# Ideal Gas Equation

A gas that follows Boyle’s law, Charles’ law and Avogadro’s law strictly at all conditions, is called ideal gas. It is assumed that intermolecular forces are not present between the molecules of ideal gas. Real gases follow these laws only under certain specific conditions when forces of interaction are practically negligible. In all other situations the real gases deviate from ideal behaviour.

The combination of various gas laws namely; Boyles law, Charles’ law and Avogadro S law leads to the development of the mathematical relation which relates four variables pressure, volume, absolute temperature and number of moles of ideal gas. The equation so formulated is called ideal gas equation.

The ideal gas equation can be derived by combining Boyle’s law, Charles’ law and Avogadro’s law as follows: According to Boyle’s law

V = 1/p                                    ( at constant T and n)

According to Charles’ law

V =T                                        (at constant P and n)

According to Avogadro’s law

V = n                                       (at constant T and P)

Combining (i), (ii) and (iii)

V = nT / p

PV = nT

PV = nRT

where, R is constant of proportionality and is known as universal gas constant.

Equation (15.1) is called ideal gas equation. This equation is applicable to any gas under those conditions when behaviour of gas approaches ideal behaviour. Since this equation relates four variables which are used to describe the state of any gas, it is also known as equation of state for ideal gas. It may be noted that out of the four variables, the two namely: pressure (P) and temperature (T) are intensive variables as they do not depend on the bulk or quantity of the gas. The other two variables, i.e., volume (V) and mole (n) are extensive variables as they depend upon the bulk or quantity of the gas.

COMBINED GAS LAW OR GENERAL GAS EQUATION

If temperature, volume and pressure of fixed amount (say n mol) of gas vary from T1V1 and P1 to T2, V2 and P1 respectively. Then ideal gas equation for two states can be

P1 V1 = nRT1 or  P1 V1 / T1 = nR

P2 V2 = nRT2 or P2 V2 / T2 = Nr

Combining (i) and (ii) we have

P1 V1 / T1 = P2 V2 / T2

Equation (15.2) is called combined gas law or general gas equation. This equation is very useful in calculations involving gas laws. The knowledge of five of the variables help in calculating the unknown sixth variable.

RELATIONSHIP BETWEEN DENSITY AND  MOLECULAR MASS

We have studied in unit one that the number of moles (n) of the substance is related to the molar mass (M) as

n  = w/M

where, w = mass in gram.

Substituting this value in the ideal gas equation we can get the relationship between density and molar mass,

PV = nRT = w / M RT

P = wRT / MV  or  P = dRT / M

Where d is the density of gas [ d = Mass / volume = w / v

P = dRT / M
NATURE AND NUMERICAL VALUES OF THE GAS CONSTANT (R)

In order to understand the significance of R, let us examine the nature of quantities in the ideal gas equation:

PV = nRT  or R = PV / nT = pressure x volume / moles x temperature = force x length / moles x temperature

Since force x length = work energy

R = work / moles x temperature

Thus, R represents work done per degree per mole.

Since work can be expressed in different systems of units, R will have different numerical values in different systems.

Different numerical values of R are given in the Table 15.1. It may be noted that although R can be expressed in different units, but for pressure volume calculations R must be taken in the same units as those used for pressure and volume.

Table 15.1. Values of R in Different Units Let us now apply ideal gas equation and combined gas laws to solve some numerical problems. Students may note that in many of these problems the conversion of temperature in celsius scale to kelvin scale has been done by adding 273instead of 27 3.15 in order to make the calculations simple.