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
