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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?

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  • $\begingroup$ It appears that you want to find a way to measure how the density profile of an air column changes as the temperature of the air changes. The best approach would be to measure air density directly instead of assuming that suspended dust particles would have the same density profile. Perhaps you could measure sound speed or refractive index of the air at different heights. $\endgroup$
    – S. McGrew
    Commented Mar 28, 2018 at 17:47
  • $\begingroup$ @S.McGrew thanks for your suggestion! Yes, you are right. it is hard to quantify "the height of dust" from the calculation. Also, since speed of sound depends on temperature, and can be measured exactly. $\endgroup$
    – Shing
    Commented Mar 28, 2018 at 18:07
  • $\begingroup$ @S.McGrew would you mind writing an answer for this question? (If I understand you correctly: we can measure temperature by measuring speed of sound at different height, and then to see if the average potential energy equal to $\tau=k_BT$) $\endgroup$
    – Shing
    Commented Mar 28, 2018 at 18:18

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As described in [https://en.wikipedia.org/wiki/Speed_of_sound], the following equation is a good approximation:

$c_{air} = 331.3(( 1 + ϑ(2⋅273.15))$ meters/sec,

where $c_{air}$ is the speed of sound in air and $ϑ^∘C$ is the temperature in degrees C.

So, by measuring sound speed vs height you can calculate temperature vs height. From temperature and pressure you can calculate density, so you should be able to determine the mass distribution from the bottom to the top of the column.

You can set up an acoustic interferometry system to measure changes in sound speed. However, a quick internet search shows that several types of sound speed sensors (many of them interferometric) are commercially available.

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