Thermodynamics and Statistical Mechanics
1
General Definitions and Equations
• First Law of Thermodynamics: Heat Q is a form of energy, and energy is conserved.
– dU = dQ − dW
• Second Law of Thermodynamics: The entropy S of a system must increase.
– ∆S ≥ 0
• Third Law of Thermodynamics: For a system with a nondegenerate ground state
S → 0 as T → 0.
– lim S = kB ln Ω0 , where Ω0 is the degeneracy of the ground state
T →0
• A system is equally likely to be in any of the quantum states accessible to it.
1
• Equipartition theorem: each “degree of freedom” of a particle contributes kB T to
2
its thermal average energy
– Any quadratic term in the energy counts as a “degree of freedom”
3
– E.g. monoatomic: 3 translational, 0 rotational, 0 vibrational → hEi = kT
2
– E.g. diatomic:
3
translational,
2
rotational,
0
vibrational
(2
at
high
T) →
5
7
hEi = kT
kT at high T
2
2
N n
• Binomial Distribution: P (n, N − n) =
p (1 − p)N −n
n
N
N!
–
=
n
n!(N − n)!
– Normalization:
N
X
P (n, N − n) = 1
n=0
– Mean: hni =
N
X
nP (n, N − n) = N p
n=0
N
X
– Variance: n2 − hni2 =
n2 P (n, N − n) − (N p)2 = N p(1 − p)
n=0
1
(x − µ)2
• Gaussian Distribution: P (x) = √
exp −
2σ 2
2πσ 2
Z ∞
P (x) dx = 1
– Normalization:
−∞
Z ∞
– Mean: hxi =
xP (x) dx = µ
−∞
Z ∞
2
– Variance (x − µ) =
(x − µ)2 P (x) dx = σ 2
1
!
−∞
Z
– P (a < x < b) =
b
P (x) dx
a
• Heat capacity, c: Q = C∆T , C = mc, where c is the specific heat capacity
• Molar heat capacity, C: Q = nC∆T , where n is the number of moles
• Latent heat of fusion (Q to make melt): Q = mLf
• Latent heat of vaporization (Q to vaporize): Q = mLv
• Ideal Gas Law: P V = nRT = N kB T
• For an ideal gas, Cp = CV + R,
• Adiomatic index: γ =
– γ=
where CV =
∂Q
∂T
,
CP =
V
∂Q
∂T
P
CP
CV
7
5
for monoatomic gases, γ = for diatomic gases
3
5
• Stirling’s Approximation: ln(n!) ≈ n ln(n) − n, for n >> 1
2
Energy equations
• Thermodynamic Identity: dE = T dS − P dV + µdN
• Helmholtz Free Energy: A = E − T S ⇒ dA = −P dV − SdT + µdN
• Enthalpy: H = E + P V → dH = T dS + µdN + V dP
• Gibbs free energy: G = A + P V = E + P V − T S → dG = V dP − SdT + µdN
∂
∂Φ
∂
∂Φ
• Maxwell relations:
=
∂xj ∂xi
∂xi ∂xj
2
3
Thermodynamic Processes
Z
• Reversible process: happens very slowly, ∆S =
dQ
T
• Irreversible process: happens very suddenly
Z
• Work: W = P dV
• Isothermal: T held constant
– dU = 0 → dQ = dW
• Isochoric: V held constant
– W = 0 → dU = dQ
• Isobaric: P held constant
– W = P ∆V
• Isoentropic: S held constant
• Adiabatic: dQ = 0
– dU = −dW
– Also isoentropic if reversible
– P V γ = constant
• Heat engine thermal efficiency: e = 1 +
QC
QH
• Refridgerator coefficient of performance: k =
|QC |
|QH | − |QC |
• Carnot cycle thermal efficiency (ideal): e = 1 −
TC
TH
• Carnor refridgerator coefficient of performance: k =
4
Microcanonical Ensemble
• S = kB ln Ω, Ω is number of microstates accessible
3
TC
TH − TC
5
Canonical Ensemble
• Partition function: Z =
X
• With degeneracies: Z =
X
e−βεs , where β =
s
1
kB T
gs e−βεs , where gs is the degeneracy of the s state
s
e−εs /kB T
Z
1 X −βεs
d ln Z
• hεi =
εs e
=−
Z s
dβ
• P (εs ) =
• A = −kB T ln Z
• S=
∂A
∂
(kB T ln Z) = −
∂T
∂T
• For N indistinguishable particles, ZN =
6
ZN
N!
Grand Canonical Ensemble
∂S • Chemical potential µ = −T
∂N E
• Grand partition fucntion: Z =
∞ X
X
N =0 s(N )
εs (n) − µN
exp −
kB T
εs (n) − µN
1
• P (N, ε) = exp −
Z
kB T
• hN i = kB T
7
∂ ln Z
∂µ
Blackbody Radiation
• Stephan Boltzmann Law: Φ = σT 4
– Φ is energy flux of star
– T is the temperature of the star
– σ = 5.67 × 10−8 JK −4 m−2 s−1 is Stephan-Boltzmann constant
4
• Planck radiation Law: uω =
~
ω3
π 2 c3 e−β~ω − 1
– uω is spectral density, energy per unit volume per unit frequency
• Wein’s Law
– λpeak =
8
2.9 × 10−3 mK
T
Particle Distributions
• Maxwell-Boltzmann (for classical, distinguishable particles): n(ε) = e−β(ε−µ)
• Fermi-Dirac (for identical fermions): n(ε) =
1
eβ(ε−µ)
+1
(
1 if ε < µ(0)
– As T → 0, n(ε) →
0 if ε > µ(0)
– Fermi energy EF = µ(0), all states filled below EF no states filled above EF
• Bose-Einstein (for identical bosons): n(ε) =
5
1
eβ(ε−µ)
−1