This problem explores under what conditions the classical limit is reached for a macroscopic cubic box of edge length a. A nitrogen molecule of average translational energy 3/2k_BT is confined in a cubic box of volume V = 1.250 m³ at 298 K. Use the result from Equation (15.25) for the dependence of the energy levels on a and on the quantum numbers n_x ,n_y , and n_z

a. What is the value of the “reduced quantum number” a = √n²_x + n²_y + n²_z for T = 298 K?

b. What is the energy separation between the levels α and α + 1? (Hint: Subtract Ε _{α +1} from Ε _α before plugging in numbers.)

c. Calculate the ratio E α + 1 – E α /kBT and use your result to conclude whether a classical or quantum mechanical description is appropriate for the particle.