Kinetic Theory of Gases

Question 1

Assumptions

Answer 1

Gases are modelled as hard spheres with no potential and negligible volume.
Spheres are in ceaseless, random motion.
Collisions between gases are brief and elastic.

Question 2

Energy contribution (kinetic) ΔEi =

Answer 2

(1/2)*mv_x^2
(Only in x direction)

Question 3

Distribution of this (partition theorem)
We use integral as 1/2mv_x^2 is sufficiently large

Answer 3

f(v_x) = partition with integral and substitution = (m/2pikT)^(1/2)*e^((-mv_x^2)/2kT)

Question 4

For 3 dimensions speed formula:

Answer 4

c = (v_x^2 +v_y^2+v_z^2)^(1/2)

Question 5

Using c for partition theorem to get F(c)

Answer 5

=(4pic^2)(m/2pikT)^(3/2)(e^((-mc^2)/2kt))
It is ^3/2 as there are 3 dimensions.

Question 6

Peak velocity, ĉ

Answer 6

dF(c)/dc = 0
ĉ = ((2kt)/m)1/2

Question 7

Mean speed, <c></c>

Answer 7

=∫F(c)cdc = ((8kT)/pim)^(1/2)
Integral between 0 and infinty

Question 8

Root mean squared speed
<c^2>^(1/2)

Answer 8

=[∫F(c)c^2]^(1/2) =((3kT)/m)^(1/2)

Question 9

Ek for rms

Answer 9

Ek = 1/2mc^2=(m/2)((3kT)/m) =(3/2)kT
(In line with equipartition theorem)

Question 10

Why are translational energy levels highly degenerate?

Answer 10

For every c there are many combinations of v_x, v_y and v_z’s