Localised and Non-Localised Systems Flashcards Preview

Year 2 Statistical Mechanics > Localised and Non-Localised Systems > Flashcards

Flashcards in Localised and Non-Localised Systems Deck (34)
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1
Q

Localised System - Helmholtz Free Energy

Equation

A

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
2
Q

Localised System - Entropy

Equation

A

S = Nkln(Z1) + NkT* [∂(lnZ1)/∂T]|v

3
Q

Localised System - Internal Energy

Equation

A

E = N k T² * (∂(lnZ)/∂T)|v

4
Q

Degeneracy

Definition

A
  • more than one state with the same energy

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

5
Q

Localised System - Ratio of Occupancy

Equation

A

-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 )

6
Q

Temperature at Which Ratio of Occupancy is α

A

T = -(εi-εj) / (k lnα) = - Δε (klnα)

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

7
Q

Localised System - Total Energy

Equation

A

E = Σ (N εi exp(-εi/kT)) / Z

summed between i=0 and i=N

8
Q

Localised System - Average Energy

Equation

A

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

9
Q

Maxwell-Boltzmann Distribution Function

A

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

10
Q

Non-Localised System

Definition

A

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

11
Q

Localised System

Definition

A

Particles are fixed in position so they are distinguishable

12
Q

Non-localised System

Single Particle Partition Function

A

-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

13
Q

Single Particle Partition Function for an Ideal Gas

A

Z1 = V * [2πmkT/h²]^(3/2)

-where h is Planck’s constant

14
Q

System Partition Function for Localised Particles

A

Z = (Z1)^N

15
Q

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

A

localised

16
Q

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

A

-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

17
Q

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

A

Ω = N! / ∏ni!

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

18
Q

System Partition Function for Non-Localised Particles

A

Z = (Z1)^N / N!

19
Q

Helmholtz Free Energy in terms of the system partition function

A

F = -kT lnZ

20
Q

de Broglie Wavelength

A

λ = h / mv

21
Q

Thermal de Broglie Wavelength

Equation

A

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

-the de Broglie wavelength expressed in terms of temperature

22
Q

Thermal de Broglie Wavelength for Particles of an Ideal Gas

A
-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)
23
Q

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

A

-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

24
Q

Give three examples of non-localised systems

A

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

25
Q

Energy Level Occupancy of an Idea Gas

A
  • 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
26
Q

Ideal Gas
Single Particle Partition Function
Equation

A

Z1 = V* [ 2πmkT/h² ]^(3/2)

27
Q

System Partition Function

Ideal Gas Derivation

A

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

28
Q

Ideal Gas

Helmholtz Free Energy

A

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

29
Q

Ideal Gas

Entropy

A

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

30
Q

Ideal Gas

Energy

A

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

31
Q

Ideal Gas

Pressure From Helmholtz Free Energy

A

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

32
Q

Ideal Gas

ln(Z)`

A

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 )

33
Q

Boltzmann Distribution Function

A

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

34
Q

Boltzmann Distribution

ni

A

ni = exp(-Ei/kT)