# Boyle’s law

This states that, for a true gas, the product of pressure and volume at constant temperature has a fixed value.

As an equation then, one can write

PV = a constant (2.3)

Where p is pressure in Pa and V is volume in m3.

Graphically, this is a family of rectangular hyperbolas, each curve of which shows how the pressure and volume of a gas varies at a given temperature. Early experiment produced this concept of gas behaviour and subsequent theoretical study seems to verify it. This theoretical approach is expressed in the kinetic theory of gases, the basis of which is to regard a gas as consisting of an assembly of spherically shaped molecules. These are taken to be perfectly elastic and to be moving in a random fashion. There are several other restricting assumptions, the purpose of which is to simplify the treatment of the problem. By considering that the energy of the moving molecules is a measure of the energy content of the gas, and that the change of momentum suffered by a molecule upon collision with the wall of the vessel containing the gas is indication of the pressure of the gas, an equation identical with Boyle’s law can be obtained.

However simple Boyle’s law may be to use, the fact remains that it does not represent exactly the manner in which a real gas behaves. Consequently one speaks of gases which are assumed to obey Boyle’s law as being ideal gases. There are several other simple laws, namely, Charles’ law, Dalton’s laws of partial pressures, Avogadro’s law, Joule’s law and Gay Lussac’s law, which are not strictly true but which are in common use. All these are known as the ideal gas laws.

Several attempts have been made to deal with the difficulty of expressing exactly the behaviour of a gas. It now seems clear that it is impossible to show the way in which pressure-volume changes occur at constant temperature by means of a simple algebraic equation. The expression which, in preference to Boyle’s law, is today regarded as giving the most exact answer is in the form of a convergent infinite series:

PV = A(l + B/V + CIV2 + D/V3 + …) (2.4)

The constants A, B, C, D, etc., are termed the virial coefficients and they have different values at different temperatures.

It is sometimes more convenient to express the series in a slightly different way:

PV = A’+ B’p + C’p2 + D’p3 + … (2.5)

At very low pressures the second and all subsequent terms on the right-hand side of the equation become progressively smaller and, consequently, the expression tends to become the same as Boyle’s law. Hence, one may use Boyle’s law without sensible error, provided the pressures are sufficiently small.

The second virial coefficient, B, is the most important. It has been found that, for a given gas, B has a value which changes from a large negative number at very low temperatures, to a positive one at higher temperatures, passing through zero on the way. The temperature at which B equals zero is called the Boyle temperature and, at this temperature, the gas obeys Boyle’s law up to quite high pressures. For nitrogen, the main constituent of the atmosphere, the Boyle temperature is about 50°C. It seems that at this temperature, the gas obeys Boyle’s law to an accuracy of better than 0.1 per cent for pressures up to about 1.9 MPa. On the other hand, it seems that at 0°C, the departure from Boyle’s law is 0.1 per cent for pressures up to 0.2 MPa.

We conclude that it is justifiable to use Boyle’s law for the expression of the physical properties of the atmosphere which are of interest to air conditioning engineering, in many cases.

In a very general sort of way, Figure 2.2 shows what is meant by adopting Boyle’s law for this purpose. It can be seen that the hyperbola of Boyle’s law may have a shape similar to the curve for the true behaviour of the gas, provided the pressure is small. It also seems that if one considers a state sufficiently far into the superheated region, a similarity of curvature persists. However, it is to be expected that near to the dry saturated vapour curve, and also within the wet zone, behaviour is not ideal. Fig. 2.2 Boyle’s law and the true behaviour of a gas.

Posted in Engineering Fifth Edition