The Helmholtz free energy

1
The Helmholtz free energy
plays an important role for systems where T, U
and V are fixed
- F is minimum in equilibrium, when U,V and T are
fixed!
by using
immediate relations
Maxwell relation
Calculation of the Helmholtz free energy (F) from
the partition function (Z) (proof by showing
that F-?ln(Z) satisfy the FU-?? U?(?F/??)V
relation, or by the use of the Renyi entropy
formula --gt see extra problem)
Immediate relations
2
. One atom in a box
Ideal gas A first look
We first calculate the partition function (Z1) of
one atom of mass M free to move in a cubical box
of volume VL3
Wave functions of possible states
Energies of possible states
(nx , ny , nz positive integers)
where
after performing the integrals
We can introduce the so called quantum
concentration, which is rouhgly the concentration
of one atom in a cube of side equal to the
thermal average de Broglie wavelength. (
)
Whenever nltltnQ --gt classical regime. An ideal gas
is defined as a gas of nonintearcting atoms in
the classical regime!
Average internal energy of one particle
Thermal average occupancy of one state (for the
classical regime, this must be ltlt1!)
3
N atoms in a box
If we have N non-interacting, independent and
distinguishable particles in a box
if the particles are identical and
indistinguishable
For an ideal gas composed of N molecules we have
total energy of an ideal gas
thermal equation of state of ideal gases
entropy of an ideal gas
Sackur-Tetrode equation
4
Equipartition of energy for ideal gases
- in an ideal gas for all possible degrees of
freedom the average thermal energy is ?/2
(kT/2) - generalization whenever the hamiltonian
of the system is homogeneous of degree 2 in a
canonical momentum component, the classical limit
of the thermal average kinetic energy associated
with that momentum will be ?/2 degrees of
freedom for one molecule - molecules composed by
one atom 3 --gt motion in the three direction of
the space - molecules composed by two atom 7 --gt
motion of the molecule in the three directions of
space rotations around the two axis
perpendicular to the line connecting the two
atoms vibrations (kinetic and potential energy
for this) degrees of freedom for the system N x
degrees of freedom for one molecule
Heat capacity at constant volume of one molecule
of H2 in the gas phase
Problems
1. Problem nr. 1 (Free energy of a two state
system) on page 81 2. Problem nr. 2 (Magnetic
susceptibility) on page 81 3. Problem nr. 3 (Free
energy of a harmonic oscillator) on page 82 4.
Problem nr. 4 (Energy fluctuations) on page 83
Extra problem
Consider a closed thermodynamic system (N
constant) with fixed temperature (T) and volume
(V). By using the Renyi entropy formula, the
expression for the probability of one state, and
the fact that FU-TS, prove, that F-kBTln(Z)