## The Equation of State of an Ideal

Two fundamental gas laws were experimentally determined for gases far removed from their point of liquefaction. These were: (a) the Boyle's law in 1660 and (b) the Charles' law in 1787 which was repeated by Gay-Lussac in 1802.

The Boyle's law states that at constant temperature, the pressure p of a gas varies inversely as its volume V, i.e., p V(p, t) = p0 V(p0, t) (2.4.5)

where t denotes the constant temperature (°C) and p0 some standard pressure. In a (p, V) diagram, the isothermals have the shape of hyperbolas.

(b) Charles and Gay-Lussac's law

According to this law, the isobaric volume co-efficient a is the same for all gases independent of temperature and has a value of 1/273. Thus, at a constant pressure po, we may write

Combining (2.4.5) and (2.4.6) we have p V(p, t)= poV(po, 0)( 1 + at) (2.4.7)

We now define a scale of temperature T which is related to the Centigrade scale t by the relation,

Using (2.4.8), we can write (2.4.7) in the form p V = poV(po,273.16)T/273.16 (2.4.9)

Now, since, according to Avogadro's law, the volume occupied by 1 mol (gramme-molecule) of a gas at a given temperature and pressure is the same for all gases, we may write (2.4.9) as pv = (povo/273.16) T = R*T (2.4.10)

where v is taken as the volume of 1 mol of the gas and R* stands for (po vo/273.16) which is constant and called the Universal Gas constant. The numerical value of R* is found to be 8.314 Joules moP1 K_1. If the volume V contains n mols of the gas, pV = nR* T (2.4.11)

It then follows that if we take R as the gas constant for a particular gas, it is related to the Universal Gas constant R* by the relation, R = R*/M, where M is the molecular weight of the gas.

Equation (2.4.11) may also be written as, p = p RT, (2.4.12)

where p is the density of the gas.