Pressure
Ingredients:
Weight
Piston
System
A thermodynamic model of a system, Ω(U,V),
which describes the number of microstates of the
system when isolated at particular values of U and
V, (i.e. in a particular macrostate).
A reservoir of energy is also a system in thermal
contact with our system of interest, which is much
‘larger’ – dU/dTR >> dU/dT
The weight and piston are a system that is in
mechanical contact with our system – it describes
the work done (f *dx) by our system in exchanging
space with a reservoir of volume at constant
pressure.
Reservoir of Energy at TR
An isolated composite system
ΩC(UT)
Conservation of Energy: UT=U + pV + UR
UR = UT -U - pV
Fundamental Postulate
When isolated at a particular UT for a sufficiently
long time, the most probable values of U and V are
those that maximize the function ΩC(U,V).
Again, for convenience we can define entropy S= log Ω
When total U is fixed, the change in the entropy of the composite is given by:
(The weight contributes nothing to the change in entropy of the composite)
Recalling conservation of energy:
We now have the entropy of the composite as a function of two
independent internal variables, U and V:
We can therefore apply the fundamental postulate: after a long
time the composite will be in the macrostate with the largest
number of microstates i.e. highest entropy.
We can extract two equilibrium conditions required to satisfy this statement
Recognizing
We have
Which defines
We now know how to count the states of a system capable of independent
variations of energy and volume:
1. Measure T, P hold V constant and add δU (slowly)
2. Measure T, P hold U constant and add δV (slowly)
What happens when we begin to consider not just U,V, but also N?
If we fix U,N and allow V to vary, at equilibrium we recover the definition of pressure
Weight
Piston
“Entropic elasticity”
Just a consequence of N things exploring a volume V – but,
the system must be dilute for each molecule to be independent!
‘Osmotic’ pressure
Consider a semipermeable rigid membrane with volume V immersed in
fluid. Again, we ask about the number of states contributed by N
molecules of solute exploring a volume:
Here the tendency for water to permeate the membrane is
balanced by the pressure exerted by the rigid walls of the
membrane on the fluid
Osmosis is just greek for ‘a pushing’, so osmotic pressure is the ‘pushing pressure.’ I
suspect we hid our ignorance about the phenomena by giving it a greek name!
Perhaps ‘entropic’ pressure would be better.
Atmosphere as a reservoir of volume at constant p
Two conservation statements:
Atmosphere
at pR
Ω(VR)
Conservation of Energy: UT=U + UR
Conservation of Volume: VT=V+VR
System
Reservoir of
Energy at TR
Equilibrium condition:
We now ask…. How do we study what happens when two systems
can exchange matter in addition to space and energy?
Vapor
Wine
We consider that the number of states of each
system is now a function of three variables:
Ω(U,V,N)
Liquid
Cheese
An isolated composite system
composed of two systems: wine
(in two phases, vapor and liquid)
and cheese. Consider the cheese
wrapped in an impermeable
membrane such that only water
can move in or out.
Again, it is convenient to define the
entropy of a system as
S = log Ω(U,V,N)
We can label the two systems A` and A`` for the
wine and cheese respectively.
For a given partition of energy, volume and water
between the wine and cheese, the number of microstates
in such a macrostate of the composite is the product of
the number of microstates in the corresponding
macrostates of the wine and of the cheese….
Vapor
Therefore the entropy of the composite is just the sum:
Wine
Liquid
Cheese
The variation in the entropy of the composite is given by the sum
When the composite is isolated, the quantities of U,V, N in the composite are fixed,
and the variations in the cheese are equal and opposite those of the wine:
We have identified three independent variables for the composite
We can now apply the fundamental postulate: Any change in the internal variables of
the isolated system must increase the entropy of the isolated system. That is,
for any combination of
we require
When
the entropy of the composite is maximized, and the wine and
cheese are in their equilibrium configurations and their partial
derivatives are equal. We already know some of these:
What about the last term? What this says is that when the gain in entropy of the wine for
adding water is greater than the loss in entropy of the cheese for losing water, water will
flow spontaneously from the cheese to the wine. We can give this tendency a label:
Where we give μ the name ‘chemical potential.’
Referring back to our inequality we see what we really mean – that when
pressure and temperature have equilibrated, if water still flows from the cheese
to the wine we say the chemical potential of the water in the cheese is higher!
We can now approach the question of how to determine S(U,V,N) experimentally
(remember, we need this function because for all our other physics we take it as given!)
For small changes in the independent variables, the entropy of the composite changes as:
We have discussed how to measure temperature and pressure, and small
flows of energy and changes in volume; counting N does not seem to
problematic, but how do we measure μ? We will here outline an approach
based a nice property of pure substances in a single phase.
The nice property is just this: for a system composed of a pure substance in a single phase,
the energy, entropy and volume of the system are all proportional to the number of
molecules of the substance. We define:
We can then prescribe a re-scaled thermodynamic function s(u,v), the entropy per
molecule. For a small variation in u and v the entropy per molecule varies as
How can we interpret these partial derivatives? Holding N constant it is easy to see
And we therefore recover our familiar definitions of the partial derivatives
We can then obtain the function s(u,v) according to
Furthermore, if we know N, we now have our thermodynamic function in hand
We may now derive an expression for the chemical potential based on
experimental variables – things we can actually measure! Recall
Changing variables and the chain rule unpacks a lot of terms out of that one partial derivative:
But this simplifies to just
Evidently, for a pure substance in a single phase,
chemical potential is the Gibbs free energy per molecule
recalling
yields
Evidently, for a pure substance of a single phase, the chemical potential is a
function of the temperature of and the pressure acting on the system:
That’s nice, but the world is dirty, and we need to measure it!
In the vapor phase, if we go back to counting states, immediately we can say a few things:
(Ideal part)
Ideal gas law
For a single species of N, we can define the volume per molecule as
and the gas law becomes
Weight
Piston
Recall
at constant temperature we find
0
We can define po as the partial pressure in equilibrium with a
liquid phase, and define μ(T, po) as zero.
Weight
Piston
When the vapor phase is in equilibrium with a vapor
phase, the pressure in vapor phase is by definition the
saturated vapor pressure.
We now show that this pressure is a function of the
temperature. For a small change in temperature and pressure
we find:
We may then regard the pressure as the function of the temperature
(but we could equally pose the inverse relationship!)
Clapeyron equation
We can obtain the form of this functional relationship with by noting that:
And define the enthalpy of vaporization ~ the energy in the molecular bonds of the liquid
We are almost ready to do an experiment!
In general however, we need to talk about water as one component of a gas composed of k
species. If the probability of any one molecule being in a location is again independent of the
location of any other molecule, the number of states of such a system is given by :
The point is that in a multi-component system, the pressure that the
chemical potential of a species depends on is the partial pressure of that
species, and not the total system pressure.
To recap, if we can find pw, we can measure T and calculate pw* for air in equilibium
with a piece of cheese, we can determine the chemical potential of the water in the
cheese. How do we find pw ?
The ‘chilled mirror’ chemo-meter
We place a mirror backed with peltier
plate connected to a current source, and
monitor the resulting temperature with a
thermocouple.
Ta
Tm
T
We then lower the temperature of the
mirror until it fogs. Saturated vapor
pressure is a known function of
temperature, so once the temperature at
which condensation occurs is known, we
know the vapor pressure and, with Ta,
can calculate the chemical potential in
the cheese.
What about the liquid phase?
For the liquid phase,
takes a different form.
We cannot count the states contributed per molecule, as the location of each
molecule is no longer independent. However, the density variations of water over
the changes in pressure typically encountered are very small, and so we can
approximate v as independent of the pressure (although it is still a function of
temperature). At constant temperature, for pure water
Equilibrating an undersaturated vapor
with a liquid results in huge potential
tensions
Pl
Mpa
p/psat
An explicit view of incompressibility
Res. of V
at pR
System = water
Res. of U
at TR
Res.
of N
at μR
Notice, what matters is not that water
transferred from the reservoir to the
system is incompressible, but that
the system is incompressible!
Using the reservoir chemical potential and pressure as our reference gives
By raising the pressure in the fluid we can balance a negative chemical
potential and bring it to equilibrium with pure water in the reference state
μ/v
p