Kinetic Theory of an Ideal Gases

KINETIC THEORY OF GASES

Fluids consist of molecules which are in random motion, which collide with each other and with the walls of the container. The intermolecular separation in a gas is larger by an order of magnitude than the intermolecular separation in a liquid. The intermolecular separation in a gas increases as the temperature increases, and decreases with increasing density. it is experimentally observed that most gases follow a universal equation of state

pV = n RT

Gas molecules are relatively free:

the interactions between them are small. This means that the total energy of the gas molecules is mostly kinetic.

A simple model of a gas, with point particles representing molecules and their collisions with the walls of the confining vessel causing the pressure exerted by the gas leads to some important conclusions:

(a) The pressure exerted by an ideal gas is given by

p = $\frac{1}{3}$n mc${}_{rms}^2$, where n = number of molecules per unit volume,

m = mass of each molecule, c_rms = the rms speed of each molecule.

(b) The total internal energy of an ideal gs is

U = Nf$\left(\frac{1}{2}{k}_{B}T\right)$ where N = number of molecules in the gas, f = number of degree of freedom, T = absolute temperature of the gas, k_B = $R/N_A$ = Boltzmann's constant.

(c) The rms speed is given by

c_rms=$\sqrt{\frac{3RT}{M}}$, where M represents the molar mass of the gas (i.e. mass of 1 mole of the gas)

(d) The pressure of an ideal gas is given by

p = $\frac{2}{3}$$\frac{U_{tran}}{V}$, where U_tran is the translational energy of the molecules of the gas.

Illustration1: Find r.m.s speed of Hydrogen molecules at room temperature (=300 k).

Solution: Mass of 1 mole of Hydrogen gas= 2 gm = 2 x10^-3 kg

$\Rightarrow$ V_rms = $\sqrt{\frac{3RT}{M}}$

= $\sqrt{\frac{3\times 8.3\times 300}{2\times {10}^{-3}}}$

= 1.93 x 10³ m/s.

2. When the container contains more than one gas, total pressure exerted by all the gases on the wall is sum of pressures exerted by each gas as it would while filling the container alone. In a way, each gas behaves independent of each other. Thus we have

P = P₁+P₂+P₃+……, where P₁, P₂, P₃ are the partial pressures of gases 1, 2 & 3 respectively

This is known as Dalton's Law of partial pressures.

3. One mole of any gas occupies a volume of 22.4 litre at standard temperature and Pressure which are 273.15 (=0°C) and 1.013x10⁵ Pa (=1atm) respectively,

Illustration 2: 4 gm Hydrogen is mixed with 11.2 litre of He at S.T.P. in a container of volume 20 litre. If the final temperature is 300 K find the pressure.

Solution: 4 gm Hydrogen = 2 moles Hydrogen

11.2 $\ell$ He at S.T.P. = 1/2 mole of He

P = P_H + P_He

= (n_H+n_He) \dfrac{RT}{V}\= (2+½) \dfrac{8.3\times (300k)}{(20\times {{10}^{-3}}){{m}^{3}}}\

= 3.12x 10⁵ N/m².

Internal Energy

Internal energy, of any body is sum total of kinetic energies and potential energies of its constituents (at molecular level). In case of an ideal gas, as there are no intermolecular forces, except during collision the possibility of potential energy is ruled out, so it is only kinetic energy. The kinetic energy of the molecules can be of three types.

(i) Translational

(ii) Rotational

(iii) Vibrational

In a way, it means that the energy of molecules is shared in various modes. These independent modes of motions are called degrees of freedom. The table given below gives the number of degrees of freedom for various types of molecules at normal temperature.

Note: At room temperature the energy associated with vibrational motion is negligibly small in comparison to translational and rotational K.E.