Kinetic Theory of Ideal Gases
THEORY
An ideal gas consists of atoms that do not exert forces on each other and
collide with the walls of the container in elastic collisions. The atoms obey
the ideal gas law:
PV = nRT,
where R is the universal gas constant, and n is given by,
n  N/NA = msample/M with M  NAmatom.
Assume there is just one atom that is inside a one-dimensional box residing
on the x-axis and the length of the box is L. The time interval t for the atom
to make one round trip is a distance of 2L, and assuming a speed of vx ,
t = 2L/vx.
Using this t, an expression for p associated with a collision of the atom
with the wall, and Newton’s second law in the form, F = p/t, gives the
force by the wall on one atom as,
F = matomvx2/L
Moving to three dimensions and averaging over all the N atoms where by
symmetry, <v2> = 3< vx2>, we multiply by N to get the total force on one
wall,
F(total) = Nmatom<v2> /3L.
But P = F(total)/L2 and V=L3 (assuming a cube) gives,
P = Nmatom<v2>/3V = nRT/V (ideal gas law).
Using n = N/NA and R = NAkB leads to,
matom<v2>/3V = kBT
or
Eintone atom = ½matom<v2> = 3/2 kBT.
If there are N atoms or n = N/NA moles, this becomes,
Eint = 3/2 nRT
(for a monatomic ideal gas = “m.i.g.”)
In summary,
(i)
(ii)
(iii)
(iv)
the internal energy of an ideal gas depends only on its absolute
temperature;
temperature is a measure of the random kinetic energy of atoms;
this equation is remarkable since it provides a connection between the
macroscopic world (n, T) and the microscopic world (Eint of a gas of
atoms),
and finally this results conforms to equipartition theorem where
Eint = no. degrees of freedom x ½ kBT.
Since the internal energy of a m.i.g. is entirely kinetic we have
Eint = ½ msample <v2> = 3/2 n R T,
which gives the root mean square speed of an atom of the gas
vRMS = [ 3RT/M]1/2 .
Caution: Keep straight n, N, NA, matom, msample, and M. These are six distinct
quantities.
EXAMPLES
[in class]