Briefly state the ergodic hypothesis and explain its importance.

Consider an ideal, classical, monatomic gas in the presence of a uniform gravitational field in the negative $z$-direction. For convenience, assume the gas is in an arbitrarily large cubic box.

(i) Compute the internal energy $E$ of the gas.

(ii) Explain your result for $E$ in relation to the equipartition theorem.

(iii) What is the probability that an atom is located at a height between $z$ and $z+d z ?$

(iv) What is the most probable speed of an atom of this gas?