Stat Mech Flashcards
What does classical thermodynamics describe?
Macroscopic systems ~in equilibrium in terms of a few measurable variables
What are state variables?
Variables that describe the current state of a system. Importantly, they do not depend on the history of the system.
Includes temperature, pressure, volume, composition, internal energy, and entropy.
Cannot all be varied independently, equations linking them are equations of state.
What is quasi-equilibrium?
Changes to conditions (eg pressure) can be made sufficiently slowly that the system remains in equilibrium at all times.
What is a reversible process?
A process where an infinitesimal change in the external conditions is enough to reverse the direction of the process;
e.g. compression of a gas in a cylinder with a frictionless piston by exerting an external force only just sufficient to move the piston one way or the other.
What is a macrostate?
The state of a macroscopic system in equilibrium specified by a handful of macroscopically-manipulable variables. It completely ignores what is going on with the individual atoms that comprise the system.
What is a microstate?
A description of the positions and momenta of all atoms (or of their combined quantum state) of a system.
The Zeroth Law of Thermodymamics
If two bodies are separately in thermal equilibrium with a third body, they are also in thermal equilibriums with one another. All three are then said to be at the same temperature.
Absolute temperature
Measured in Kelvin, defined to be 273.16 K at the triple point of water:
T = lim(P→0) PV/(PV)ₜᵣᵢₚₗₑ × 273.16 K
Low pressure limit taken so real gases approach ideal behaviour.
Zero of the Kelvin scale is absolute zero. At this temperature, pressure of an ideal gas would vanish, because (classically) the motion of its molecules would cease.
First Law of Thermodynamics
Any change in internal energy, E, of a system is due to the amount of heat added to it and the work done on it.
∆E = Q + W
dE = đQ + đW
Energy and work in a reversible process.
dE = đQʳᵉᵛ + đWʳᵉᵛ
Adiabatic processes
No heat transfer
Q = 0
System in a cycle
Returns to its initial state
∆E = 0
Work done in reversible processes
Compression of a fluid:
đWʳᵉᵛ = −P dV
Stretching a wire of tension Γ by dl:
đWʳᵉᵛ = Γ dl
Increasing the area of a film of surface tension γ by dA:
đWʳᵉᵛ = γ dA
Increasing the magnetic field B imposed upon a paramagnetic sample:
đWʳᵉᵛ = −m · dB = −VM · dB
where M is magnetisation per unit volume, and m is total magnetic moment of the sample.
To calculate the internal energy change for irreversible processes (such as free expansion of a gas), it is necessary to find a reversible process linking the same initial and final states of the system.
Second law of thermodynamics
No process which would decrease the entropy of a system will happen spontaneously.
OR
If left alone, systems always head towards the same equilibrium as determined by external constraints.
OR
In an isolated system, entropy can only increase (or stay the same)
Kelvin and Planck statement
“It is impossible to construct an engine which, operating in a cycle, will produce no other effect than the extraction of heat from a reservoir and the performance of an equivalent amount of work”
Clausius statement
“It is impossible to construct an refrigerator which, operating in a cycle, will produce no other effect than the transfer of heat from a cooler body to a hotter one.”
Entropy
dS = đQʳᵉᵛ/T
For an isolated system, during any spontaneous change
dS ≥ 0
Fundamental thermodynamic
relation
dE = TdS − PdV
(dE = đQʳᵉᵛ + đWʳᵉᵛ)
E : energy
T : absolute temperature
S : entropy
P : pressure
V : volume
Carnot engine
Qₕ is removed from a hot reservoir and Q꜀ is discarded to a cold one in such a way that the combined entropy change is zero.
Qₕ/Tₕ = Q꜀/T꜀
The difference is available to do work
W = Qₕ − Q꜀
From the fundamental thermodynamic relation, derive functions for fixed properties.
dE = TdS − PdV
⇒E(S, V) with
(∂E/∂S)|ᵥ = T and (∂E/∂V)|ₛ = −P
dS = 1/T dE + P/T dV
⇒S(E, V) with
(∂S/∂E)|ᵥ = 1/T and (∂S/∂V)|ₑ = P/T
Helmholtz free energy
System in contact with heat bath, spontaneous change at constant volume, Q absorbed.
∆E = Q.
System starts and ends at temperature T. Total change in entropy has two parts, ∆S for system and −Q/T for surroundings.
∆Sₜₒₜ = ∆S − Q/T = 1/T (T∆S − ∆E) ≥ 0
⇒ ∆(TS − E) ≥ 0
In approach to equilibrium, quantity TS − E is maximised.
Helmholtz free energy, F, is negative of this, so it is minimised.
F = E − TS
⇒ dF = −SdT − PdV
(∂F/∂T)|ᵥ = −S and (∂F/∂V)|ₜ = −P
Gibbs free energy
Approach to equilibrium minimises Gibbs free energy, G, of a system.
G = E − TS + PV
⇒ dG = −SdT + VdP
(∂G/∂T)|ₚ = −S and (∂G/∂P)|ₜ = V
Gibbs free energy and chemical potential
µ = (∂G/∂N)|ₜ,ₚ
G = ΣμᵢNᵢ
μᵢ is chemical potential of each species
Nᵢ is number of molecules of each species
Entropy for an ideal gas as a function of temperature and pressure
S(T, P) = S(T₀, P₀) + Cₚ ln(T/T₀) − nRln(P/P₀)
Minimising chemical potential through pressure
Chemical potential is higher at higher pressures, and gas will diffuse from higher to lower pressure. Partial pressures of each species will minimise despite mechanical pressure. For example, If two ideal gases are at different concentrations on either side of a rigid membrane, but only one can pass through, the partial pressure of the mobile one will equalise even if that increases the mechanical total pressure on one side.
Comparing the chemical potential or Gibbs free energy per molecule at two different pressures but constant temperature.
µ(T₀, P₂) − µ(T₀, P₁) = kᴮT₀ln(P₂/P₁)
What is an ensemble?
The set of all allowed microstates.
Types of ensemble
Microcanonical (NVE)
Total energy and number of particles are fixed. The system must remain totally isolated (unable to exchange energy or particles with its environment) to stay in statistical equilibrium.
Canonical (NVT)
Energy is not known exactly but number of particles is fixed. In place of energy, the temperature is specified. It can describe a closed system which is/has been in weak thermal contact with a heat bath. The system must remain totally closed (unable to exchange particles with its environment) but may come into weak thermal contact with other systems that are described by ensembles with the same temperature.
Grand canonical (μVT)
Neither the energy nor particle number are fixed. Instead, the temperature and chemical potential are specified. It can describe an open system: one which is in, or has been in, weak contact with a reservoir (thermal contact, chemical contact, radiative contact, electrical contact, etc.). It will remain in statistical equilibrium if the system comes into weak contact with other systems that are described by ensembles with the same temperature and chemical potential.
What is an ensemble average?
If the value of some quantity X in the iₜₕ microstate is Xᵢ, and the probability that the system is in that
microstate is pᵢ, then the value of X in the macrostate is the ensemble average.
⟨X⟩ = Σ pᵢXᵢ
Number of ‘allowed’ or ‘accessible’ microstates
Allowed or accessible means having the same constraints, such as volume, particle number and total energy, as the macrostate.
pᵢ = 1/Ω
Σpᵢ = Ω 1/Ω = 1
For the example of blue/green counters
Ω = N!/n!(N − n)!
N!/n! (N − n)! are binomial coefficients, written ᴺCₙ
Entropy from microstates
More microstates exist with higher entropy, as a system evolves it goes to more likely microstates, hence entropy incerases.
S = kᴮ lnΩ
Total magnetic moment of an ideal paramagnet
Lattice of N sites, at each the spin points either up
or down. Total magnetic moment is the sum of the individual moments. If n↑ spins are pointing up and n↓ = N − n↑ are pointing down:
m = n↑µ + n↓(−µ) = µ(2n↑ − N)
where µ is magnetic moment.
Energy of a magnetic moment in an external magnetic field
−µ · B
Spin-up atoms will have energy −µB, and spin-down atoms will have energy µB
Total energy in paramagnet
E = n↑(−µB) + n↓(µB) = −Bm
Macrostate probability distribution for spin-1/2 paramagnet
For n↑ = 1/2 (N + m/µ)
Ω(n↑) = N!/n↑!(N − n↑)!
Plotting Ω(n) (normalised to 1 at the peak) as a function of n/N, get a gaussianesque shape with a sharper peak for higher N.
For large N, curve approximated by
Ω(n) ∝ exp(−(n−N/2)²/(N/2))
Mean N/2
Standard deviation σ = √N/2’
Tends to gaussian, central limit theorem
Temperature, pressure, and chemical potential from entropy
(∂S/∂E)|ᵥ,ₙ = 1/T
(∂S/∂V)|ₑ,ₙ = P/T
(∂S/∂N)|ₑ,ᵥ = −µ/T
They give
dS = 1/T dE + P/T dV − µ/T dN
which is the fundamental thermodynamic relation rearranged
Fundamental thermodynamic relation in spin-1/2 paramagnet
Instead of −PdV we have −mdB
dE = TdS − mdB + µdN
dS = 1/T dE + m/T dB − µ/T dN
(∂S/∂E)|ᴮ,ₙ = 1/T
(∂S/∂B)|ₑ,ₙ = m/T
(∂S/∂N)|ₑ,ᴮ = −µ/T
Temperature and magnetic moment from entropy for an isolated spin-half paramagnet
S = kᴮ ln Ω(E, B) = kᴮ ln(N!/n↑!(N − n↑)!)
Large N, Stirling’s approximation:
ln(n!) = nln(n)−n
S = kᴮ(Nln(N) − n↑ln(n↑) − (N−n↑)ln(N−n↑))
1/T = (∂S/∂E)ᴮ,ₙ
= (∂S/∂n↑)ₙ (∂n↑/∂E)ᴮ,ₙ
= kᴮ/2µB ln(n↑/n↓)
m/T = (∂S/∂B)ₑ,ₙ
= (∂S/∂n↑)ₙ (∂n↑/∂B)ₑ,ₙ
= −kᴮE/2µB² ln(n↑/n↓)
⇒ m = −E/B
Derive the ideal gas equation from first principles
Isolated system of N atoms in a box of volume V . Box is subdivided into many tiny cells of volume ∆V , so there are V/∆V cells total. Each atom can be in any cell, V/∆V microstates for each atom, and (V/∆V)N microstates for the gas as a whole.
S = Nkᴮln(V/∆V)
Change in entropy when volume changes (at fixed energy)
∆S = Nkᴮ ln(Vf/Vi)
From entropy,
P/T = (∂S/∂V)ₑ,ₙ=
⇒ P = NkᴮT/V
⇒ PV = NkᴮT
When can you treat atoms independently?
When they don’t interact AND they’re in a non-isolated system (energy isn’t limited)
Derive the Boltzmann distribution for a system in contact with a heat bath at temperature T.
System in contact with a heat bath at temperature T, probability it is in the ith microstate, energy εᵢ
System S in contact with heat reservoir R, the whole forming a single isolated system with energy E₀
Eᵣ = E₀ − ε
Eₛ = ε
Likelihood of a particular partition of energy depends on the number of microstates of whole system S + R that correspond to that partition. System and reservoir are independent, so total number of microstates factorises
Ω = ΩₛΩᵣ
Specify the ith microstate (with energy εᵢ). Probability pᵢ of finding the system in that microstate is proportional to the number of compatible microstates Ω(E₀, εᵢ) of the whole
system. Specified state of S, Ωₛ = 1, only the microstate of the reservoir is unspecified, Ω(E₀, εᵢ) = Ωᵣ(E₀ − εᵢ)
Using relation between Ω and entropy,
pᵢ ∝ Ωᵣ(E₀ − εᵢ) = exp{Sᵣ(E₀ − εᵢ)/kᴮ}
R ⟩⟩ S, so εᵢ ⟨⟨ E₀. Can expand Sᵣ about Sᵣ(E₀) and keep only the lowest terms:
Sᵣ(E₀ − εᵢ) = Sᵣ(E₀) − εᵢ(∂Sᵣ/∂E)ᵥ,ₙ + ½εᵢ²(∂²Sᵣ/∂E²)ᵥ,ₙ + . . .
(derivatives are evaluated at E0)
Derivative of S w.r.t. E is just the inverse of temperature. Dropping third term as negligibly small:
pᵢ ∝ exp{Sᵣ(E₀)/kᴮ − εᵢ/(kᴮT)}
∝ exp{−εᵢ/(kᴮT)}
(since Sᵣ(E₀) is a constant, independent of the microstate we are interested in)
Call constant of proportionality 1/Z, final result:
pᵢ = exp(−εᵢ/kᴮT) / Z
Where normalisation constant Z is found by saying total probability is one:
Σⱼ pⱼ = 1
Z = Σⱼ exp(−εⱼ/kᴮT)
This is also the partition function
Partition function
Z = Σⱼ exp(−εⱼ/kᴮT)
Energy from partition function
⟨E⟩ = −1/Z (∂Z/∂β)ₙ,ᵥ
or
⟨E⟩ = −∂lnZ/∂β
where
β ≡ 1/kᴮT
Heat capacity from energy
Cᵥ = (∂⟨E⟩/∂T)ᵥ,ₙ
Fluctuations in energy
(∆E)² = ⟨E⟩² − ⟨E⟩²
Since ⟨E⟩² = Σᵢ εᵢ² pᵢ,
⟨E⟩² = 1/Z (∂²Z/∂β²)ₙ,ᵥ
(∆E)² = 1/Z (∂²Z/∂β²) − (∂lnZ/∂β)² = ∂²lnZ/∂β²
= −∂T/∂β ∂⟨E⟩/∂T = (kᴮT)² Cᵥ/kᴮ
Normal macroscopic system, average energy is of order NkᴮT and heat capacity is of order Nkᴮ.
∆E/E ≈ 1/√N’
Derive Gibbs entropy
Gibbs entropy of system is entropy of ensemble divided by the number of copies, ν, in the ensemble.
⟨S⟩ = Sᵥ/ν
Ensemble has νᵢ copies in the ith microstate, number of ways of arranging these is
Ωᵥ = ν!/ν₁!ν₂!ν₃!…
Stirling’s approximation,
ln Ωᵥ = νlnν − ν − Σᵢ (νᵢ (lnνᵢ − νᵢ)
= Σᵢ νᵢ (lnν − lnνᵢ) (using ν = Σᵢνᵢ)
= −Σᵢ νᵢ (lnνᵢ/ν)
= −ν Σᵢ pᵢ lnpᵢ
Ensemble entropy is Sᵥ = kᴮlnΩᵥ and system entropy is
⟨S⟩ = −kᴮ Σᵢ pᵢ lnpᵢ
