THE TRANSLATIONAL PARTITION FUNCTION

An expression for translational partition function can be derived as follows. According to definitions,

   q trans = ∑ ɡtrans  e−∈trans/kT

For each translational energy level, there can be only one energy state, i.e ɡ trans = 1.

q trans = ∑e −  ∈trans 

                            kT

The particles can move along anyone of the three coordination. Therefore the net translational partition function can be resolved into three translational partition functions such that,

qtrans = qtrans (x).qtrans (y).qtrans (z)

Where qtrans (x), qtrans (y) and qtrans (z)

are the three Cartesian coordinates. In the equation.

    qtrans = ∑ e⁻∈⨯/kT

It's assumed that the particle is moving along x coordinate only. According to wave mechanics for a particle moving in one dimension, the energy is given by

            ∈χ =  n²h² 

                    8ml²x

Where n = integer (1,2,3......), h = Planck's constant, m = mass of the particle, lₓ= width of the box in which the particle is moving.

   qtrans (x) = ∑ e⁻ⁿ²ʰ²/8ml²ₓ kT

The summation can be evaluated by integration,

        qtrans(x) = ∫₀∞ e⁻   n²h²    

                                        8ml²ₓ KT dn

Put h²/8ml²ₓ kT = a;

qtrans(x) = ∫₀∞ e⁻ᵅⁿ² dn =  1  √π/a

                                               2

Substitute the value of a, we get 

qtrans (x) =1/2   √         π              = 1/2

                              h²/8ml²ₓ kT

√ 8ml²ₓ kT   = (2πmkT)¹/₂   lₓ 

      h²                                     h

The net translational partition function qtrans is given by 

qtrans = qtrans (x).qtrans(y).qtrans (z) =

(2πmkT)³/₂ lₓ.lᵧ.lz : but lₓ.lᵧ.lz = V

       h²

qtrans = (2πmkT)³/₂ v

                   h³

For an ideal gas V = RT/P, hence 

qtrans = (2πmkT) ³/₂ . RT

                    h³                P