The quantum states available for gas atoms of energy ? in a cubical box of length L correspond to integer values for each n_x, n_y and n_z, according.

In a three-dimensional Euclidean space with coordinates n_x, n_y and n_z, each unit volume will contain one quantum state.

The total number of quantum states g with energy less than f is equal to the volume of the positive octant of a sphere of radius r = L(8m?_i)^½/ h.

(a) Show that

g' = (4ΠV(2m∈')^3/2) / (3h³)

(b) In a volume of 1 cm³ of helium gas at 300 K and 1 atm pressure, ? is about 10^-5 J. Calculate g.

(e) Calculate the number N of helium atoms. (d) Show that g.