I was calculating the average potential energy of gas particles in an uniform gravitational field. I have obtained the expected answer - $\left< U\right>=\tau$, but I am not very sure "what exactly it means". Therefore, I am looking for experimental ways to test this result (if I know how to test it, I probably know what it means).
Basically, I started with: $$N\left< U_{potential} \right>=\int^\infty_0mgh\mathbb{N}(h)\ dh\tag{1}$$
Where $N$ is the total number of air particles, $\mathbb{N}$ is the number of air particle per unit height.
In order to obtain the average value, we need to find $\mathbb{N}(h)$ - consider at any height $h_i$, we have chemical potential $\mu_i$:
$$\mu_i=\mu_{external}+\mu_{internal}\tag{2}$$
which is just $\mu_i=\mu_{gravitional\ potential}+\mu_{ideal\ gas}$:
$$\mu_i=mgh_i+\tau \ln (n/n_Q)\tag{3}$$
Where $\tau$ is fundamental temperature, and $n$ is concentration $\big(n(h)*(Area)=\mathbb{N}(h)\big)$ and $n_Q$ is quantum concentration.
Assuming temperature is constant. When all the air particles are in diffusive equilibrium : $\mu_i=\mu_j=constant$. From $(3)$ we derive:
$$\mathbb{N}(h)=n_0\exp [-mgh/\tau]\tag{4}$$
By normalization condition: $N=\int_{all} \mathbb{N}(h)\ dh$ implies $n_0=Nmg/\tau$.
Plug these things back into $(1)$ we have at last:
$$\tag{5}\left< U_{potential} \right>=\tau$$
Now it seems to me that the potential energy increase (the average height) as the air particles' temperature increase. If it is true, can I simply heat up a column of air with some very tiny dusts, then I measure if the height of the dusts increase?
