Lecture 9:
The Third Law
Unattainability of absolute zero
(and other things)
Aims:
The third Law:
S→0 as T → 0.
T =0 cannot be reached.
Adiabatic demagnetisation.
Gibbs’ entropy.
S = −k
i
pi ln pi
Microscopic reversibility
Ehrenfest’s urn:
Irreversible behaviour from reversible rules.
Recap. and summary of key results.
March 04
Lecture 9
1
The Third Law
Third Law (Nernst 1906)
The entropy of any system is zero at the
absolute zero of temperature.
The main point is that all systems tend to the
same entropy as T → 0.
Originally a highly contentious statement
whose opponents either:
believed it to true but inapplicable, or,
believed it to be applicable, but false.
The arguments are largely irrelevant today.
Now regarded as true, applicable and useful.
We can give an experimental justification. E.g.
In a reaction A + B → AB the entropy change
∆SAB(T) = SAB(T) - SA(T) - SB(T) can be determined
directly and by individual measurements on the
components (e.g. by calorimetry).
T C (T )
v
S A (T ) = S A (0 ) +
dT +
Li Ti
etc.
i
T
0
The results are consistent if
SAB(0) = SA(0) = SB(0) = 0
Also consistent with a statistical picture since
we must be in the ground state, where g=1, at
T=0. So, k lng = 0.
March 04
Lecture 9
2
Unattainability of absolute zero
Consequence of the third Law
It is impossible to attain absolute zero in a finite
number of operations.
This is an alternative statement of the law
Black holes MUST radiate!
The third law says they must have a
temperature, T>0 and Stefan’s Law gives the
amount radiated at tempereature, T.
Cooling can be achieved by:
Thermal contact with something cooler
Evidently no use in achieving absolute zero for
the first time!
Adiabatic expansion, or other energy lowering
process.
∆S = 0 in an adiabatic process.
Hence T = 0 cannot be achieved if all systems
have S = 0 at T = 0.
To consider the argument in more detail look at
an important way of achieving low
temperatures: adiabatic de-magnetisation.
March 04
Lecture 9
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Adiabatic demagnetisation
Paramagnetic salt
Manipulate energy levels in a B-field.
First, cool the salt in a strong field (helium gas
provides thermal conductivity).
Second, isolate the salt (evacuate helium gas).
Finally, cool adiabatically, by reducing the field.
March 04
Lecture 9
4
Approaching T = 0
Entropy changes in demagnetisation
Entropy vs T for the
high and low field
regimes follow from the
temperature dependence
of heat capacity in a twolevel system.
(see lecture 8)
Consider two cases:
Middle figure, S(T=0)
are different in the low
and high field states.
T=0 is possible
Lower figure, S(T=0)
is the same in low and
high filed states.
T=0 is impossible
(in a finite number
of operations).
Hence, the third law
precludes the possibility
of reaching T=0 in a
finite number of steps.
March 04
Lecture 9
5
Back to basics - Entropy
Boltzmann entropy:
Boltzmann distribution applies to both large
and small systems.
The entropy (k ln g) on which it is based is
problematic, as discussed in lecture 4, since in
a perfect quantum system (especially a small
one) g must be a small number.
Willard Gibbs (1839-1903) formulation
Gives entropy directly, in terms of the
occupation probabilities, pi = exp(-βεi).
Start from dS = dQrev/T = dU/T.
d S dU
=
= β dU
k
kT
= d( βU ) − U d β
d ln Z
= d( βU ) +
dβ
dβ
integrating gives
S k = βU + ln Z
U
U == -1/Z
-1/Z dZ/d
dZ/dββ
AA
N.B. the integration constant is zero since βU
and lnZ both vanish as T→0.
March 04
Lecture 9
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Gibbs’ Entropy
We know
U=
piε i
;
i
i
pi = exp(− βε i ) Z
pi = 1 ;
Substitution into A gives
S k=β
=−
S k =−
i
i
i
piε i + ln Z
i
pi
pi (− βε i − ln Z )
Gibbs’
Gibbs’
Entropy
Entropy
pi ln pi
Applicable to all systems, large and small.
The Boltzmann form follows for large systems
with g states all of roughly the same energy,
since pi = 1/g.
g
S = −k
i =1
1
ln g = k ln g
g
Σp
Σpi i ==11
Gibbs’ form also opens up applications in
information theory.
March 04
Lecture 9
7
Microscopic reversibility
Origin of irreversibility
The laws of physics are reversible yet the
second law is founded on the observation of
irreversibility
Maxwell’s demon.
Ehrenfest’s Urn
The origin of irreversibility from reversible rules.
A simple, discrete model (c.f. its continuous
analogue in Q.8, examples sheet I).
2N balls in two containers, at each step one is
chosen randomly and transferred to the other
container.
Probability of transfer A→B: (N+k)/2N.
Probability of transfer B→A: (N-k)/2N.
March 04
Lecture 9
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Ehrenfest’s Urn
Microscopic reversibility
Consider the events immediately preceding,
and following the state where k additional balls
are in A. There are 4 possibilities, α, β, γ, δ.
Reversing time (changing arrow directions)
makes no change. α and δ are unchanged on
time reversal while β and γ are complements.
Simulation shows that when k≠0, the system
tends to its equilibrium state of k=0.
Pα = ( N − k + 1) 2 N
Pβ = ( N − k + 1) 2 N
Pγ = ( N + k + 1) 2 N
Pδ = ( N + k + 1) 2 N
March 04
Lecture 9
(N + k ) 2N
(N − k ) 2N
(N + k ) 2N
(N − k ) 2N
9
Trend towards equilibrium
For N>>k>>1, these equations simplify to
N −k N +k
Pα : Pβ : Pγ : Pδ = 1 :
:
:1
N +k N −k
Thus when k>0, γ is overwhelmingly probable
and k tends to reduce.
k
iteration
March 04
Lecture 9
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Statistical mechanics:
summary of argument
Boltzmann entropy
S = k ln g. (originally an educated guess)
Definition of temperature
d ln g
1
=β =
dU
kT
Boltzmann distribution
exp(− ε i / kT )
pi =
Z
Partition
Z = exp(− ε i / kT )
Partition function
function
i
1 dZ
d ln Z
U =−
=−
Z dβ
dβ
F = U − TS = − kT ln(Z )
Gibbs’ entropy
S = −k
pi ln pi
i
March 04
Lecture 9
Classical
Classical variables
variables
in
in terms
terms of
of the
the
Partition
Partition function,
function, ZZ
11
Overview
Zeroth Law
Concept of Temperature.
First Law
conservation of energy.
dU = d Q + dW
Second Law
Concept of Entropy.
dS =
d Qrev
T
; ∆S ≥ 0
Third Law
Impossibility of achieving T=0.
As quoted in “The American Scientist” (1964)
1st Law: You can’t win, you can only break even.
2nd Law: You can only break even at absolute
zero.
3rd Law: You cannot reach absolute zero.
You can neither win nor break even.
March 04
Lecture 9
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