Gaseous State
6.0 Kinetic theory of gases
6.0 Kinetic theory of gases
The theory that attempts to describe the behaviour of gases is known as kinetic molecular theory. There are seven assumptions or postulates in kinetic theory of gases.
- Gases consist of large number of identical particles (atoms or molecules) that the vacant space between constituent species is so large that the actual volume of the molecules is negligible in comparison to the empty space between them.
- Gaseous particles do not exert force on each other or on the wall of the container except during collision .
- Particles of a gas are always in constant and random motion .
- Particles of a gas move in all possible directions in straight lines .
- Collisions of gas molecules are perfectly elastic. There may be exchange of energy between colliding molecules, their individual energies may change, but the sum of their energies remains constant.
- At any particular time, different particles in the gas have different speeds and hence different kinetic energies. This is because after collision speed of particles changes .
- A molecule's variable speed implies its variable kinetic energy. In kinetic theory, it is assumed that average kinetic energy of the gas molecules is directly proportional to the absolute temperature.
Types of velocity:
1. Root mean square velocity: This velocity is the root of mean of squares of velocities of individual gas molecules. Mathematically,$${C_{rms}} = \sqrt {(C_1^2 + C_2^2 + C_3^2 + .... + C_n^2)/n} $$It can also be expressed as, (for 1 mole of gas)$${C_{rms}} = \sqrt {3RT/M} = \sqrt {3PV/M} $$
2. Average velocity: This velocity is the mean velocity of individual gas molecules. Mathematically,$${C_{av}} = ({C_1} + {C_2} + {C_3} + ....{C_n})/n$$It can also be expressed as,(for 1 mole of gas)$${C_{av}} = \sqrt {8RT/\pi M} = \sqrt {8PV/\pi M} $$
3. Most probable velocity: This velocity is the velocity possessed by maximum number of molecules of a gas. It can be expressed as, (for 1 mole of a gas) $${C_{mp}} = \sqrt {2RT/M} = \sqrt {2PV/M} $$
Note: Ratio of all three velocity is $${C_{rms}}:{C_{av}}:{C_{mp}} = \sqrt 3 :\sqrt {8/\pi } :\sqrt 2 $$ Hence, it is clear that magnitude wise rms velocity has greatest value followed by average velocity and then most probable velocity.
Question 7. Calculate the temperature at which rms velocity of oxygen is same as most probable velocity of nitrogen at ${600^ \circ }C$ .
Solution: $$\begin{equation} \begin{aligned} {600^ \circ }C = (600 + 273.15)K = 873.15K \\ Now,\sqrt {\frac{{3RT}}{{{M_{{O_2}}}}}} = \sqrt {\frac{{2RT}}{{{M_{{N_2}}}}}} \\ \Rightarrow \frac{{3 \times T}}{{32}} = \frac{{2 \times 873.15}}{{28}} \\ \Rightarrow T = 665.26K \\\end{aligned} \end{equation} $$
