Kinetic Theory

• Equation of state of a perfect gas

• Work done in compressing a gas

• Kinetic theory of gases −

  • Assumptions

  • Concept of pressure

• Kinetic interpretation of temperature −

  • rms speed of gas molecules

  • Degrees of freedom

  • Law of equi-partition of energy (statement only) and application to specific heat capacities of gases

  • Concept of mean free path

  • Avogadro's number

SUMMARY

1. The ideal gas equation connecting pressure (P), volume (V) and absolute temperature (T ) is PV = μ RT = kB NT. Real gases satisfy the ideal gas equation only approximately, more so at low pressures and high temperatures.

2. The temperature of a gas is a measure of the average kinetic energy of a molecule, independent of the nature of the gas or molecule. In a mixture of gases at a fixed temperature the heavier molecule has the lower average speed.

3. The law of equipartition of energy states that if a system is in equilibrium at absolute temperature T, the total energy is distributed equally in different energy modes of absorption, the energy in each mode being equal to kB T. Each translational and rotational degree of freedom corresponds to one energy mode of absorption and has energy kB T. Each vibrational frequency has two modes of energy (kinetic and potential) with corresponding energy.

4. Using the law of equipartition of energy, the molar specific heats of gases can be determined and the values are in agreement with the experimental values of specific heats of several gases. The agreement can be improved by including vibrational modes of motion.

POINTS TO PONDER

1. Pressure of a fluid is not only exerted on the wall. Pressure exists everywhere in a fluid. Any layer of gas inside the volume of a container is in equilibrium because the pressure is the same on both sides of the layer.

2. We should not have an exaggerated idea of the intermolecular distance in a gas. At ordinary pressures and temperatures, this is only 10 times or so the interatomic distance in solids and liquids. What is different is the mean free path which in a gas is 100 times the interatomic distance and 1000 times the size of the molecule.

3. The law of equipartition of energy is stated thus: the energy for each degree of freedom in thermal equilibrium is kB T. Each quadratic term in the total energy expression of a molecule is to be counted as a degree of freedom. Thus, each vibrational mode gives 2 (not 1) degrees of freedom (kinetic and potential energy modes), corresponding to the energy 2 kB T = kB T.

4. Molecules of air in a room do not all fall and settle on the ground (due to gravity) because of their high speeds and incessant collisions. In equilibrium, there is a very slight increase in density at lower heights (like in the atmosphere). The effect is small since the potential energy (mgh) for ordinary heights is much less than the average kinetic energy mv2 of the molecules.