Localised and Non-Localised Systems

Question 1

Localised System - Helmholtz Free Energy

Equation

Answer 1

F = - N k T ln(Z1)

• where N is the number of particles
• k is Boltzmann’s constant
• T is temperature
• Z1 is the single particle partition function

Question 2

Localised System - Entropy

Equation

Answer 2

S = Nkln(Z1) + NkT* [∂(lnZ1)/∂T]|v

Question 3

Localised System - Internal Energy

Equation

Answer 3

E = N k T² * (∂(lnZ)/∂T)|v

Question 4

Degeneracy

Definition

Answer 4

• more than one state with the same energy

- denoted g, where gi = the number of states with energy εi

Question 5

Localised System - Ratio of Occupancy

Equation

Answer 5

-the ration of occupancy of two states i and j is given by the Boltzmann factor for state i divided by the Boltzmann factor for state j:
ni/nj = exp(-εi/kT) / exp(-εj/kT)
ni/nj = exp(- (εi-εj)/kT )

Question 6

Temperature at Which Ratio of Occupancy is α

Answer 6

T = -(εi-εj) / (k lnα) = - Δε (klnα)

Δε>0 => α<1
Δε<0 => α>1
-fewer in the high energy states

Question 7

Localised System - Total Energy

Equation

Answer 7

E = Σ (N εi exp(-εi/kT)) / Z

summed between i=0 and i=N

Question 8

Localised System - Average Energy

Equation

Answer 8

E = Σ (εi exp(-εi/kT)) / Z
summed between i=0 and i=N
-this is just the total energy divided by N

Question 9

Maxwell-Boltzmann Distribution Function

Answer 9

f(E) = exp(-Ei/kT) / Σ exp(-Ei/kT)
where the sum is between i=0 and i=N
-gives an e^(-x) shaped surve

Question 10

Non-Localised System

Definition

Answer 10

The particles are moving around all the time so they are indistinguishable from each other
Energy levels are common to the entire container

Question 11

Localised System

Definition

Answer 11

Particles are fixed in position so they are distinguishable

Question 12

Non-localised System

Single Particle Partition Function

Answer 12

-replace sum with an integral over the density of states
∫ g(ε) e^(-ε/kT) dε
-where the integral is over all energies (between 0 and infinity)
-and g(ε) is the density of states

Question 13

Single Particle Partition Function for an Ideal Gas

Answer 13

Z1 = V * [2πmkT/h²]^(3/2)

-where h is Planck’s constant

Question 14

System Partition Function for Localised Particles

Answer 14

Z = (Z1)^N

Question 15

Is the two level paramagnet a localised or non-localised system?

Answer 15

localised

Question 16

Why is does the method for localised systems given an overcount for the number of states in an equivalent non-localised system?

Answer 16

-for indistinguishable particles (a non-localised system) the number of states is overcounted because interchanging two particles does not produce a different state unlike in the case of a localised system

Question 17

Number of ways of distributing N particles in a non-localised system

Answer 17

Ω = N! / ∏ni!

-the number of ways of distributing N particles with ni particles in each state

Question 18

System Partition Function for Non-Localised Particles

Answer 18

Z = (Z1)^N / N!

Question 19

Helmholtz Free Energy in terms of the system partition function

Answer 19

F = -kT lnZ

Question 20

de Broglie Wavelength

Answer 20

λ = h / mv

Question 21

Thermal de Broglie Wavelength

Equation

Answer 21

Λ = [2πħ²/mkT]^(1/2) = [h²/2πmkT]^(1/2)

-the de Broglie wavelength expressed in terms of temperature

Question 22

Thermal de Broglie Wavelength for Particles of an Ideal Gas

Answer 22

-starting from the de Broglie wavelength, sub in until you have Λ expressed in terms of temperature:
λ = h / ρ
E = ρ² / 2m = 3/2 kT 
-rearrange for ρ
ρ = √(3mkT)
-sub in for Λ :
Λ = h / √(3mkT) = √(h²/3mkT)

Question 23

What is the condition for choosing between quantum statistics and Boltzmann statistics?

Answer 23

-quantum mechanical interactions become important when the particles have spin and their wave functions overlap, this occurs when Λ becomes close to the mean interparticle distance, d = [N/V]^(1/3)
-if:
Λ &laquo_space;d then classical Boltzmann statistics applies
but, if:
Λ ≈ d then quantum statistics is necessary

Question 24

Give three examples of non-localised systems

Answer 24

1) ideal gas
2) electron gas
3) Planck spectrum

Question 25

Energy Level Occupancy of an Idea Gas

Answer 25

• for any macroscopic system, ε &laquo_space;kT
• since there are so many more states that particles, statistically it is highly highly unlikely to find more than one particle in a given energy state
• each state is either occupied by one particle or unoccupied

Question 26

Ideal Gas
Single Particle Partition Function
Equation

Answer 26

Z1 = V* [ 2πmkT/h² ]^(3/2)

Question 27

System Partition Function

Ideal Gas Derivation

Answer 27

Ω = N! / ∏ni! , gives the number of ways N particles with ni particles in each state
-for an ideal gas every state contains either 1 or 0 particles 0! = 1! = 1 so ni! = 1 for all i, thus
Ω = N!
Z = Z1^N / Ω = Z1^N / N! = 1/N! ( V [2πmkT/h² ]^(3/2)] )^N

Question 28

Ideal Gas

Helmholtz Free Energy

Answer 28

F = -kTln(Z) = /NkTln( [mkT/2πℏ²]^(3/2) * Ve/N )

Question 29

Ideal Gas

Entropy

Answer 29

S = E-F / T = 3Nk/2 + Nk ln([mkT/2πℏ²]^(3/2) * Ve/N )])
= Nk (5/2 + 3/2*ln(mkT/2πℏ²) + lnV - lnN)
-also known as the Sackur-Tetrode equation

Question 30

Ideal Gas

Energy

Answer 30

E = kT² (∂lnZ/∂T)|v = NkT² ∂/∂t [ln( [mkT/2πℏ²]^(3/2) * Ve/N )]
E = NkT² ∂/∂T [3/2 lnT] = 3/2 NkT
-this is equal to the classical expression for translational kinetic energy of a monatomic ideal gas

Question 31

Ideal Gas

Pressure From Helmholtz Free Energy

Answer 31

p = - (∂F/∂V)|T,N = NkT ∂/∂V( lnV + ln( [mkT/2πℏ²]^(3/2)*e/N ))
-terms that don’t depend on V go to 0
p = NkT/V
-which is the ideal gas law

Question 32

Ideal Gas

ln(Z)`

Answer 32

ln(Z) = ln(1/N! ( V [2πmkT/h² ]^(3/2)] )^N)
= -NlnN - N + ln( V[2πmkT/h² ]^(3/2)] )^N
= N ln( [mkT/2πℏ²]^(3/2) * Ve/N )

Question 33

Boltzmann Distribution Function

Answer 33

f = ni / N = exp(-Ei/kT) / Σexp(-EI/kT)

Question 34

Boltzmann Distribution

ni

Answer 34

ni = exp(-Ei/kT)