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Statistical Thermodynamics Crib Sheet

The Molecular Partition Function

The molecular partition function, q, can be thought of as a measure of theaverage number of thermally accessible states available to a system at a giventemperature, at equilibrium. It can be defined as a sum over states, j withenergies j :

q =j

ej/kBT

or as a sum over energy levels, i, where each level has degeneracy gi (i.e. thereare gi states with energy i):

q =i

giei/kBT

The energies in these expressions are measured relative to the ground-state, sothat 0 = 0.

It is usually a good approximation to factorise the partition function in termsof the various modes of molecular motion:

q = qtransqrotqvibqelec

The Translational Molecular Partition Function

For translational motion of a molecule with total mass m inside a one-dimensionalbox of length Lx, the energy levels are:

Enx =h2n2x

8mL2x, nx = 1, 2, 3, . . .

Factorising the total translational partition functions: qtrans = qxqyqz, andapproximating the summation for each dimension as an integral gives:

qtrans =V

3where =

h2mkBT

for a box of volume V = LxLyLz. is called the thermal wavelength.

The Rotational Molecular Partition Function

For a diatomic molecule treated as a rigid rotor, the energy levels are:

EJ = hcBJ(J+ 1) with gJ = 2J+ 1, where B =h

82Iccm1

and J = 0, 1, 2, . . . is the rotational quantum number. In the high-temperaturelimit, where the summation over J can be replaced by an integration:

qrot =T

rotwhere rot =

hcB

kB


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is the symmetry number, equal to 1 for heteronuclear diatomics, and 2 forhomonuclear diatomics.At low temperatures (and for H2 even at room temperature), where the ro-tational temperature rot is comparable to or greater than T, the summationmust be performed numerically, (taking into account any nuclear spin effectsexplicitly):

qrot =J

(2J+ 1)exp hcBJ(J + 1)

kBT

The Vibrational Molecular Partition Function

For each normal mode of a molecule (in the simple harmonic approximation),there are energy levels given by:

Ev = hc(v +1

2), where =

1

2c

k

cm1

The vibrational partition function can be factorised into each normal mode:qvib = qvib,1qvib,2qvib,3 . . . where each qvib,k is an infinite geometric progres-

sion: relative to the v = 0 level,

qvib,k = 1 + exp hck

kBT

+ exp

2hck

kBT

+ . . .

This can be evaluated analytically using the formula S = a/(1 r). For adiatomic molecule (and for each normal mode of a polyatomic molecule),

qvib =1

1 exp(vib/T)where vib =

hc

kB

At room temperature, for most diatomic molecules vib T, and qvib 1.

The Electronic Molecular Partition Function

The excited electronic states of most molecules are so much higher in energy thanthe ground electronic state (relative to kBT) that the approximation qelec = g0is usually a good one. There are some notable exceptions, though: for example,the NO molecule has a doubly-degenerate ground state (g0 = 2) and a low-lyingdoubly-degenerate excited state (1 = 121cm

1, g1 = 2). qelec can be treatedas a two level system:

qelec = g0 + g1e1/kBT

The Canonical Partition Function

A system ofN particles within a fixed volume V can be imagined to be replicatedN times (where N can be very large, and is unrelated to N), with each (closed)system in thermal contact with the others, so that there is a fixed temperature,T. These N systems constitute the canonical ensemble. The number of systemswithin the ensemble with energy Ei is ni, distributed according to:

ni

N=

eEi/kBT

Qwhere Q =

i

eEi/kBT


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Q is the canonical partition function; for large N this distribution is overwhelm-ingy the most likely and defines the properties of the system. Q is entirely gen-eral, and does not rely on the particles being independent - they may interactwith each other.When the particles are taken to be non-interacting, there is a simple relationshipbetween Q and q:

Q = qN for distinguishable particlesQ = qN/N! for indistinguishable particles

Identical but distinguishable particles (such as those in a crystal lattice, whichcan be referred to by their (x, y, z) co-ordinates) have Q = q1q2q3 . . . qN = q

N.For indistinguishable particles (such as those of an ideal gas), there is no way oftelling between different particles in the same state, so this product over-countsthe number of states by a factor of N!.The Stirling approximation is useful in manipulating equations involving N!:

ln N! Nln NN for large N

Some Thermodynamic Formulae

The internal energy of a system is

U = U(0) + kBT2

ln QT

V

where U(0) is its value at T = 0 K (this information is not contained in thepartition function). For non-interacting particles, in terms of q:

U = U(0) + N kBT2

ln qT

V

The entropy of the system (relative to S(0), the residual entropy at 0 K) is givenby:

S =U U(0)

T+ kB ln Q

An important use of this formula is in calculating the entropy of n moles of anideal gas (for which only translational motion contributes) - the result is theSackur-Tetrode equation:

S = nR ln e5/2V

nNA3

The Helmholtz Free energy has a simple relationship to Q, known as the Massieubridge):

A = A(0) kBT ln QFrom this equation, expressions for many other thermodynamic functions canbe obtained. For example,

S = A

T

V

, p = A

V

T

, =A

n

V,T


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From these, U = A + T S gives the internal energy, G = A +pV the Gibbs freeenergy, etc...A full set of relations for systems of constant composition, undergoing reversiblechange with no non-expansion work is given below:

dU = dq + dw

dU = T dS

pdV T = USV

, p =

U

VS

H = U + pV dH = T dS+ V dp T =H

S

p

, V =H

p

S

G = H T S dG = V dp SdT V =G

p

T

, S = G

T

p

A = U T S dA = pdV SdT p = A

V

T

, S = A

T

V

The Maxwell Equations

A differential df of a function f(x, y):

df =f

x

y

dx +f

y

x

dy

is exact if 2fyx

= 2f

xy

dU, dH, dG, and dA above are exact differentials, because U, H, G, and A arestate functions. Therefore,

2UV S

= 2U

SV

T

V

S

= p

S

V

The other three Maxwell equations are derived in a similar way:

Tp

S

=V

S

pV

T

p

= S

p

T p

T

V

= S

V

T

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