School of Physics and Astronomy
Junior Honours Thermodynamics
GJA 2016-2017
Lecture TOPIC 18 A miscellany of applications
Superfluid Helium Fountain effect
Below 2.17K, superfluid helium II is a liquid which
consists of two components: a normal fluid, and a superfluid of atoms in the same quantum state with zero
viscosity and zero entropy.
Consider two vessels, (“reservoir”, and “system”) connected by a plug through which only the superfluid component can flow1 . Since the superfluid components are
in contact, He II must have the same µ in both reservoir,
and system. i.e. ∆µ = 0 = v∆P − s∆T , where ∆ is the
difference between reservoir, and system.
If the system is then heated, to keep the chemical potentials equal there will be a pressure difference between
system and reservoir ∆P/∆T = s/v > 0. This high pressure in the system is sufficient to generate
a fountain of helium II, forever spraying into the air and falling back to the reservoir.
The superfluid “quantum ground state” extends through a large region of space, including
liquid in each vessel, and the fountain itself. The second law isn’t violated because heat is being
supplied - it’s just a fancy heat engine.
The “ammonia fountain” looks similar, but is a more prosaic example of reduction in chemical potential overcoming gravitational potential. Ammonia in water has a very low chemical
potential, so water can be sucked up a tube against gravity into an ammonia atmosphere, with
inertia producing the fountain. However, once equilibrium is reached, the fountain stops.
Osmosis
Osmosis exploits the fact that particles flow down gradients of chemical potential, not concentration. High
concentration usually means high chemical potential. But not always.
Some biological systems have a
special type of semipermeable membrane which allows heat transfer, sustains a pressure differences, but only
allows some particle flow - say type
w (water) but not type s (sugar).
Consider a system comprising two regions A and B separated by such a semipermeable
membrane, and allowed to equilibrate. Since water can flow freely, equilibrium requires µAw =
µBw , and partial pressure PAw = PBw . But the sugar particles are not able to equilibrate, µAs 6= µBs ,
and consequently total pressures in A and B differ.
1 The
zero viscosity of the superfluid component of helium II makes such a plug possible
1
The simplest case treats both particles as ideal gases with sugar particles only in region
A. Now the “osmotic pressure”: the pressure difference between the two regions, is simply
the partial pressure PAs . If the particles interact (as real water and sugar do!) things get more
complicated. If the interaction is attractive.
• The interaction lowers the pressure (like a in the van der Waals gas), Raoult’s Law fails.
• The attraction lowers the chemical potential of the water, causing it to flow from the low
pressure region to the high pressure region until the pressure increase raises µ to balance
the attraction. So one side contains more sugar and more water.
• Biological cell are like this, the osmotic pressure keeping them stiff and rigid. It their
membranes fail, plants wilt. If placed in a hypertonic environment (too much external
solute) cells can have water sucked out of them.
Brinicles and ice cream makers - Adding salt to ice water
Consider mixing hot salt water and ice
initially at its melting point (0oC) together.
What final temperature will one get? To answer this consider the equilibrium situation
of ice in contact with a salt solution (noting
ice contains no salt). The chemical potential
of H2 O in the ice, µI , and in the salt solution, µL , must be equal at equilibrium. We
can Taylor expand µL (p, T, X) about X = 0,
T = 0 where X is the molar fraction of ions,
Na+ +Cl− in the solution (the molar fraction
of both ions are identical since we are only
adding NaCl) with X = 2NNa+ /(2NNa+ + NL ).
Working at constant pressure (atmospheric) we can drop P giving;
∂ µ ∂ µ ∂ µ L
L
L
dT +
dX = −sL dT +
dX
dµL =
∂ T P,X
∂ X P,T
∂ X P,T
Where in the last equality we used s = −(∂ µ/∂ T ) p from TOPIC 17. For the ice
dµI =
∂ µ I
∂T
P
dT = −sI dT
Moving along the co-existence (ice in equilibrium with salt water) line in the {X, T } plane we
must have a Clausius-Clapeyron type relation (dµL )saltwater-ice = (dµI )saltwater-ice , ⇒
dT dX
saltwater-ice
=
∂ µ L
∂X
T
1 T ∂ µL =
sL − sI
L ∂X T
with L the latent heat of melting (of pure ice into pure water evaluated at the point X = 0). If
2NNa+
we assume that the salt solution is ideal (X = NL +2N
):
+
Na
µL (X, T ) = µL (X = 0, T ) + RT ln(1 − X)
⇒
∂ µ L
∂X
T
=
−RT
= −RT
1−X
(assume much more water than salt, X << 1)
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So finally,
dT dX
saltwater-ice, X=0
=−
RT 2
L
Remarkably, the final temperature of the mixture is lower than the initial temperature of
either component. This result is independent of the properties of salt (other than that it forms
an ideal solution).
Given the concentration of salt added we can estimate the drop in temperature. L = 6010 Jmol−1 ,
T = 273.15 K. The maximum solubility of salt in water is X=0.16 at the lowest temperature
achievable. Therefore within the above approximations (eg assuming L is constant) we estimate
an achievable drop in temperature of 16.5 K. The lowest freezing point of salt solution is actually -21.1o C indicating small errors in some of the above approximations. Below -21.1o C one
has pure water and pure hydrated salt with no salt solution.
This cooling effect can be used in ice-cream makers.
Electrons and Fermi Energy
E=0
In the metal, electrons occupy energy states up to the
W1
“Fermi Level” Adding an electron to an isolated metal
W2
means putting in the lowest unoccupied energy state, thus Ef
∆µ
the chemical potential for the electron is µ = EF .
Εf
Different metals have different in work functions, W :
the energy needed to remove an electron from the Fermi
level to “far away” - which gives us a reference potential
µ 0 appliable to both metals.
Metal 1
Metal 2
Thus µ1 = µ 0 −W1 6= µ2 = µ 0 −W2 . When the metals are brought into contact, the system
lowers its Free Energy by flow of electrons from one metal to the other. This equates to a
potential difference (“Voltage”), equal to the difference in work functions divided by electron
charge V = (εW,1 − εW,2 )/e. A “Pile” of such pairs forms a battery.
Hurricanes
A hurricane is a self assembling heat engine. The hot reservoir is the ocean surface (Th ≈
300K), the cold reservoir is the top of the atmosphere, (Tc ≈ 200K) work is done creating
winds. Elements of the cycle can be idealised as a Carnot cycle:
Isothermal expansion - the eye, is at low pressure. Air spirals across the ocean towards it
being heated and absorbing water vapour as it goes.
Adiabatic expansion Hot air/water vapour rises from the eye to even lower pressure at high
altitude, cooling as it goes.
Isothermal compression Water vapour condenses, releasing latent heat. Rain falls.
Adiabatic compression In theory, air drops back to sea level.
The spiralling is caused by Coriolis forces. This stabilises the structure, but is not an essential part of the thermodynamics. When the hurricane comes ashore, it loses contact with its heat
source and the engine stops. In the absence of land, this would take a long time (e.g. Great Red
Spot on Jupiter).
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Breaking the Law - Virial theorem and Self-gravitating clusters
Clausius derived the virial theorem...
“For a self-interacting system the average kinetic
energy is equal to half the average potential energy”
Where PE is negative and KE positive. This is similar to the equipartition theorem - energy
is equally divided between all degrees of freedom. So we can write: KE = - PE/2 = -U
If we add some heat energy to the system, increasing the PE (to a smaller negative value),
then the KE must decrease. If we also equate the KE to a temperature, then the temperature
decreases as we add heat. Negative heat capacity!.
This applies to systems such as self-gravitating star clusters and black holes floating in a
vacuum (P=0), where the “Heating” might be the addition of a fast-moving star. Note that such
systems are inhomogeneous, so our derivation of positive heat capacity from the second law
doesn’t apply. They also expand, so while the intensive entropy may decrease, the total entropy
does not. 2
Breaking the Law - Fluctuation Theorem
The “fluctuation theorem” can be derived from statistical mechanics. It relates the probability
that a system will produce entropy A to the probability that it will spontaneously reduce entropy
by -A,
p(−A)/p(A) = e−At
This was first shown in computer simulation (1993), proven analytically in 1994 and finally
demonstrated experimentally in 2002. The experiment involved moving a tiny bead in an optical
trap slowly through a solution of other beads and measuring the force needed to move it: this
force fluctuates depending on whether more of the other beads hit it from behind or in front.
Velocity
Movement of the bead in the trap is regarded as work, movement of other beads as heat.
On rare, and unpredictable, occasions work is done on the trap bead from the environment,
violating the Kelvin statement. The resolution is that the Second Law is really statistical, and
that on average work has to be done to move the trap-bead.
2 see
paper by Mr and Mrs Lynden-Bell MNRAS, 181, 405 (1977)
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