How do you calculate the number of microstates in a two-state system?

For N distinguishable objects with n in one state and N−n in the other, use W = N!/[n!(N−n)!].

Why does the equal distribution usually have the highest entropy in a two-state system?

It has the greatest multiplicity—the largest number of possible microstates—so S = k ln W is maximized.

Which ensemble describes a system that exchanges both energy and particles with its surroundings?

The grand canonical ensemble, where temperature, volume, and chemical potential are fixed.

Free Course Image Statistical Thermodynamics

Free online course Statistical Thermodynamics

Duration of the online course: 3 hours and 52 minutes

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Build real intuition for entropy and microstates in this free statistical thermodynamics course, with practice problems and a certificate-ready foundation.

In this free course, learn about

Core terms of statistical thermodynamics: microstate, macrostate, multiplicity W, entropy
How to count microstates for two-state systems (coin-toss model) and identify most probable macrostate
Compute average occupation numbers and distributions that maximize multiplicity for large N
Use Stirling’s approximation to estimate n!, ln(n!), and log10(n!) for large numbers
Multiplicity vs N behavior: sharp peak near equilibrium for N=10, 50, 100, 1000
Counting microstates in multi-state and fixed-total-energy systems (3 particles, multiple levels)
Entropy from multiplicity: S = k ln W; compare entropies of macrostates and coin/box examples
Entropy change in mixing/free expansion and moving molecules between volumes; isothermal ΔS formulas
Improbability vs impossibility: probabilities of all molecules in one region (binomial/volume ratios)
Microstate counting for N distinguishable molecules in boxes split into 2, 3, 6, or N partitions
General method: distribute N distinguishable particles among n energy levels; compute W for a macrostate
Ensembles: identify the grand canonical ensemble for exchange of both energy and particles
Quantum degeneracy concepts (e.g., p-orbital degeneracy) and its effect on state counting
Particle-in-a-box energy levels in 1D/3D; scaling with volume and estimating quantum numbers for He

About the free online course

Statistical thermodynamics is where the rules of probability and the behavior of enormous numbers of particles meet to explain the laws you see in everyday thermodynamics. In this free online course, you will learn to translate between microscopic pictures of matter and macroscopic quantities such as multiplicity, probability, and entropy, so the second law stops feeling like a slogan and starts feeling inevitable.

The lessons guide you from the first essential vocabulary and purpose of the field into the core habit of thinking in terms of microstates and macrostates. By working through classic two-state models and coin-toss analogies, you will build a reliable intuition for what it means for a distribution to be likely, why a system naturally drifts toward the most probable macrostate, and how sharply that probability peak grows as the number of particles increases. Along the way, you will use tools like Stirling’s approximation to handle the huge numbers that appear in real systems, and you will see why approximation is not a shortcut but the key that makes large-N physics readable.

As you progress, entropy becomes more than a definition: it becomes a count. You will connect entropy to multiplicity and use that connection to reason about mixing, free expansion, and the difference between something being impossible versus merely astronomically unlikely. Thought experiments with molecules in compartments make the statistical meaning of equilibrium tangible, while guided exercises help you practice turning a physical situation into a counting problem and then into a thermodynamic conclusion.

The course also bridges into quantum ideas that underlie modern statistical physics, including energy levels, degeneracy, and particle distributions across quantized states. By relating confinement, energy spacing, and volume scaling in simple potential well models, you gain a clearer picture of how microscopic constraints shape macroscopic behavior. If you are studying physics as a school subject, preparing for exams, or strengthening fundamentals for chemistry, engineering, or materials science, this course offers a rigorous, practice-driven path to confident reasoning in statistical thermodynamics.

Course content

Video class: Physics 32.5 Statistical Thermodynamics (1 of 39) Basic Term and Concepts 06m
Exercise: Counting microstates for a two-state assembly
Video class: Physics 32.5 Statistical Thermodynamics (2 of 39) Purpose and Objective Statistical Thermodynamics 05m
Exercise: Why is statistical thermodynamics essential for analyzing macroscopic assemblies?
Video class: Physics 32.5 Statistical Thermodynamics (3 of 39) Understanding Statistical Thermodynamics 1 04m
Exercise: For four fair coin tosses, what is the thermodynamic probability W_k (multiplicity) for the macrostate with two heads and two tails?
Video class: Physics 32.5 Statistical Thermodynamics (4 of 39) Understanding Statistical Thermodynamics 2 05m
Exercise: Which macrostate has the highest number of microstates for four coins with two energy levels?
Video class: Physics 32.5 Statistical Thermodynamics (5 of 39) The Average Occupation Number 03m
Exercise: Average occupation number of heads in a four-coin system
Video class: Physics 32.5 Statistical Thermodynamics (6 of 39) Calculate the Number of Microstates 06m
Exercise: How many microstates exist for the macrostate with 10 coin tosses yielding 5 heads and 5 tails?

Video class: Physics 32.5 Statistical Thermodynamics (7 of 39) Stirling's Approximation Explained 09m
Exercise: Using the large n factorial approximation, what is log10 of 100 factorial approximately equal to
Video class: Physics 32.5 Statistical Thermodynamics (8 of 39) Stirling's Approximation: Summery 05m
Exercise: Estimate the order of magnitude of 100! using Stirling approximation and base-10 conversion
Video class: Physics 32.5 Statistical Thermodynamics (9 of 39) Number of Microstates Analyzed N=10 07m
Exercise: Two state system with 10 objects maximum microstates
Video class: Physics 32.5 Statistical Thermodynamics (10 of 39) Number of Microstates Analyzed N=50 07m
Exercise: Relative microstate height 10 percent from the maximum for N = 50 two state system
Video class: Physics 32.5 Statistical Thermodynamics (11 of 39) Number of Microstates Analyzed N=100 09m
Exercise: In a two-state system with N = 100 objects, which distribution maximizes the number of microstates W?
Video class: Physics 32.5 Statistical Thermodynamics (12 of 39) Number of Microstates Analyzed N=1000 10m
Exercise: Multiplicity peak in a two-state system with N=1000
Video class: Physics 32.5 Statistical Thermodynamics (13 of 39) Number of Microstates in a Multi-State System 07m
Exercise: Counting microstates in a multi state system
Video class: Physics 32.5 Statistical Thermodynamics (14 of 39) Number of Microstates in a Multi-Energy State Sys 05m
Exercise: In a system with 3 distinguishable particles and four energy levels 0, e, 2e, 3e constrained to total energy U equal 3e, how many microstates are possible

Video class: Physics 32.5 Statistical Thermodynamics (15 of 39) Definition of Entropy of a Microstate 05m
Exercise: Which macrostate of four fair coins has the highest entropy?
Video class: Physics 32.5 Statistical Thermodynamics (16 of 39) Definition of Entropy of a Microstate: Example 04m
Exercise: For 6 molecules in a two-compartment box of equal size, what is the entropy change ΔS when going from all 6 in one compartment to the most probable state with 3 in each?
Video class: Physics 32.5 Statistical Thermodynamics (17 of 39) Microstates 07m
Exercise: Which statement best describes the thermodynamic probability W for a macrostate in 100 distinguishable coin tosses?
Video class: Physics 32.5 Statistical Thermodynamics (18 of 39) 6 Molecules in a Box (Divided in Half) 08m
Exercise: For six distinguishable molecules in a box divided into two halves, which macrostate has the highest entropy?
Video class: Physics 32.5 Statistical Thermodynamics (19 of 39) 6 Molecules in a Box: Microstates in Detail 04m
Exercise: A box is divided into two compartments by a partition. When the partition is removed and an ideal gas expands freely into the full volume, what happens to the internal energy and entropy of the gas
Video class: Physics 32.5 Statistical Thermodynamics (20 of 39) 6 Molecules in a Box: Divided in 3 Equal Sections 05m
Exercise: Entropy from number of microstates
Video class: Playlist Organizer 01m
Exercise: Which ensemble is appropriate for a system that can exchange both energy and particles with a reservoir

Video class: Physics 32.5 Statistical Thermodynamics (21 of 39) 6 Molecules in a Box: Divided in 3 Equal Sections 04m
Exercise: How many microstates exist for 6 distinguishable molecules in a box divided into 3 equal sections, counting all possible configurations?
Video class: Physics 32.5 Statistical Thermodynamics (22 of 39) 6 Molecules in a Box: Divided in 3 Equal Sections 05m
Exercise: How many total microstates exist for 6 distinguishable molecules distributed among 3 equal partitions?
Video class: Physics 32.5 Statistical Thermodynamics (23 of 39) 6 Molecules in a Box: Divided in 6 Equal Sections 05m
Exercise: Total microstates for 6 distinguishable molecules in 6 partitions
Video class: Physics 32.5 Statistical Thermodynamics (24 of 39) N Molecules in a Box: Divided in N Equal Sections 04m
Exercise: Microstates for n molecules in n partitions (one per partition)

Video class: Physics 32.5 Statistical Thermodynamics (25 of 39) What is Sterling's Approximation? S = k ln n! 04m
Exercise: Using Stirling's approximation, which value is closest to ln(100!)?
Video class: Physics 32.5 Statistical Thermodynamics (26 of 39) What is Entropy of 1 mol of Gas Distributed? 04m
Exercise: In applying Stirling approximation to compute S = k ln W for one mole with W = N!, which term can be neglected in ln(N!) for N ≈ Avogadro number?
Video class: Physics 32.5 Statistical Thermodynamics (27 of 39) Entropy Change for Moving N Molecules 04m
Exercise: For an isothermal compression of an ideal gas where N molecules move from volume V1 to smaller volume V2, which expression gives the entropy change Delta S
Video class: Physics 32.5 Statistical Thermodynamics (28 of 39) Improbability vs Impossibility 05m
Exercise: For 10 ideal gas molecules in volume V1, what is the probability that at an instant all are found in the subvolume V1 divided by 2
Video class: Physics 32.5 Statistical Thermodynamics (29 of 39) Probability Compared to Macrostates 06m
Exercise: Probability that six molecules are all in one half of a box
Video class: Physics 32.5 Statistical Thermodynamics(30 of 39) 6 Distinguishable Molecules in a Box with 2 Halves 06m
Exercise: Total microstates for six distinguishable molecules in a two-compartment box

Video class: Physics 32.5 Statistical Thermodynamics (31 of 39) General Counting Method for w 07m
Exercise: Microstates for distributing N distinguishable particles among n energy levels
Video class: Physics 32.5 Statistical Thermodynamics (32 of 39) Energy Level Example 1 05m
Exercise: For three distinguishable particles in levels n0 n1 n2 n3 with total energy 3 quanta, how many microstates correspond to the macrostate with one particle in n0, one in n1, and one in n2
Video class: Physics 32.5 Statistical Thermodynamics (33 of 39) Energy Level Example 2 07m
Exercise: Total microstates for three distinguishable particles with total energy 4 quanta across levels 0, 1, 2, 3
Video class: Physics 32.5 Statistical Thermodynamics (34 of 39) Energy Level Example 3 08m
Exercise: Total microstates for six distinguishable particles with total energy 4 quanta across five energy levels
Video class: Physics 32.5 Statistical Thermodynamics (35 of 39) What is a Degenerate Quantum State? 04m
Exercise: Degree of degeneracy for p orbitals at the same energy
Video class: Physics 32.5 Statistical Thermodynamics (36 of 39) The One-Dimensional Potential Well 04m
Video class: Physics 32.5 Statistical Thermodynamics (37 of 39) The Three-Dimensional Potential Well 03m
Exercise: How do energy levels scale with volume in a cubic 3D infinite potential well

Video class: Physics 32.5 Statistical Thermodynamics (38 of 39) Find the Quantum Number of Volume L^3 of He Ex1 04m
Exercise: For helium at 298 K in a 0.10 m cubic box, estimate the 1D quantum number n using de Broglie wavelength with v_rms and n = L divided by half wavelength.
Video class: Physics 32.5 Statistical Thermodynamics (39 of 39) Find the Quantum Number of Volume L^3 of He Met 2 02m
Exercise: Estimate the effective quantum number n for a helium atom confined in a cubic box of side 0.10 m at 298 K using E = 3/2 k T and E = n^2 π^2 ħ^2 / (2 m V^{2/3})

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