![Reference: 'Thermal walls in computer simulations', R. Tehver, Phys Rev E 1998,
<http://pre.aps.org/abstract/PRE/v57/i1/pR17_1> , which can also be downloaded from
<http://polymer.chph.ras.ru/asavin/teplopr/ttk98pre.pdf>
In the above paper, one finds that molecules emitted from a wall of temperature T have a
velocity distribution of
[1]
where v is the velocity normal to the wall. In our case, let us take a ground plane as a
thermal wall with temperature T1, have gravity present so that the normal component of
velocity reduces with altitude, and find how this distribution varies as a function of
altitude.
First we note that for a molecule of mass m which reaches a plane 2 at height h, its
kinetic energy will be reduced by the gravitational potential energy.
[2]
The velocity at height h for this particular molecule can then be found as
[3]
Now let us deal with the distribution of velocities. First note that the number of particles
which can be found within velocities va and vb at ground level is given by
[4]
This is not hard to solve, but for convenience can be solved via ‘The Integrator’, e.g.
<http://integrals.wolfram.com/index.jsp?expr=%28m%2F%28kT%29%29*x*exp%28-
0.5*m*x%5E2%2F%28kT%29%29&random=false>
This gives the result
[5]
Now let us assume that va is large enough so that all the particles reach height 2. We do
not know the equation for the distribution of velocities, or the temperature, at height 2,
but we do know that va will be reduced according to Eqn(3) to
[6]
and vb will be reduced to
[7]](https://image.slidesharecdn.com/analyticvelocitydistributionundergravity-120606135908-phpapp01/85/Analytic-velocity-distribution-under-gravity-1-320.jpg)
![and that the particle number will remain constant so that
[8]
We then write an equation for the unknown distribution at level 2 to get
[9]
From Eqns(6,7), we have a relation between va and vc, and between vb and vd, so solving
for va and vb, and substituting in, one gets
[10]
Fortunately, we already know a function that when integrated between its limits gives
such a dependence on v, it is of course 1. Hence we can relate 2 to 1 as
[11]
showing that the distribution at altitude 2 has the same characteristic temperature as the
distribution at altitude 1, the functional form of the distribution is the same, the only
difference being a reduction in amplitude. This reproduces the standard result of an
isothermal column of gas under gravity, with a density drop off according to the
barometric formula.
Note on assumptions:
By using the 1 distribution, one circumvents the complaint that the equilibrium volume
distribution in a gravitational field is assumed to be that of the MB form. The only
requirement in the above derivation is that there is thermal continuity between the gas
and the ground at the interface between the two, and this requirement is independent of
the presence or absence of a gravitational field.
The density drop off is of course due to those particles which leave ground level but do
not reach altitude 2. As these molecules have v<v1min, they do not affect the condition
ncd=nab which transfers molecules with va>v1min to altitude 2 (v1min being the minimum
velocity needed for a molecule leaving ground level to reach altitude 2).
Other assumptions are usual for an ideal gas: no collisions, no Van der Waals forces, etc.](https://image.slidesharecdn.com/analyticvelocitydistributionundergravity-120606135908-phpapp01/85/Analytic-velocity-distribution-under-gravity-2-320.jpg)
Analytic velocity distribution under gravity
- 1. Reference: 'Thermal walls in computer simulations', R. Tehver, Phys Rev E 1998, <http://pre.aps.org/abstract/PRE/v57/i1/pR17_1> , which can also be downloaded from <http://polymer.chph.ras.ru/asavin/teplopr/ttk98pre.pdf> In the above paper, one finds that molecules emitted from a wall of temperature T have a velocity distribution of [1] where v is the velocity normal to the wall. In our case, let us take a ground plane as a thermal wall with temperature T1, have gravity present so that the normal component of velocity reduces with altitude, and find how this distribution varies as a function of altitude. First we note that for a molecule of mass m which reaches a plane 2 at height h, its kinetic energy will be reduced by the gravitational potential energy. [2] The velocity at height h for this particular molecule can then be found as [3] Now let us deal with the distribution of velocities. First note that the number of particles which can be found within velocities va and vb at ground level is given by [4] This is not hard to solve, but for convenience can be solved via ‘The Integrator’, e.g. <http://integrals.wolfram.com/index.jsp?expr=%28m%2F%28kT%29%29*x*exp%28- 0.5*m*x%5E2%2F%28kT%29%29&random=false> This gives the result [5] Now let us assume that va is large enough so that all the particles reach height 2. We do not know the equation for the distribution of velocities, or the temperature, at height 2, but we do know that va will be reduced according to Eqn(3) to [6] and vb will be reduced to [7]
- 2. and that the particle number will remain constant so that [8] We then write an equation for the unknown distribution at level 2 to get [9] From Eqns(6,7), we have a relation between va and vc, and between vb and vd, so solving for va and vb, and substituting in, one gets [10] Fortunately, we already know a function that when integrated between its limits gives such a dependence on v, it is of course 1. Hence we can relate 2 to 1 as [11] showing that the distribution at altitude 2 has the same characteristic temperature as the distribution at altitude 1, the functional form of the distribution is the same, the only difference being a reduction in amplitude. This reproduces the standard result of an isothermal column of gas under gravity, with a density drop off according to the barometric formula. Note on assumptions: By using the 1 distribution, one circumvents the complaint that the equilibrium volume distribution in a gravitational field is assumed to be that of the MB form. The only requirement in the above derivation is that there is thermal continuity between the gas and the ground at the interface between the two, and this requirement is independent of the presence or absence of a gravitational field. The density drop off is of course due to those particles which leave ground level but do not reach altitude 2. As these molecules have v<v1min, they do not affect the condition ncd=nab which transfers molecules with va>v1min to altitude 2 (v1min being the minimum velocity needed for a molecule leaving ground level to reach altitude 2). Other assumptions are usual for an ideal gas: no collisions, no Van der Waals forces, etc.