MATH 261, Spring 2001 Due Date: Name(s):
Honors Project 14: The Energy of Non-Interacting Gas Particles
A general expression for the energy of a collection of N non-interacting gas particles is
U =
k
Q
Q
T
where k is a fundamental physical constant (Boltzmann's constant), T is temperature, and Q is a special
function called the partition function, which is of the form Q = 1
N! qN
where q is a molecular partition
function that must be derived or determined.
Problems:
1. Find U for a collection of N particles if the molecular partition function
q =
V
h3
(2mkT)3/2
where V is the volume of the gas, m is the mass of a gas particle, and h is a fundamental physical
constant (Planck's constant).
2. The heat capacity CV is the heat energy required to change the temperature of a substance when the
volume is constant, and is given by
CV =
U
T
.
Find CV for a collection of N particles.
[Note on notation: The heat capacity is sometimes written
CV =
U
T V
.
The reason is that when several variables are used in a problem, ambiguity about the meaning of partial
derivatives can evolve from ambiguity about which variables are assumed to be independent and which
are dependent. For example, if
u = x + y and v = x - y
and x and y are assumed to be independent, then u/x = 1. However, if x and v are assumed to
be independent then, since y = x + v, it follows that u = 2x + v so u/x = 2. One way to clarify
which variables are assumed to be independent is to use the notation
u
x y
to denote the partial
derivative of u with respect to x, x and y being the independent variables. Thus in the above example,
u
x y
= 1 and
u
x v
= 2. While this notation is useful, it is important to remember when using it
that subscripts make a statement about which variables are independent: they do not represent partial
derivatives.]
Contributor: Dr. Clifford Dykstra
Department of Chemistry, IUPUI