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Thursday, March 13, 2008

AP Physics B– Kinetic Theory of Gases – Equations to be Remembered

You will have to remember certain basic formulae for solving multiple choice questions within the permitted time. The essential things you need to remember in kinetic theory of gases are given below:

(1) Pressure exerted by a gas, P = (1/3) nmc2 = (1/3)ρc2 where ‘n’ is the number of molecules per unit volume, ‘m’ is the molecular mass, ‘c’ is the r.m.s. speed of the gas molecule and ρ’ is the density of the gas.

(2) A very useful expression for the pressure of a gas is P = nkT where ‘k’ is Boltzman’s constant and T is the absolute temperature (Kelvin scale).

(3) Root mean square (R.M.S.) speed of gas molecule, c = √(3P/ρ) = √(3kT/m).

This can be rewritten in terms of the molar mass M and the universal gas constant R as

c = √(3RT/M)

(4) Since translational motion along three directions only are possible in our three dimensional space, the average translational kinetic energy of any type of gas molecule is (3/2)kT

(5) If the molecule has ‘f’ degrees of freedom, the average kinetic energy per molecule is (f/2 )kT.

Note the following points in this context (bearing in mind that the energy per degree of freedom is ½ kT):

(i) A mono atomic gas molecule has 3 degrees of freedom and has translational kinetic energy only [equal to (3/2)kT ].

(ii) A diatomic gas molecule has 5 degrees of freedom (three translational and two rotational) and hence the total average kinetic energy per molecule is (5/2 )kT.

(iii) Tri-atomic and polyatomic gas molecules have 6 degrees of freedom (three translational and three rotational). The total average kinetic energy per molecule is ( 6/2 )kT = 3kT.

(6) The kinetic energy. per mole in all the above cases is N times the kinetic energy of a molecule where N is the Avogadro number. Since Nk=R, the average K.E. per mole is (3/2 )RT for mono atomic, (5/2 )RT for diatomic and 3RT for triatomic and polyatomic gas molecules.

(7) The molar heat capacity (molar specific heat) at constant volume (CV) is obtained by putting T = 1 (corresponding to a temperature rise of 1K) in the above expressions. The values are therefore (3/2 )R for mono atomic gas, (5/2 )R for diatomic gas and 3R for tri atomic and polyatomic gases.

(8) The molar heat capacity (molar specific heat) of a gas at constant pressure (CP) is given by

CP = CV + R. This is Meyer’s relation.

Therefore, the values of CP are (5/2)R for mono atomic gas, (7/2)R for diatomic gas and 4R for tri and poly atomic gases.

(9) The ratio of specific heats of a gas is γ = CP/CV

The ratio of specific heats ‘γ’ is related to the number of degrees of freedom ‘f’' as

γ = 1+ (2/f)

[You should note that in the above discussion, the vibrational modes of the molecules have not been considered. Even though the above values are in agreement with the values obtained from experiment in the case of several gases, there are discrepancies in the case of certain diatomic gases and several polyatomic gases. In a more rigorous treatment, the vibrational modes also are to be taken into account; but the above discussion is sufficient at the moment].

In the next post, questions on kinetic theory will be discussed. Meanwhile, find some useful posts in this section here

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