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3 Laws of Thermodynamics
O)
1)
2)
If two systems are in equilibrium with a third, they are in equilibrium with each other.
Conservation of energy ΔEth = W + Q
The entropy of an isolated system never decreases. The entropy either increases until it reaches
equilibrium, or if it’s in equilibrium, it stays the same.
⇒ given two system w/σ1 > τ2, heat will be spontaneously transferred from system 1 to 2.
⇒ heat cannot be completely converted into work.
Thermodynamic basics
Partition function:
Probability of being in a state w/energy
The fundamental assumption: a closed system is equally likely to be any of the quantum states
accessible to it.
g(N, U) ≡ The multiplicity of a system with N particles and energy
S = kBσ = kBlogg(N, u)
Specific case: Hermonic oscillator: g(N, n) =
where N = #oscillators, n = quantum #
Specific case: N magnets with Sp in excess Zs = N – N : g(N, s) =
↑
where
=
↓
where B is the magnetic field and M is the magnetic moment
Kinds of energy:
τdσ = du + pdV – µN
Helmholtz Free Energy
(isothermal)
F = u – τσ = – τlogz
(isobaric)
dF = du – τdσ + µdN
= –σdτ – PdV + µdN
Enthalpy
H = µ + pV
dH = τdσ + Vdp – µdN
(isobaric, isothermal) Gibbs Free Energy
G = F+pV = u+pV – τσ
dG = –σdτ + Vdp + µdN
,
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Distributions
Fermi – Dirac : Average occupancy of an orbital w/energy ε, for fermions
Bose – Einstein: Average occupancy of an orbital w/energy ε, bosons
Plank distribution: Thermal average number of photons in a single mode a
Ideal gas
PV = nRT = NkBT
µ = KEavg =
(
Per atom in a
monatomic gas
for each degree of freedom, note that f ∃ potential energy, each of those
degrees of freedom gets
Heat capacity, constant volume :
as well by the Equipartition Theorem)
(τ = kBT)
Partition function of an atom in a box. Z1 =
na =
Partition function of N atoms in a box :
Entropy S = kBσ = kBN
Chemical potential :
Average occupancy of an orbital of energy ε
where λ = e /
µ
τ
Free energy:
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Reversible isothermal
Reversible isentropic
Irreversible extension into
vacuum
u2 – u1
σ2 – σ1
W
Q
O
O
O
O
O
O
Diatomic Gas:
2-D ideal gas
u = kBT,
Cv = NkB
Cp = 2NkB
Van DerWalls – attemps to modify the ideal gas law to take into account interactions between atoms
or molecules
where a is a measure of the long-range attractive part (adds to internal pressure) of the interaction
and
b is a measure of the short-range repulsion (volume of molecules themselves)
Critical points : Pc =
Vc = 3Nb,
at this point, there is no separation between the vapor
and liquid phases (a horizontal point of inflection)
τ>τc
τ=τc
( K σ K1 p290 Fig. 10.10)
τ<τc
v1
(For a given
v2
, τ < τc, V < V1 ⇒ liquid V > V2 ⇒ gas,
V1 < V < V2 ⇒ both show that sum of volume of
liquid G gas = V)
Phase Diagram
Solid
Liquid
Critical point
Triple point
T
Triple point : The one value of T and P for which all three phases
can happily coexist. Happily.
Critical point: below this point a phase change between liquid &
gas. Above this point
phase change (fluid Δ continuously
between high & low density)
Diffusion
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,
Maxwell velocity distribution
Fick’s Law
(diffusion constant)
Fourier’s Law
Carnot cycle and Work in general
Work done on a system =
= –(area under pV curve)
Energy in: heat from resevoir RH (@ τH)
Energy out: heat to resevoir RL (@ τL < τH)
For a reversible engine, ΔσH = ΔσL (if τH τL, only work may be transferred)
≠
efficiency:
1) compress isothermally (Q ↓)
2) compress isentropically (τ ↑)
3) expand isothermally (Q ↑)
4) expand isentropically (τ↓)
(for a heat pump, reverse order)
V
P
2
3
1
4
1
4
2
3
V
for the carnot cycle, efficiency is at a maximum
or
(engine)
(pump/refrigerator)
For an ideal gas, isothermal process ⇒ QH = ω = NτH log
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τ
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isentropic process ⇒ ω = Δµ =
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