1,I2)   (j = 3,4,...,n)

Turning now to the extensive properties it is evident that the choice of only two of these is insufficient to determine the state of a system, even if it is a pure substance. Thus if we fix both the volume and mass of a quantity of hydrogen, it is still possible to make simultaneous changes of pressure and of' hotness'. An extensive property of a pure phase is usually determined by the choice of three of its properties, one of which may be conveniently chosen as the mass (thereby determining the quantity of the pure phase in question) and the other two as intensive properties. For example, if E1, E2, ... , Er are extensive properties, then any one of them will usually be determined by the same two intensive variables, E1 and E2, as chosen previously, together with the total mass m. Thus

Ej = m ⋅ f(I1,I2)   (j = 3,4,...,n)

As an example condier the familiar ideal gas law

PV = nRT

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