Localised and Non-Localised Systems Flashcards
Localised System - Helmholtz Free Energy
Equation
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
Localised System - Entropy
Equation
S = Nkln(Z1) + NkT* [∂(lnZ1)/∂T]|v
Localised System - Internal Energy
Equation
E = N k T² * (∂(lnZ)/∂T)|v
Degeneracy
Definition
- more than one state with the same energy
- denoted g, where gi = the number of states with energy εi
Localised System - Ratio of Occupancy
Equation
-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 )
Temperature at Which Ratio of Occupancy is α
T = -(εi-εj) / (k lnα) = - Δε (klnα)
Δε>0 => α<1
Δε<0 => α>1
-fewer in the high energy states
Localised System - Total Energy
Equation
E = Σ (N εi exp(-εi/kT)) / Z
summed between i=0 and i=N
Localised System - Average Energy
Equation
E = Σ (εi exp(-εi/kT)) / Z
summed between i=0 and i=N
-this is just the total energy divided by N
Maxwell-Boltzmann Distribution Function
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
Non-Localised System
Definition
The particles are moving around all the time so they are indistinguishable from each other
Energy levels are common to the entire container
Localised System
Definition
Particles are fixed in position so they are distinguishable
Non-localised System
Single Particle Partition Function
-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
Single Particle Partition Function for an Ideal Gas
Z1 = V * [2πmkT/h²]^(3/2)
-where h is Planck’s constant
System Partition Function for Localised Particles
Z = (Z1)^N
Is the two level paramagnet a localised or non-localised system?
localised
Why is does the method for localised systems given an overcount for the number of states in an equivalent non-localised system?
-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
Number of ways of distributing N particles in a non-localised system
Ω = N! / ∏ni!
-the number of ways of distributing N particles with ni particles in each state
System Partition Function for Non-Localised Particles
Z = (Z1)^N / N!
Helmholtz Free Energy in terms of the system partition function
F = -kT lnZ
de Broglie Wavelength
λ = h / mv
Thermal de Broglie Wavelength
Equation
Λ = [2πħ²/mkT]^(1/2) = [h²/2πmkT]^(1/2)
-the de Broglie wavelength expressed in terms of temperature
Thermal de Broglie Wavelength for Particles of an Ideal Gas
-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)
What is the condition for choosing between quantum statistics and Boltzmann statistics?
-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 «_space;d then classical Boltzmann statistics applies
but, if:
Λ ≈ d then quantum statistics is necessary
Give three examples of non-localised systems
1) ideal gas
2) electron gas
3) Planck spectrum
Energy Level Occupancy of an Idea Gas
- for any macroscopic system, ε «_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
Ideal Gas
Single Particle Partition Function
Equation
Z1 = V* [ 2πmkT/h² ]^(3/2)
System Partition Function
Ideal Gas Derivation
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
Ideal Gas
Helmholtz Free Energy
F = -kTln(Z) = /NkTln( [mkT/2πℏ²]^(3/2) * Ve/N )
Ideal Gas
Entropy
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
Ideal Gas
Energy
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
Ideal Gas
Pressure From Helmholtz Free Energy
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
Ideal Gas
ln(Z)`
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 )
Boltzmann Distribution Function
f = ni / N = exp(-Ei/kT) / Σexp(-EI/kT)
Boltzmann Distribution
ni
ni = exp(-Ei/kT)