D O C . 2 9 0 T H E O R Y O F R A D I O M E T E R F O R C E S 2 8 9
(1)
molecules, where n signifies the number of molecules per unit volume. In order for
us to do justice to the fact of heat flow, we must assume that the velocity of the mol-
ecules in the direction of the positive x-axis is somewhat larger than u; the cor-
respondingly defined must be correspondingly somewhat smaller than u. The
heat flow σf is given by the surface element
. (2)
If we take into account the relation
as well as the circumstance that for the molecular velocities and , the tem-
peratures at the locations where the last collision took place are decisive (λ = free
path length), then instead of (2),
. (2a)
results.
Now we consider instead of the surface element a small body of surface exten-
sion σ. The molecules impinging on this body in the x -direction yield an excess
momentum K in the direction of the positive X-direction
. (3)
If one neglects the circumstance that the colliding molecules upon leaving the body
have again a momentum effect on the body, which amounts to a certain fraction of
the one just calculated, then K is also the motive force acting on the body. From (2)
and (3) we get, taking into account that and differ only slightly from u,
(3a)
where p is the gas pressure. For this formula, as in (2), f naturally means only that
portion of the heat flow which is based on the translational motion of the mole-
cules.
This force K will move the particle, when it is free, in the direction of the positive
x-axis. In order to be able to know the velocity v of this motion, we only need to
calculate the frictional force K′ which is exerted by the gas on the particle when it
is moved with the velocity v through the gas. This frictional force essentially arises
1
6
--nσu -
u+
u–
σf
1
6
--nσu§
-
m
2
--- -u+ 2
m
2
--- -u–¹
2·
–
©
=
1
2
--mu2 -
3
2
--κT - =
u+ u–
f
n
2
--κλu -
∂x
∂T
–=
K
1
6
--nσu( - mu+ mu–) – =
u+ u–
K
σf
u
-----
1
2
-
λ∂T
T∂x
- σ –--p-- = =
[p. 3]