Download Homework 4

MAE 216: Statistical Thermodynamics
University of California, Davis, Winter Quarter 2014
Homework 4, Due Weds Feb 19
1) Maxwell Boltzmann Distribution for speed:
Recall the Maxwell Boltzmann speed distribution derived in class.
a) Using this probability density function, calculate the formula for the average speed, C, for a monoatomic
gas with atomic mass m.
h i1/2
b) Now calculate the formula for the root mean squared speed, C 2
.
c) Now consider the Maxwell Boltzmann distribution for Helium. What is the average speed for a Helium
atom at 0o C? What is the average speed at 200o C?
d) Plot the distribution of particle density, dN/N dC, for Helium versus C for T=0o C and for T=200o C.
e) For a gas of Helium at T = 200o C, what is fraction of the particles have speeds greater than C = 2000
m/s? (You can leave your answer expressed in terms of an integral expression.)
2) Maxwell Boltzmann Distribution for velocity:
In class we also derived the Maxwell Boltzmann velocity distribution.
a) From the statement of unconditional probability we now that p(Vx ) =
R
R
Vy
Vy
p(Vx , Vy , Vz )dVy dVz .
Using this, derive the distribution for the fraction of particles with x-velocities Vx (i.e. derive
from
dNVx ,Vy ,Vz
N
dNVx
N
starting
).
b) Calculate the average x-velocity, Vx .
h i1/2
c) Now calculate the formula for the root mean squared x-velocity, Vx2
.
h i1/2
d) What is Vx2
for an atom of Helium at T = 200o C?
3) The 3D gas in a gravitational field (kinetic and potential energy):
Consider them a monoatomic gas trapped in a box that extends from 0 < x < L, 0 < y < L, and 0 < z < L.
The gas has kinetic energy due to velocities Vx , Vy , and Vz , but it also has gravitational potential energy,
mgz. (Set gravitational PE equal to zero when z = 0.)
a) What is the partition function for an atom in this 3-D gas in a gravitational field?
b) What is the expected energy per atom? (Hint, in class we used
want to evaluate ln z and then take the derivative.)
c) What is the expected energy per atom as T → ∞?
1
∂ ln z
∂T
=
1 ∂z
z ∂T .
Here, in contrast, you may
d)The expressions above are for an individual atom. What is the partition function for a gas of N such
atoms? (Treat the particles as independent.)
e) What is the expected energy for the gas of N atoms?
f) Does the expected velocity in the ẑ-direction depend on height z?
4) Equipartition of energy predictions
a) Using the predictions from equipartition of energy, what would be the value of the heat capacity at
constant volume (Cv ) for unit mass of Helium gas? (Give answer in units of kJ/kg o K.)
b) Look up the real value for Cv per kg of Helium at standard temperature and pressure. What is the
percentage error between the equipartition prediction and the real value? (Values can be found, e.g., at
http://www.engineeringtoolbox.com/spesific-heat-capacity-gases-d 159.html)
c) Using the predictions from equipartition of energy, what would be the value of the heat capacity at
constant volume (Cv ) for O2 gas? (Give answer in units of kJ/kg o K.)
d) Look up the real value for Cv per kg of O2 at standard temperature and pressure. What is the percentage error between the equipartition prediction and the real value? (Values can be found, e.g., at
http://www.engineeringtoolbox.com/spesific-heat-capacity-gases-d 159.html)
Extra credit: Tchebycheff’s inequality:
A central component of probability theory is the ability to bound the likelyhood of observing a random
variable with a value increasingly greater than the mean. Tchebycheff’s derived a powerful formula showing
how the probability decays for any probability density p(x) with (−∞ < x < ∞), mean λ, and variance
σ 2 . It says that the probability of outcomes that are more than nσ greater than λ is less than 1/n2 , that is,
Z
1
p(x)dx ≤ 2 .
n
|x−λ|≥nσ
Way have seen that large systems of particles often follow Gaussian distributions. Many other quantities of
interest are described by power law distributions, p(x) = (γ − 1)x−γ , where 2 < γ < 3. Such distributions
describe the spread of wealth in the US, the number of friends people have on Facebook, the popularity of
web pages, and the distribution of earthquake sizes. Typically the values of x cannot take on negative values.
Show that if 2 < γ < 3 the variance σ 2 → ∞. Will we be able to use Tchebycheff’s inequality to
bound expected values in this case?
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