THE UNIVERSITY OF BRITISH COLUMBIA

PHYSICS 455

Lecture # 8 :

Wed. 22 Jan. 1997

Helmholtz Free Energy

LOGISTICS: Brian Turrell will take the Friday lecture this week, as I will be in Toronto at a CIAR meeting.

HANDOUTS: Assignment 4

READING: Finish up Ch. 3.

I. F = - log as an ``ENERGY OFFSET'':

We have shown that the probability of finding a small system in state | > while in thermal equilibrium with a heat reservoir at temperature is P = exp(-/)/Z = exp(-/) / exp(-F/) = exp(-[ - F]/). That is, if we measure energies relative to F (as a ``zero'' of energy) then P requires no normalization for the Boltzmann factor.
This might look a little dangerous: if < F, we could get P > 1! This can never happen, though, since Z is dominated by the smallest _min: Z = exp(-F/) > exp(-_min/) so F < _min and therefore no can ever be smaller than F.

II. THERMODYNAMIC DEFINITION: F = U -

Pressure: to see the motivation for the thermodynamic definition of F, we first need the formal mathematical statement of the First Law of Thermodynamics, and for that we need a fundamental understanding of pressure. So...
A long skinny box (cross sectional area A and length x, so volume V = A x) contains a particle in a certain quantum state | > for which there are n_x half-wavelengths in the length x and therefore a momentum in the x direction given by p_x = h/. This produces a pressure p on the ends of the box. If one end moves a distance dx under the force pA the work done on the particle is dW = - p A dx = - p dV. In this process the wavelength gets longer but the particle stays in the same state (though the state itself changes). Thus there is no change of entropy in this process. All that changes is the volume and the energy. Mathematically, dU = (U/V) dV for such an adiabatic (isentropic) process.
Comparing this with the work done on the particle (or a gas of independent, non-interacting particles obeying the same rules) implies that p = - (U/V) which is (one of) the formal definition(s) of p: the volume rate of change of energy at constant entropy.
For the microcanonical ensemble, N is fixed and so the entropy is a function only of U and V. Thus d = (/U)_V dU + (/V)_U dV.
If we specify constant entropy processes (d = 0) and recall the original definition of temperature as well as the definition of pressure derived above, then we find p = (/V)_U, an alternate definition of pressure.
We now relax the requirement of constant and out equation reads d = (1/) dU + (p/) dV. This can be rearranged to give the differential form of the First Law of Thermodynamics:
dU = d - p dV
where dU is the change in the total energy of the system, d is the heat energy added to the system and - p dV is the mechanical work done on the system. You may recall having seen this before.
Now we are ready to understand the implications of the thermodynamic definition of the Helmholtz free energy: F = U - :
Taking the exact differential yields dF = dU - d - d = d - p dV - d - d or dF = - d - p dV. That is, the free energy is constant in any reversible process at constant temperature and volume.

III. USES of the FREE ENERGY:

One has just been described. An even stronger statement can be obtained from the same sort of argument used in deriving the Boltzmann probability distribution: if we treat the entire energy U_s of a small system as an infinitesimal reduction of the energy U_r or a large reservoir relative to the combined energy U of both, the net entropy = _r(U_r) + _s(U_s) = _r(U) - (1/)U_s + _s(U_s) = _r(U) - (1/)F_s. The first term on the right is a constant, and in equilibrium must be a maximum, so
F is always minimum in thermal equilibrium.

dF = dU - d - d = d - p dV - d - d or
dF = - d - p dV

For a process at constant , the above relation gives p = - (F/V), an alternate definition of pressure.
For a process at constant V, we get = - (F/)_V, an alternate definition of entropy.
Note that the ``formula'' for the free energy, F = - log , can be derived from the above definition of the entropy and the thermodynamic definition of F using the expression for the energy U in terms of a temperature derivative of log Z. Check it for yourself.
Since the order of partial derivatives can be interchanged, taking the appropriate partial derivatives of the two preceding relationships yields the first of the Maxwell relations:
(/V) = (p/)_V

Expanding the above expression for p gives the formula for pressure at a fixed temperature:
p = -(U/V) + (/V)
where the first term can be thought of as the ``energy pressure'' and the second term as the ``entropy pressure.''

IV. EXAMPLES:
Any time we have a handy formula for the partition function Z, we can easily find F and, from that, as a function of temperature.

F and in the Two-State System:
We have Z = 1 + e^-/ giving F = - log(1 + e^-/) From this and the definition of the entropy as the temperature derivative of F at constant volume, we obtain (after a little calculus)
= log(1 + e^-/) + (/) / (e^/) + 1).
This entropy vanishes as 0, as expected, and levels off at log 2 (the maximum possible entropy for a 2-state system) at high temperature, again as expected. Note that such limited systems don't hold much heat energy!
F and for the Harmonic Oscillator:
We have Z = 1/(1 - e^-/) giving F = log(1 - e^-/). Thus
= - log(1 - e^-/) + (/) / (e^/) - 1).
This entropy also vanishes as 0, as expected, but grows without limit as the temperature increases: the higher the temperature, the more (higher energy) modes can be involved, adding to the entropy.

V. Preface to IDEAL GASES:
We have defined pressure in the abstract:
p = -(U / V) = (/ V)_U = -(F / V)
What we need now to do any practical calculations is a specific result - a formula for p in terms of N, V and , for instance. Such a relationship is called an equation of state. On Friday Brian Turrell will introduce the most important one of all, that for an ideal gas.
Last modified: Tue Feb 4 23:14:26 PST