a) (3 points) For an ideal gas of N atoms, show that the energy i of the ith atom with x, y, and z velocity vector components vix, viy, and viz is given by:

i = 1/2 mv^2 ix + 1/2 mv^2 iy + 1/2 mv^2 iz,

b) (3 points) Use the definition of the linear momentum vector with components pix, piy, and piz to show that the energy per particle i is equal to the per particle kinetic energy ki , and is given by:

i = ki = (p^2 ix / 2m + p^2 iy / 2m + p^2 iz / 2m) . Eq. (1)

c) (4 points) Use Eq. (1) and the equipartition theorem to show that the total energy E of the system is equal to the total kinetic energy K, and is given by:

E= K = 3/2 Nk_BT