Topic 5

Question 1

What is statistical mechanics?

Answer 1

A way of deriving relationships in a system based off of microscopic properties

Statistical physics is about working out the balance between these factors to determine equilibrium for given conditions

Question 2

How is entropy related to the number of microstates?

Answer 2

𝑆=𝑘_𝐵 ln⁡W

Question 3

How can you calculate entropy changes?

Answer 3

• Macroscopic using the ideal gas law and adiabatic expansion
• Microscopic using the lattice gas model

Question 4

What is the microscopic view of Joule expansion/change in entropy?

Answer 4

• Use a microscopic model that allows us to count the number of microstates
• Use the lattice model
• We have M lattice sites and N atoms
• Calculate the number of microstates

*

Question 5

What is the microscopic view of Joule expansion/change in entropy?

Answer 5

• Use a microscopic model that allows us to count the number of microstates
• Use the lattice model
• We have M lattice sites and N atoms
• Calculate the number of microstates

*

Question 6

What is the macroscopic view and how does it compare of Joule expansion to the microscopic view?

Answer 6

For an ideal gas
Δ𝑆=𝑛𝑅 ln⁡(𝑉_𝑓 /𝑉_𝑖 )

∆𝑆≈𝑛𝑅[ln⁡(𝑉_𝑓/𝑉_𝑖 ) - 𝑁𝑣_0)/2 (1/𝑉_𝑖 −1/𝑉_𝑓 )]

The first term is equal:
- When the fraction of sites being occupied is very small then the first term would be the dominate term.

The second term is NOT equal:

• Expressions are not identical
• Lattice gas model considers that each site can only be occupied by 1 atom at most which start to distinguish it from an ideal gas
• This may bring the idea of the VdW gas because in the VdW gas that atoms and molecules do not have infinitesimal small volumes they actually have a finite volume which has an impact on the equation of state

Question 7

What is the macroscopic view of the Joule expansion of a Van der Waals gas?

Answer 7

• Use an isothermal path between the initial and final state(We are free to use any path because entropy is a state function)
  -We can’t assume that dU=0 for an VdW gas
  For the ideal gas case: U only depends on temperature because the internal energy is all in its momentum of the atom of the molecules and there is no interaction between them
  For a VdW: The internal energy is a consequence of momentum and interaction between molecules
  since one of the contribution comes from the interaction between molecules i.e potential between particles , so as you increase the distance the attraction changes, because of joule expansion depends on volume then we can’t say dU=0

Question 8

How does the macroscopic view of VdW gas compare to the microscopic view of a lattice gas?

Answer 8

Macroscopic view of a VdW gas

∆𝑆≈𝑛𝑅[ln⁡(𝑉_𝑓/𝑉_𝑖) −𝑁𝑣_0(1/𝑉_𝑖 −1/𝑉_𝑓 )]

Microscopic view of a lattice gas

∆𝑆≈𝑛𝑅[ln⁡(𝑉_𝑓/𝑉_𝑖) - 𝑁𝑣_0)/2(1/𝑉_𝑖 −1/𝑉_𝑓 )]

First-term: Equal
Second term:
-Lattice gas has a prefactor of 1/2 this relates to the difference between V_f and V_i
- They are not identical because the expression for VdW gas is determined from the equation of state of VdW which is written empirically and does not consider molecular properties, what we expect to happen only use the finite volume

Question 9

What is an ensemble?

What is a microcanonical ensemble?

Answer 9

Ensemble is the system we are in and its interaction in the environment

A system that completely isolated from the outside. There is no exchange.
Constant NVE. used to make prediction of the behaviour of systems with a large number of particles

Question 10

What is a canonical ensemble?

Answer 10

A system that is connected to an infinite heat bath

Constant NVT

Question 11

Why is energy fixed in a microcanonical ensemble?

Answer 11

The insulation
There is no heat exchanged of energy in terms of the first law
No exchange allowed in or out

Question 12

How do you apply the microcanonical ensemble in order to determine the relationship between that microscopic property and 𝑇?

Answer 12

> Need to work out how energy is distributed
Depends on the model for our system(Model is given in question)
A model to predict 𝑆 and 𝑈 in terms of some microscopic property

• Define the parameter which characterizes the system
• Calculate energy as a function of the parameter
• Calculate entropy as a function of the parameter
• Relate temperature and the parameter

Question 13

The limiting behaviour of the crystal defect model

Answer 13

Low T:: internal energy dominate behaviour. Fewer defects as energy cost outweighs entropy gain

High 𝑇: entropy dominates behaviour
More defects as entropy gain outweighs increased energy

Question 14

How do you find the limiting behaviour of a model?

Answer 14

Consider the behaviour at high and low T

Question 15

What is Helmholtz free energy?

Answer 15

F = U - TS

• Minimising 𝐹 with respect to the parameter that characterises a system corresponds to finding equilibrium at a fixed temperature
• 𝐹 is the maximum work that can be extracted from a system at constant 𝑇 and 𝑉

Question 16

What is a canonical ensemble?

Answer 16

A system connected to a heat reservoir at constant temperature. Constant NVT.

Infinite(give E to microsystem to reach equalibirum) heat reservoir at fixed T:
𝐸_𝑅 = 𝐸_0 − 𝜀_𝑖

Imposes the same T (zeroth law of thermodynamics)

Connected to a microsystem:

The energy of a given microstate, 𝑖, of the microsystem is
𝜀𝑖

Question 17

What is a canonical ensemble?

Answer 17

A system connected to a heat reservoir at constant temperature. Constant NVT.

Infinite(give E to microsystem to reach equalibirum) heat reservoir at fixed T:
𝐸_𝑅 = 𝐸_0 − 𝜀_𝑖

Imposes the same T (zeroth law of thermodynamics)

Connected to a microsystem:

The energy of a given microstate, 𝑖, of the microsystem is
𝜀𝑖

Question 18

What is the Boltzmann factor?

Answer 18

The probability, 𝑝_𝑖, of the microsystem being in microstate 𝑖 with particular energy 𝜀_𝑖

The expression is dependent upon the amount of heat borrowed by the reservoir

Assume that this probability is proportional to the number of microstates of the reservoir Ω(𝐸_𝑅)(Number of ways) when it gives up an energy 𝜀_𝑖 to the microsystem

𝑝_𝑖∝Ω(𝐸_𝑅)

Question 19

What is the Boltzmann factor?

Answer 19

The probability, 𝑝_𝑖, of the microsystem being in microstate 𝑖 with particular energy 𝜀_𝑖

The expression is dependent upon the amount of heat borrowed by the reservoir

Assume that this probability is proportional to the number of microstates of the reservoir Ω(𝐸_𝑅)(Number of ways) when it gives up an energy 𝜀_𝑖 to the microsystem

𝑝_𝑖∝Ω(𝐸_𝑅)
𝑝_𝑖∝exp⁡(−𝜀_𝑖/(𝑘_𝐵 𝑇))=1/𝑍 exp⁡(−𝜀_𝑖/(𝑘_𝐵 𝑇))

𝑍 normalises the probabilities

Question 20

What is the partition function (Z)?

Answer 20

The normalisation for the probability is known as the partition function

It is the sum is over all the microstates of the system.

𝑍=∑_𝑖exp⁡(−𝜀_𝑖/(𝑘_𝐵 𝑇))
OR
𝑍=∑_𝑖 exp⁡(−𝛽𝜀_𝑖 )

where 𝛽=1/(𝑘_𝐵 𝑇)

It can help to calculate all macroscopic properties of the system

Question 21

For the canonical ensemble the internal energy of the microsystem can be given as?

Answer 21

The sum of all possible microsystem multiplied by probability
𝑈=−(𝜕 ln⁡𝑍)/𝜕𝛽

Question 22

For the canonical ensemble the specific heat of the microsystem can be given as?

Answer 22

𝐶_𝑉=1/(𝑘_𝐵 𝑇^2 )(𝜕^2 ln⁡𝑍)/(𝜕𝛽^2 )

Question 23

How can you relate Helmholtz free energy to the partition function?

Answer 23

𝐹=−𝑘_𝐵 𝑇 ln⁡𝑍

Question 24

How can you relate entropy to the partition function?

Answer 24

𝑆=−𝑘_𝐵 ∑_𝑖𝑝_𝑖 ln(𝑝_𝑖)

Question 25

How do you write the partition for two energy level?

Answer 25

We can use the canonical ensemble to determine the population of each of the two energy levels

The partition function is
=∑_𝑖𝑒^(−𝛽ℇ_𝑖)=𝑒^(+𝛽Δ\/2)+𝑒^(−𝛽Δ\/2)=2 cosh⁡(𝛽Δ/2)

where
±∆/2 is the energy the energy associated with the spin state of an electron placed in a magnetic field of strength, 𝐵

Question 26

How do you work out the N-particle partition function for N distinguishable particles?

Answer 26

𝑍_𝑀=𝑍_𝑆^𝑀

Question 27

How do you work out the N-particle partition function for N indistinguishable particles?

Answer 27

𝑍_𝑀=(𝑍_𝑆^𝑀)/𝑀!

Question 28

The equation that proves the consistently with CE and MCE?

Answer 28

𝑆=𝑘_𝐵 ln⁡𝑊=−𝑀k_𝐵 ∑_𝑖(𝑝_𝑖 ln⁡𝑝_𝑖 )

Question 29

What is the partition function for discrete states?

Answer 29

𝑍=∑_𝑖exp⁡(−𝛽𝜀_𝑖 )

Question 30

What is the partition function for continuous states?

Answer 30

𝑍=∫(𝜀_𝑚𝑖𝑛 to 𝜀_𝑚𝑎𝑥)(exp⁡(−𝛽𝜀) 𝑑𝜀