Entropy of Ideal Gas 1
5
Entropy of Ideal Gas
We introduce applications as we develop the foundations of statistical mechanics.
This chapter treats ideal gas using S  k ln W . In the process, we develop some
relations with thermodynamics and show how to extract macroscopic information
from entropy.
Number of States W
Free particles in a box serve as a microscopic model of a monatomic ideal gas.
Our plan is to find an expression for the entropy of N such particles. Our development of
the number of states W is crude, but will produce accurate thermodynamic relations.
Consider the uncertainty relation xp  h
p
where x and p are variances in position and
momentum and h is Planck’s constant. We take the
view that for each linear dimension the “space” of
position and momentum can be divided into
rectangles of area h. A three-dimensional box
h h h
containing an ideal gas particle has a volume
h h
Lx Ly Lz p x p y p z in position-momentum space in
obvious notation. Dividing each Lp product by h
will give the number of cells (states) available for
that dimension. It follows that the states available
to the particle is given by
W1  Lx Ly Lz px p y pz / h3 .
The average momenta are equal in each direction and related to the total momentum p
such that
px2  p 2y  pz2  3 px2  p 2 .
3
The number of cells in the position-momentum volume can now be written 3 2 Vp3 / h 3
where V is the volume of the box. The average energy of one particle is U / N  p 2 2m
so the number of states available to one particle in terms of U and V is
3
 2mU  2

W1  V 
2 
 3N h 
1
Repeat the detailed steps in the development of W1 above.
The number of states W available to N ideal gas molecules is then W1N , but since the
molecules are identical, they can be rearranged in N! indistinguishable ways. We have
W  W1N / N!
x
Entropy of Ideal Gas 2
2
Show that the entropy of a monatomic ideal gas according to the W of problem 1
is
3
  2mU  2 
   Nk ln N
S  Nk ln V 
2 
  3N h  
where N! is approximated by the leading term of Sterling’s formula.
Applications to Macro Variables: Thermodynamics
Now that an expression for entropy is known, all macroscopic properties of the ideal gas
can be found. Substitute dW  PdV into dU  TdS  dW ,
dU  TdS  PdV
(1)
The negative sign on the work term is appropriate for the gas applying pressure to the
vessel. We will see that once S is known as a function of U and V, all macro-state
equations can be derived using Eq.(1).
3
Solve Eq.(1) for dS and compare the result with the mathematical identity,
 S 
 S 
dS (U ,V )  
 dU  
 dV
 U V
 V U
1
P
 S 
 S 
Show 
  (many books define temperature with this) and 
 
 U V T
 V U T
4
Substitute S from problem 2 into the expressions from problem 3 to show
3
U  NkT and PV  NkT
2
These results are the state equations for monatomic ideal gas. They represent the
macroscopic properties of the gas.
Much of the literature, especially in chemistry, describes quantities in moles
where one mole is NA=6.031023 particles. Rather than Boltzmann’s constant k, these
authors use the gas constant R where
N Ak  R
5
Let n represent the number of moles in N particles, N  nN A and rewrite the state
equations of problem 4 replacing N and k with n and R. Show the heat capacity
3
( C  U T ) is C  nR .
2
Adiabatic Processes
Often, systems are either insulated from heat transfer or a process takes place too
quickly for a significant heat transfer to occur. These are adiabatic processes and we
characterize them with the condition dS  0 or S = constant.
Entropy of Ideal Gas 3
6
For the entropy of problem 2 require that S = constant while N is fixed. Show that
2
2
2
TV 3  constant (or equivalently, T1V1 3  T2V2 3 ).
7
A Helium balloon at 310 K rises in the atmosphere and swells in the reduced
pressure increasing its volume by 10%. Assume the process is adiabatic and find
the new temperature. [ans 291 K]
8
(a) Using the method of this chapter, calculate an expression for the entropy of an
ideal gas constrained to a two-dimensional area A.
 AmU 
 AmU 
 kN ln N  Nk ln  2 2  ]
[ans. S  Nk ln 
2 
 Nh 
N h 
(b) Find the heat capacity of this gas in terms of number of moles and the gas
constant. [ans. C  nR ]
(c) Show that an adiabatic condition for this gas is T1 A1  T2 A2 .
Summary
This chapter illustrated some of the predictive power that follows from knowing
the form of entropy. Later chapters will introduce more systematic ways of calculating
entropy, energy, and other thermodynamic functions.