Law of Equipartition of Energy

A gas in equilibrium has uniform density and pressure throughout its volume. This is strictly true only if there are no external influences. A gas column under gravity, for example, does not have uniform density (and pressure). As you might expect, its density decreases with height. The precise dependence is given by the so-called law of atmospheres

${\mathrm{n}}_2={\mathrm{n}}_1\exp \left[-\mathrm{mg}\left({\mathrm{h}}_2-{\mathrm{h}}_1\right)/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}\right]$
where ${\mathrm{n}}_2, {\mathrm{n}}_1$ refer to number density at heights ${\mathrm{h}}_2$ and ${\mathrm{h}}_1$ respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:
${\mathrm{n}}_2={\mathrm{n}}_1\exp \left[-{\mathrm{mgN}}_{\mathrm{A}}\left(\mathrm{\rho }-\mathrm{\rho }\text{'}\right)\left({\mathrm{h}}_2-{\mathrm{h}}_1\right)/\left(\mathrm{\rho RT}\right)\right]$
where $\mathrm{\rho }$ is the density of the suspended particle, and $\mathrm{\rho }\text{'}$ that of the surrounding medium.
[ ${\mathrm{N}}_{\mathrm{A}}$ is Avogadro’s number, and $\mathrm{R}$ the universal gas constant.]

From a certain apparatus, the diffusion rate of hydrogen has an average value of $28.7 {\mathrm{cm}}^3{\mathrm{s}}^{–1}$. The diffusion of another gas under the same conditions is measured to have an average rate of $7.2 {\mathrm{cm}}^3{\mathrm{s}}^{–1}$. Identify the gas. [Hint: Use Graham’s law of diffusion: $R_1/R_2={\left(\frac{M_2}{M_1}\right)}^{1/2},$ where $R_1,R_2$ are diffusion rates of gases 1 and 2, and $M_1$ and $M_2$ their respective molecular masses. The law is a simple consequence of kinetic theory.]

A metre long narrow bore held horizontally (and closed at one end) contains a $76 \mathrm{cm}$ long mercury thread, which traps a $15 \mathrm{cm}$ column of air. What happens if the tube is held vertically with the open end at the bottom?

Estimate the average thermal energy of a helium atom at (i) room temperature ($27 °\mathrm{C}$), (ii) the temperature on the surface of the Sun ($6000 \mathrm{K}$), (iii) the temperature of $10$million Kelvin (the typical core temperature in the case of a star).