(1) Statistical Interpretation of Entropy
25 C
What is the physical origin of irreversibility?
Example: Free expansion of gas
Left: Vacuum, Right: Full of gas
Gas will be evenly distributed
You will not see the reverse
process in billion years!
Total N particles. Same volume in each side.
Question: How many different ways are possible
to have NL particles on the left and NR particles
on the right (N = NL + NR)?
* Example: N=4, NL = 3, NR = 1
4 possible “microstates”
In general, for a system of N particles with NL particles
on the left and NR particles on the right, the number of
possible microstates, W(NL,NR), is given by the following
formula.
N!
W(NL,NR) =
NL! NR!
Check: N=4, NL = 3, NR = 1
W(3,1) =
4!
3! 1!
=4
* Question: How does W(NL,NR) look like as the total
number of particles increases?
N=4
W(NL,NR)
NL =4, NR=0
NL =3, NR=1
NL =2, NR=2
NL =1, NR=3
NL =0, NR=4
1
4
6
4
1
Probability (completely even) = 6/16 = 3/8
Probability (completely uneven) = 2/16 = 1/8
With 4 particles, we will often see the case
where all particles are on the same side.
N= 4
N=10
N=100
If N=1000, W(500,500) = 2.5 × 10299
Conclusion: As N -> 1023 , any departure from
even distribution (equilibrium) is virtually impossible.
Once the system reaches an equilibrium, there will be
no apparent change in the system.
(1) In free gas expansion example, once the valve opens,
NL changes from its original value (NL=N) to NL = N/2,
which maximizes the W(NL,NR) .
(2) Isolated system moves to the “direction” that
maximize W.
(3) W is related to the directionality of process,
which can be measured in terms of “entropy” .
S = kB ln W (Boltzmann, ~1890)
kB= 1.38 J/K “Boltzmann Constant”
(4) NL reaches a value that maximizes the entropy
of the system (isolated).
(5) Entropy Change: dS
dSTOTAL > 0 : Irreversible Change
dSTOTAL = 0 : Equilibrium