The ideal gas equation connecting pressure
and absolute temperature
where is the number of moles and N is the number of molecules. and are universal constants.
Real gases satisfy the ideal gas equation only approximately, more so at low pressures and high temperatures.
In equilibrium, the total energy is equally distributed in all possible energy modes, with each mode having an average energy equal to
. This is known as the law of equipartition of energy
. Accordingly, each translational and rotational degree of freedom of a molecule contributes
to the energy while each vibrational frequency
contributes , since a vibrational mode has both kinetic and potential energy modes.
Kinetic theory of an ideal gas gives the relation
where n is number density of molecules, m the mass of the molecule and v2 is the mean of squared speed. Combined with the ideal gas equation it yields a kinetic interpretation of temperature.
For each atom average energy is
. Water molecule has three atoms, two hydrogen and one oxygen. So it has
This is the value observed and the agreement is very good. In the calorie, gram, degree units, water is defined to have unit specific heat. As 1 joules and one mole of water is 18 grams, the heat capacity per mole is
This tells us that 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.
The translational kinetic energy
This leads to a relation
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. Each translational and rotational degree of freedom corresponds to one energy mode of absorption and has energy . Each vibrational frequency has two modes of energy (kinetic and potential) with corresponding energy equal to
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.
The mean free path l is the average distance covered by a molecule between two successive collisions:
A vessel contains two nonreactive gases: neon (monatomic) and oxygen (diatomic). The ratio of their partial pressures is 3:2. Estimate the ratio of (i) number of molecules and (ii) mass density of neon and oxygen in the vessel. Atomic mass of , molecular mass of
Partial pressure of a gas in a mixture is the pressure it would have for the same volume and temperature if it alone occupied the vessel. (The total pressure of a mixture of non-reactive gases is the sum of partial pressures due to its constituent gases.) Each gas (assumed ideal) obeys the gas law. Since V and T are common to the two gases, we have. Here 1 and 2 refer to neon and oxygen respectively. Since .
By definition where N1 and N2 are the number of molecules of 1 and 2, and NA is the Avogadro's number. Therefore, .
(ii) We can also write where m1 and m2 are the masses of 1 and 2; and M1 and M2 are their molecular masses. (Both m1 and M1; as well as m2 and M2 should be expressed in the same units).
If and are the mass densities of 1 and 2 respectively, we have
Uranium has two isotopes of masses 235 and 238 units. If both are present in Uranium hexafluoride gas which would have the larger average speed ? If atomic mass of fluorine is 19 units, estimate the percentage difference in speeds at any temperature.
At a fixed temperature the average energy = is constant. So smaller the mass of the molecule, faster will be the speed.
The ratio of speeds is inversely proportional to the square root of the ratio of the masses. The masses are 349 and 352 units. So,