Chem 2 - The Second Law of Termodynamics: Entropy Microstates and the Boltzmann Equation II

1. The Second Law of
Thermodynamics:
Entropy, Microstates, and the
Boltzmann Equation (Pt. 2)
By Shawn P. Shields, Ph.D.
This work is licensed by Shawn P. Shields-Maxwell under a Creative Commons
Attribution-NonCommercial-ShareAlike 4.0 International License.

2. The Second Law of Thermodynamics
The total entropy change for any
spontaneous process is positive
(+S)
This increase in entropy involves
an evolution from less probable
configurations to more probable.

3. Entropy and Microstates
Entropy is related to the number of
possible arrangements or
“microstates.”
Microstates are instantaneous
properties of the system.
Examples include instantaneous
velocities and positions of molecules
(atoms, etc.)

4. Entropy and Microstates
The number of microstates can be
increased by
Increasing the volume
Increasing the temperature
How?

5. Example 1: Counting Microstates
Suppose we have four
distinguishable particles in one
side of a two-compartment
container.
How many possible ways are there
to distribute them between the
two sides?

6. The Possible Microstates for a Sample of Four Gas Molecules in Two Bulbs of
Equal Volume" from http://2012books.lardbucket.org/books/principles-of-
general-chemistry-v1.0m/s22-03-the-second-law-of-thermodynami.html
Microstates
Highest probability to
observe state III
(Largest number of
microstates)

7. Boltzmann's Equation Relates the
Microscopic to the Macroscopic
The relationship between the number of
microstates and the entropy is
S = 𝑘 𝐵 ln Ω
Where  is the number of microstates and
kB (Boltzmann constant) = 1.38  1023 J/K

8. Boltzmann's Equation and S
The change in entropy for a process can be
calculated as
ΔS = 𝑘 𝐵 ln
Ω2
Ω1
Where 1 is the initial number of microstates
and 2 is the final number of microstates
(after the process)

9. Example 2: Calculating the Number of
Microstates
Four distinguishable particles are initially
sealed in the right side of a two-
compartment container.
Suppose the compartment is opened, and
the particles are allowed to distribute
throughout both compartments.
How many microstates are there initially,
and finally?
Calculate S for the process.

10. Example 2: Calculating the Number of
Microstates and S
How many microstates are there
initially?
There is only one possible
arrangement for all 4 particles in the
right compartment.
Edited “The Possible Microstates for a Sample of Four Gas Molecules in Two Bulbs of Equal Volume" from
http://2012books.lardbucket.org/books/principles-of-general-chemistry-v1.0m/s18-06-reaction-rates-a-microscopic-v.html
BE SURE TO CHANGE LINK!!!!!!!!

11. Example 2 (continued): Calculating the
Final Number of Microstates
How many microstates are there in the
final state? The volume was doubled:
2# 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠
= Ω 𝑓
Ω 𝑓 = 24
= 16
This is the same answer…
16 microstates (or possible arrangements).

12. Example 2 (continued): Calculating S
Calculate S for this process
Δ𝑆 = 𝑘 𝐵 ln
Ω2
Ω1
∆𝑆 = ln 1.38 × 10−23
ln
16
1
= 3.83 × 10−23
J
The entropy increased 3.8310-23 J when we
doubled the volume.

13. Next up…
Predicting Entropy Changes
Qualitatively, plus Examples
(Pt 3)