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Thermal and Statitiscal Physics Handout
Lecture notes for the Thermal and Statistical Physics course.
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Thermal and Statistical Physics
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Thermal and Statistical Physics
Part II Michaelmas 2020
Lecture Overheads
F. Malte Grosche
September 27, 2020
Statistical Mechanics
’Ludwig Boltzmann, who spent much of his life
studying statistical mechanics, died in 1906, by his own hand.
Paul Ehrenfest, carrying on the work, died similarly in 1933.
Now it is our turn to study statistical mechanics.
Perhaps it will be wise to approach the subject cautiously.’
####### [David Goodstein — States of Matter, 1975, Dover N.]
F. M. Grosche (Cavendish Laboratory) Thermal and statistical physics 3 / 146
These boots are made for walking
Thermodynamics
Statistical mechanics
Quantum
mechanics
Thermodynamics
A warning
The slides are often complemented by blackboard work.
It is important to take copious notes in the lectures.
And: use the textbooks!
F. M. Grosche (Cavendish Laboratory) Thermal and statistical physics 7 / 146
The laws of thermodynamics
If two systems are separately in thermal equilibrium with a third, then
they must also be in thermal equilibrium with each other.
Energy is conserved when heat is taken into account.
Heat does not flow from a colder to a hotter body. [Clausius]
or
You can’t cool an object to below the temperature of its surroundings,
and extract work in the process. [Kelvin]
Internal energy
Mechanical energy (kinetic, potential) accounts for contributions to the
overall energy, which are attributed to a few macroscopic degrees of freedom.
Internal energy takes care of contributions to the overall energy, which
cannot be attributed to a few macroscopic degrees of freedom.
We think of the internal energy as residing with internal, microscopic
degrees of freedom.
This introduces the difficulty that we can put this energy in – but we do not
have enough information about internal degrees of freedom to be able to
extract all of it again.
Once energy is sunk in microscopic motion within the system, it is difficult to
get it out again. But that is what a steam engine does, so it is not
impossible. Thermodynamics lets us calculate this.
Thermodynamic variables
Systems are characterised by thermodynamic variables.
These occur in conjugate pairs of thermodynamic force (p, f , μ 0 H, T ) and
thermodynamic displacement (V , `, M, S).
Thermodynamic (or generalised) forces are always intensive: independent of
the size of the system.
Thermodynamic (or generalised) displacements are always extensive:
proportional to the size of the system.
The product of a pair of conjugate variables contributes to a
thermodynamic potential and has dimensions of energy.
Functions of state
We can find the magnitude of the change in a function of state in going between
two equilibrium states, even if the path taken was irreversible,
by choosing any convenient reversible path between the
initial and final states and integrating along it.
T 0
∮ d Q /T ≤ 0 in general
∫A
B
dQ rev /T ≥∫A
B
dQ irrev /T
for process on right
Define dQ rev /T =dS
If no heat is exchanged, dS≥ 0
irreversible
process
dQrev
reversible
process
U is a function of state, independent of reversibility
Irreversible: Q h , Q c (< 0 ) reduced
Extracted work also reduced.
Net effect on U same as reversible
cycle.
For reversible changes only:
dW ̄ (rev) = p dV + μ dN + (other relevant work terms, e. magnetic)
dQ ̄ (rev) = T dS
But generally true for reversible and irreversible changes:
dU = T dS p dV + μ dN + (other relevant work terms)
Maximum entropy principle
In approaching equilibrium, the entropy of a closed system increases:
dS =
dU + pdV
P
i μ i dN i
T
0
Two systems in thermal contact:
dS tot > 0 =) dU 1 > 0 for T 2 > T 1 and vice versa.
Two systems with particle-exchange:
dS tot > 0 =) dN 1 > 0 for μ 2 > μ 1 and vice versa.
Availability – general form of maximum entropy principle
System and reservoir (i. large system, which does not change its temperature, p
or μ).
dS tot = dS + dS R 0
= dS +
dU R + p R dV R μ R dN R
T R
=
T R dS dU p R dV + μ R dN
T R
,
Define availability A such that dA = T R dS tot :
dA = dU T R dS + p R dV μ R dN
= (T T R )dS (p p R )dV + (μ μ R )dN 0
Reservoir properties (T R , p R , μ R ) are constant =)
A = U T R S + p R V μ R N
The availability of a system in contact with a reservoir
is minimised in equilibrium
Availability for system at constant T , V , N – Helmholtz free energy
Thermal contact with reservoir fixes T = T R , dT = 0, while V and N are fixed.
(dA) T ,V ,N = (dU T R dS + p R dV μ R dN) T ,V ,N
= d(U TS) T ,V ,N ⌘ (dF ) T ,V ,N ,
F is the Helmholtz free energy.
Example: movable barrier between two gases at same temperature T.
F = F 1 + F 2 = (U 1 T 1 S 1 ) + (U 2 T 2 S 2 )
T 1 = T 2 = T , dV 2 = dV 1 , dU = TdS pdV =)
dF = (p 1 p 2 )dV 1 = 0 ) p 1 = p 2.
Phase equilibria – Gibbs free energy
Liquid-vapour transition:
At the phase transition, liquid and vapour phases coexist.
μ ` = μ v
Phase equilibria – Clausius Clapeyron equation
Phase coexistence lines in the (p, T ) plane
dm 2
dm 1
dμ 1 = dμ 2
v 1 dp s 1 dT = v 2 dp s 2 dT
=)
dp
dT
=
s 1 s 2
v 1 v 2
=
T s
T v
⌘
L
T v
Ideal gas mixtures
Entropy of mixing:
S = Nk B
X
####### i
c i ln c i
(c i = p i /p is concentration, p i is partial pressure)
Chemical potential:
μ i (p i , T ) = μ 0 i (p, T ) + k B T ln c i
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Thermal and Statitiscal Physics Handout
Module: Thermal and Statistical Physics
5 Documents
Students shared 5 documents in this course
University: University of Cambridge
Was this document helpful?
Thermal and Statistical Physics
Part II Michaelmas 2020
Lecture Overheads
F. Malte Grosche
September 27, 2020
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