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Kinetic Theory : Specific Heats Of A Gas |
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When 1 mol of a gas is heated at constant volume from T° to (T+ 1)° abs., the heat absorbed is called the molecular heat at constant volume, Cv = Mcv, where M = molecular weight, cv = specific heat at constant volume. When the gas is heated at a constant pressure of 1 atm. it expands, doing work against the atmospheric pressure, and the heat absorbed is called the molecular heat at constant pressure, Cv = Mcv. If the gas is ideal, no heat absorption results from the change of volume alone, and the difference of molecular heats, (Cp - Cv), will be equal to the external work done, viz. (pressure) x (increase of volume);  In a monatomic gas the heat absorbed increases exclusively the kinetic energy of translation of the molecules, and for 1° rise of temperature this increase of energy will be:  Hence, Cv = 2.982 g. cal. But Cp = Cv + R = 4.970 g. cal., hence the ratio of specific heats for a monatomic gas is Cp/Cv = cp/ cv = γ = 4.970/2.982 = 1.667. If the gas molecule contains more than one atom, part of the heat supplied at constant volume is used up in increasing the kinetic energy of rotation of the molecule, considered as a rigid body; in addition, the energy of vibration of the atoms may be increased if the molecule is not a rigid structure. If this total extra energy is denoted by E, per 1° rise of temperature, we shall have:  The value of Cp/Cv for a gas, the molecules of which contain more than one atom, is found to be less than 1.667, as the table below shows.
Even in the case of gases containing the same number of atoms in the molecule (O2, Cl2; SO2, H2S), γ has different values; the lower values indicate the presence of additional rotations or vibrations in the molecules to which they refer. The value Cp/Cv = 1.667 was found for mercury vapour by Kundt and Warburg in 1876, and thus the monatomic character of the mercury molecule, inferred from vapour densities and on chemical grounds, was confirmed. The method was also used by Ramsay in the case of argon, etc., elements which form no chemical compounds and for which no other method was available in the determination of the atomic weight. In this case, also, the molecules were found to be monatomic. The values of y for a gas is usually determined either by methods depending on the formula for the velocity of sound:  for an ideal gas,or methods depending on the formula for the adiabatic expansion of the gas: pvγ = const., or  = const.Measurements of the specific heats of diatomic gases give values for the moments of inertia of the molecules and thence, since the moment of inertia is of the form Σmr2, of r, the distance between the atoms. At low temperatures the rotational energy of the hydrogen molecule falls off, and at about -230° C. this part of the energy has become zero. The molecular heat then assumes the value Cv = 2.982 for mon-atomic gas. This is explained by the hypothesis that the energy of rotation (and also that of the vibration of the atoms, if this form of energy is present) is regulated by the quantum law, and decreases with temperature much more rapidly in the regions of low temperatures than the value ½RT, the classical expression for the energy of each degree of freedom of rotation, or RT for each vibration. The effect is measurable only with hydrogen. |
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