(M₁ and M₂ are the respective molecular weights of the two gases) and d₁ and d₂ are their respective vapour densities.)

Knowing the experimental gas laws, it is of interest to develop a theoretical model based on the structure of gases, which can co-relate the experiment. Fortunately, such a theory has been developed and is known as kinetic theory of gases.

Kinetic Theory of Gases

The word ‘Kinetic’ means ‘Motion’. Gaseous molecules are assumed to be in constant motion. A theory with the help of which the various gas laws can be derived mathematically is known as the kinetic theory of gases.

The main postulates of the theory are as follows:

Þ      A gas is made of extremely tiny particles called molecules. The molecules of any given gas are identical and have the same mass,

and the molecules are assumed to be dispersed in a lot of vacant space.

Þ      The individual molecules are relatively far apart from each other and they exert very little attraction for each other except under collision of molecules and near the liquification point. The real volume of the gas molecules at ordinary temperatures and pressures is very small in comparison to the total volume of the gas. Here we are talking about real gases or non-ideal gases since ideal gases cannot be liquefied.

Þ      The gaseous molecules are in continuous random, straight line motion with very high speeds in all directions. They collide frequently and this may bring about a change in the direction of movement and a redistribution of energy between the colliding molecules. The collisions are perfectly elastic (i.e. no loss of energy) but only redistribution of energy may occur.

Þ      The force of gravity has negligible effect on the speed of the gas molecules.

Þ      The pressure exerted by a gas is due to collisions made by gas molecules on the walls of the container. Gases not only distribute themselves throughout the total volume of the container but also exert uniform pressure on every point of the container.

Þ      The average K.E. of the molecules is directly proportional to the absolute temperature of the gas.

Hence, the volume occupied by one mole of an ideal gas at standard temperature (273.15 K) and pressure (101.325 K Pa) has a fixed volume (22.414 dm³). This indicates that the number of molecules contained in one mole of any real gas should be a constant quantity. This number is found to be 6.023 × 10²³ and is known as Avogadro number.

Important

Ideal gas is a gas which follows all the above gas laws under all conditions of temperature and pressure.

Real gases generally do not obey the gas laws, exactly, under all conditions of temperature and pressure.

The Ideal Gas Equation

Combination of Boyle’s and Charle’s laws. When temp. (T1) is kept constant and pressure is changed from p1 to p2, Let the new volume be V ^‘.

Important

Value of the proportionality constant k depends on:

(a)      Quantity of gas and

(b)         Units, in which p, V and T, are expressed.

On the basis of Avogadro’s hypothesis, 1 mole of all gases under similar conditions of temp. and  pressure occupies the same volume. Hence k will have the same value for 1 mole of any gas taken.

pV = kT. …(5)

(k is replaced by R called the molar gas constant).

For n moles of gas considered  (5) becomes

PV = nRT. …(6)

Eq. (6) is called the ideal gas equation showing the effect on the third variable when two of the three variables are changed simultaneously for a given amount of a gas. The units of R varies with the units of the other parameters (p, V, T).

e.g. R has the following values

0.0821 litre atm/ K /mole

5.28 × 10¹⁹ ev / K / mole

8.314 Joules / K / mole

1.99 cal / K / mole

0.002 k cal / K / mole

8.314 × 10⁷ erg / K / mole

Illustrations

1.   A two litre flask, containing O2 at 1 atm pressure is at a constant temperature at 27°C. The gas pressure is reduced to 10^–6 atm by attaching the flask to a vacuum pump. Assuming ideal behaviour, answer the following:

(a) What will be the volume of the gas which is left behind?

(b) What will be the no of molecule given in the problem?

Sol. Given that V₁ = 2 l, p₁ = 1 atm, T = 27°C = 300 K

We have the following results

(a)         The volume of oxygen left behind will be the same i.e. 2 l.

(b)         The number of moles of oxygen left behind is given by

(1)     The forces of attraction, which hold the molecules together.

(2)     The thermal energy of these molecules which tend to increase the intermolecular distances.

If the thermal energy of the molecules is much greater than the forces of attraction, the state of matter that result is called the gaseous state. On the other hand, if the forces of attraction are greater than the thermal energy, we have the matter in the liquid state. When these forces of attraction are much more greater than the thermal energy compared to the liquid state, we have matter in its solid state condition. However, on the application of heat, the thermal energy of the molecules can be increased and as a result the intermolecular forces of attraction would relatively decrease simultaneously.

Molecules in the gaseous state possess high energy and have almost no force of attraction. They are far apart and show a great uniformity in behaviour, irrespective of their chemical nature, colour or odour. They are highly compressible and can also be expanded without limit. They also produce pressure on the walls of any container uniformly in all directions. They diffuse rapidly through one other to form a homogeneous mixture, and their separation is also not very easy.

Gas laws

Boyle’s law

At constant temperature, the volume of a sample of gas of definite mass varies inversely with its pressure.

i.e    when temperature is kept constant for a given mass of gas

where V = volume and  p = pressure

Introducing a constant k, we have

pV = k = constant                                …(1)

The value of the proportionality constant k depends upon the following factors:

(1)           Nature of a gas

(2) Temperature of the gas, and

(3)           The mass of the gas

Hence, at constant temperature, for a given mass of a gas, Boyle’s Law states

p₁V₁ = p₂V₂ = k = constant                  …(2)

Eq. (1) can be represented graphically as shown in figure given below.

The general term isotherm (i.e. at const. Temp.) is used to describe the above curves.

OBJECTIVE

Gaseous state is the simplest state of matter and shows the greatest uniformity in behaviour. Let us in this chapter know more about this ever accompanying companion. In this chapter we will study the behaviour of different gases at different physical condition of pressure and temperature. The study includes different laws related to behaviour of gases e.g., Boyle’s law, Charles’ law, Ideal gas law etc. to comprehend the ideal gas equation. We shall also

• know the meaning of absolute scale of temperature
• learn about Dalton’s law of partial pressures and Graham’s law of diffusion
• understand the kinetic gas equation and average kinetic energy of gaseous molecules
• learn the definition of root-mean-square, average and most probable speeds
• learn about the compression factor ?!
• know about van der Waals equation of state

PRE-REQUISITE

Relative atomic mass of an element

The ratio of average mass per atom of the natural isotopic composition of the element to of the mass of an atom of nuclide ¹²C is known as the relative atomic mass of the element.

Relative molecular mass of Compound

The ratio of average mass per molecule of the natural isotopic composition of the compound to  mass of an atom of nuclide ¹²C is known as relative molecular mass of the compound.

Mole of a substance

One mole of a substance contains as many particles (atoms, molecules, ions) as there are atoms in exactly 12 gm of the nuclide ¹²C. This number is approximately equal to 6.023 × 10²³.

or

The minimum weight of any chemical species which contains 6.023 × 10²³ molecules is called one mole.

or

At NTP or STP (Normal/standard temperature and pressure are 0°C and 1 atm, respectively) one mole occupies 22.4 litre of volume.