Theory of three dimensional motion

Question

Partition function; that the translational energy of 1 mol of molecules is 3/2 RT will come as no surprise. But the calculation of this result further illustrates the use of quantized states and the partition function to obtain macroscopic properties. The partition function is:

 
q_trans = Σ _exp [- (n²_x + n²_y + n²_z) h²/ (8ma²)/kT]  

= Σ _exp [- n²_x h²/ (8ma²)/kT] Σ _exp [- n²_y h²/ (8ma²)/kT] × Σ_exp [- n²_z h2/ (8ma²)/kT]

= Σ _exp [-n²_x h²/(8ma²)/kT] Σ _exp [-n²_y h²/(8ma²)/kT] × Σ_exp [-n²_z h²/(8ma²)/kT]

= q_x q_y q_z

Each of the three partition function terms is like the one-dimensional term. We therefore can use:

q_x = q_y = q_z = √∏/2 [kT/h²/(8ma²)] ^½ 

to obtain, with V = a₃,

q_trans = q_x q_y q_z = (2∏mkT/h²)^3/2 V

The Three dimensional translation energy: the three dimensional translation energy is derivative with respect to temperature can be used to reach an expression for the normal energy of three dimensional translational motions. Although qtrans depends on the particles and the volume of the container, the thermal energy (U - U₀)_trans has, for 1 mol of any gas in any volume the value 3/2 RT.

Distribution over quantum states: the distribution expressions for three dimensional motions can be derived by following the same procedure as we do for one dimensional motion before. First, however, we see that we can use one "effective" quantum number n in place of the three dimensional quantum numbers are n_x, n_y, and n_z.

It is enough for us to deal with a quantity that shows the sum of the square of the equation of quantum numbers rather than with the individual values. We introduce the variable n defined by n² = n²_x + n²_y + n²_z.

Then the allowed energies are given instead of the more detailed manner than the previous one which we have done above. In using the effective quantum number n, we must recognize that there are number of states all with the same value of the energy. The display of states as point shows that, for large n, the additional number of states included when n increases by 1 is equal to 1/2πn². Thus, if we use n as an effective quantum number, we must use g_n = 1/2πn².

Distribution over Quantum states: the distribution expressions for dimensional motion can be derived by following the same procedure as we did for one dimensional motion. First, however, we see that we can use one 'effective" quantum number n in place of the three quantum numbers n_x, n_y and n_z.

(n²_x + n²_y + n²_z) (h²/8ma²)

It is enough for us to deal with a quantity that shows the sum of the squares of the quantum numbers rather than with the individual values. We introduces the variable n defined by n² = n²_x + n²_y + n²_z. then the allowed energies are given by n²h²/(8ma²) instead of the more detailed, but no more useful, expression involving n_x, n_y and n_z.

In using the effective quantum number n, we must recognize that there are a number of states all with the same value of n, or of energy ε_n. The number of states at this energy is the degeneracy gn. The display of states as points shows that, for large n, the additional number of states included when n increases by 1 is equal to ½ ∏n². Thus if we use n as an effective quantum number we must use g_n, ½ ∏n² as the degeneracy.