Ideal gas, potential energy help

Question

Let us consider an ideal gas in a gravitational field which is uniform with strength g. The gas is trapped in a highly realistic h=10km vertical column of height h with perfectly insulating walls. The bottom of the container is in contact with a metal plate maintained at temperature T.

The hamiltonian applying to the gas particles is:

H= (p²/2m) + mgz

I want to find the mean potential energy of the particles in the gas and thereby find the mean elevation of the particles assuming the gas is helium, T=300K, h=10km and the cylinder is on earth.

Please if someone can help me, that would be awesome

Answer 1

I'd have to differ on the answer here. Richard P., the adiabatic lapse rate arises from a volume of gas expanding against a gradually decreasing surround pressure. Thus, such a gas packet does work on the surroundings as it rises, and experiences a loss of sensible heat as a result. Here, in spite of perfectly insulating walls, one must expect both a temperature decrease (molecules slow their vertical component as they rise against the gravitational potential) from ~290 K at sealevel to ~220 K at 10 km, and a pressure decrease (from ~760 to ~200 mm Hg). One can immediately appreciate that the temperature decrease in insufficient to account solely for all the pressure decrease; there is a substantial actual density decrease also! (Actual atmosphere data: https://www.avs.org/AVS/files/c7/c7edaedb-95b2-438f-adfb-36de54f87b9e.pdf ). This must be taken into account when integrating for mean potential energy of the molecules.

Actually the above data are for air (mostly N2): the He of the problem will be somewhat less concentrated towards the cylinder bottom than is N2, i.e. the exponential decreases in temperature and pressure will be spread to higher altitudes, but only by a scale factor proportional to (m1/m2)^0.5.

Sorry I can't offer you a better closed-form total solution; the pressure for Earth's atmosphere is considered to follow an exponential decay with 7 km as the 1/e-producing scale factor.(reference: http://www.regentsprep.org/regents/math/algtrig/atp8b/exponentialresource.htm)

 

Anyway, hope this helps you start to get a grip on the problem!

Answer 2

I think your question is related to the dry adiabatic lapse rate.   Thermodynamics  is required for analysis but the result is simple.   The temperature of the gas falls (as a function of height)  with slope Γ_d =  g/c_p   where c_p  is the  specific heat at constant pressure and g is the acceleration due to gravity.  Γ_d is called the adiabatic lapse rate.  For an ideal gas

 

  c_p =(1/MM)  Υ R/(γ -1)   where γ  is the  heat capacity ratio (equal to 5/3 for monatomic gases such as argon),  MM is the molar mass,  and R is the gas constant.  The temperature in the column is therefore  T(h) =  T - (g/c_p ) h. 

 

  The mean kinetic energy  per atom is  (3/2)k_BT.   The potential energy is mgh.