Calculate the number of helium atoms

Question

Response to the following problem:

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;.

(c) Calculate the number N of helium atoms.

(d) Show that g » N.