Statistical Thermodynamics

Question 1

Number of ways of putting n different balls into n boxes

Answer 1

No. of microstates = n!

Question 2

Number of ways of putting r different balls into n boxes (r<n)

Answer 2

No. of microstates = n!/(n-r)!

Question 3

Number of ways of putting r identical balls into n boxes

Answer 3

No. of microstates = n!/r!(n-r)!

Question 4

Ensemble average

Answer 4

1. Take a single system and follow it over a long time; so that the system goes through all possible configurations
2. Take a large number if identical but randomly configured systems and average over these at a single snapshot in time (this is the ensemble average)

Question 5

Micro canonical ensemble

Answer 5

Isolated system with fixed U,V,N

Question 6

Canonical ensemble

Answer 6

System is in contact with a thermal reservoir. V,N,T fixed for the system

Question 7

Grand canonical ensemble

Answer 7

System can exchange heat and particles with the reservoir. V,T, mu fixed for the system

Question 8

Principle of equal equilibrium probability

Answer 8

When a thermally isolated system comes into thermal equilibrium then the state probabilities of any set of mutually accessible states become equal. In a micro canonical ensemble there is equal probability of any state being occupied

Question 9

Ergodic hypothesis

Answer 9

Given enough time the systems will explore all possible microstates and will spend an equal amount of time in each of them.
Implication: in equilibrium an isolated system will adopt the macro state with the most microstates

Question 10

Boltzmann equation for entropy

Answer 10

S = kb * ln Omega

Question 11

Boltzmann distribution function

Answer 11

P(Ei) proportional to exp(-Ei/kbT)

Question 12

Properties of Boltzmann Distribution Function

Answer 12

Lower T- lower energy states more likely to be occupied
Infinite temperature- all states equally likely

Question 13

Boltzmann-Gibbs Entropy

Answer 13

S = -Kb Sumi

Question 14

In 3D the number of states between k and k+dk

Answer 14

g(k)dk = 4pik^2dk * V/(2pi)^3

Question 15

The partition function for indistinguishable atoms: What conditions are required for the approximation of N! to hold?

Answer 15

As long as we can neglect any atoms which have the same quantum numbers. Okay as long as the number of possible states is much larger than the number of atoms

Question 16

Sackur-Tetrode Equation

Answer 16

S = Nkb [ln(V/lambda^3 *N) +5/2]

Question 17

Gibbs Paradox (Question)

Answer 17

What if the particles have some property X which we can’t measure (but someone else could). So they appear to be identical but they are not. Would the entropy change when we are able to measure the difference.

Question 18

Gibbs Paradox (answer)

Answer 18

Yes, entropy depends on our knowledge of the system. The way to interpret the extra entropy is that after mixing A,B to return the system to its original state we would need to do work on the system equal to at least T*DeltaS

Question 19

Fermions

Answer 19

Particles with half integer spin

Question 20

Pauli Exclusion Principle

Answer 20

We can only have 1 fermion in any single quantum state

Question 21

Bosons

Answer 21

Particles with integer spin

Question 22

Classical limit

Answer 22

At low occupancy the fermi-Dirac, Bose-Einstein and Boltzmann distribution functions all become equal, because there is very little chance of multiple occupancy so the quantum nature of the particles is irrelevant