properties are additive, the value for the whole being the sum of values for the parts. Intensive properties are independent of the size of a system, for example temperature, pressure, and viscosity. They are not additive, for example, the temperature of a system is not the sum of the temperature of its parts. In general, an extensive property can be converted to an intensive one by dividing it by some other extensive property. For example, density is the mass per volume and is an intensive property. It is generally more convenient to work with intensive rather than extensive properties. For a single component system not undergoing reaction, specification of three variables (two intensive, one extensive) is generally sufficient to determine the rest, and specification of any two intensive variables is generally sufficient to determine the remaining intensive variables.
A final definition is that of a pure substance. A pure substance is one that cannot be separated into fractions of different properties by the same processes as those considered. For example, in many processes, the compound H2O can be considered a pure substance. However, if electrolysis were involved, this would not be the case.
2.3 EQUATIONS OF STATE
Equations of state describe the relationship that exists among the state variables of a system. We will begin by considering the ideal gas law and then very briefly consider two more complex equations of state for gases.
2.3.1 Ideal gas law
The simplest and most fundamental of the equations of state is the ideal gas law.* It states that pressure, volume, temperature, and the number of moles of a gas are related as:
where P is pressure, V is volume, N is the number of moles, T is thermodynamic, or absolute temperature (which we will explain shortly), and R is the ideal gas constant† (an empirically determined constant equal to 8.314 J/mol-K, 1.987 cal/mol-K, or 82.06 cc-atm/deg-mol). This equation describes the relation between two extensive (mass-dependent) parameters, volume and the number of moles, and two intensive (mass-independent) parameters, temperature and pressure. We earlier stated that if we defined two intensive and one extensive system parameter, we could determine the remaining parameters. We can see from eqn. 2.5 that this is indeed the case for an ideal gas. For example, if we know N, P, and T, we can use eqn. 2.5 to determine V.
The ideal gas law, and any equation of state, can be rewritten with intensive properties only. Dividing V by N we obtain the molar volume,
The ideal gas equation tells us how the volume of a given amount of gas will vary with pressure and temperature. To see how molar volume will vary with temperature alone, we can differentiate eqn. 2.6 with respect to temperature, holding pressure constant, and obtain:
(2.7)
which reduces to:
(2.8)
It would be more useful to know the fractional volume change rather than the absolute volume change with temperature, because the result in that case does not depend on the size of the system. To convert to the fractional volume change, we simply divide the equation by V:
Comparing eqn. 2.9 with eqn. 2.5, we see that the right-hand side of the equation is simply 1/T, thus
(2.10)
The left-hand side of this equation, the fractional change in volume with change in temperature, is known as the coefficient of thermal expansion, α:
(2.11)
For an ideal gas, the coefficient of thermal expansion is simply the inverse of temperature.
The compressibility of a substance is defined in a similar manner as the fractional change in volume produced by a change in pressure at constant temperature:
(2.12)
Geophysicists sometimes use the isothermal bulk modulus, KT, in place of compressibility. The isothermal bulk modulus is simply the inverse of compressibility: KT = 1/β. Through a similar derivation to the one we have just done for the coefficient of thermal expansion, it can be shown that the compressibility of an ideal gas is β = 1/P.
The ideal gas law can be derived from statistical physics (first principles), assuming the molecules occupy no volume and have no electrostatic interactions. Doing so, we find that R = NAk, where k is Boltzmann's constant (1.381 × 10–23 J/K), NA is the Avogadro number (the number of atoms in one mole of a substance), and k is a fundamental constant that relates the average molecular energy, e, of an ideal gas to its temperature (in Kelvins) as e = 3kT/2.
Since the assumptions just stated are ultimately invalid, it is not surprising that the ideal gas law is only an approximation for real gases; it applies best in the limit of high temperature and low pressure. Deviations are largest near the condensation point of the gas.
The compressibility factor is a measure of deviation from ideality and is defined as
(2.13)
By definition, Z = 1 for an ideal gas.
2.3.2 Equations of state for real gases
2.3.2.1 Van der Waals equation
Factors we need to consider in constructing an equation of state for a real gas are the finite volume of molecules and the attractive and repulsive forces between molecules arising from electric charges. The van der Waals equation
