Theoretical Chemistry Tutorial: Partition Function.

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The partition function

is defined as Q = _i g_i ^{-_i / kT}.
 

Summation has to include all allowed energy levels; g_i is the degeneracy factor, which describes how often the i-th energy level _i occurs, k is the Boltzmann constant and T is the temperature (  numeric value of the Boltzmann constant).
 

If energy differences between neighbouring energy levels are sufficiently small, the sum can be replaced by an integral. A good estimate for "sufficiently small energy differences" is the characteristical temperature , which is the temperature at which Q=1. For temperatures T>> one can integrate, for temperatures around or below one has to explicitely sum up the terms.
 

Examples of characteristical temperatures
type of movement	compound
translation		<<1 K
rotation	H2	87.5 K
	HCl	15.4 K
	N2	2.89 K
	I2	0.05 K
vibration	H2	5986 K
	HCl	4152 K
	N2	3353 K
	I2	306,8 K

 

The partition function of a single particle

is composed from several contributions, according to the contributions to the energy: translation, rotation, vibration, and electronic excitation. The energy contributions add up. In the definition of the partition function, however, energies are part of the exponent; therefore, the individual partition functions have to be multiplied:

E = E_t + E_r + E_v + E_el
 
Q = Q_t · Q_r · Q_v · Q_el .

 

The translational partition function

is the product of three basically equal terms, each of which corresponds to one translational degree of freedom.

Q_t = q_t³
 
q_t = _i=0, ^{-_i /k T}
 

The energy levels are approximated by the quantum mechanical result for a particle in a box _n = h² n² / ( 8 m a² ) ; because of the low characterical temperature, the sum can be replaced by an integral in most cases:

q_t = ^{- h² n² / ( 8 m a² k T )} d n = ( 2 m k T / h² ) · a
 
Q_t = ( 2 m k T / h² ) · V .
 

The translational partition function does not only depend on quantities of the partical under scrutiny, it also depends on the volume. In contrast to molecular properties, the volume is an external quantity; the translational partition function therefore is sometimes call outer partition function.
 

The rotational partition function of a linear molecule

can be deduced as follows:

Q_r = _l=0, g_l ^{-_l /k T} .
 

The energy levels of the rigid rotator are _l = ² l·(l+1) / ( 2 µ r² ), the degeneracy factor is g_l = 2l+1. Replacing the sum by an integral yields

Q_r = (2l+1) ^{- ² l·(l+1) / ( 2 µ r² k T )} dl .
 
 

The coordinate transformation z = l·(l+1) with dz = (2l+1) dl  simplifies this integral to

Q_r = ^{- ² z / ( 2 µ r² k T )} d z = 2 µ r² k T / ² .
 
 

For heteronuclear molecules (like HCl), the selection rule is l = ±1, whereas for homonuclear molecules (like H₂, O₂), only even or odd values of l may occur, depending in the nuclear spin. The above partition function therefore has to be divided by a factor of 2 for homonuclear molecules, since only each second energy level has to be considered. In a general formula, a symmetry number is introduced, which has a value of 1 for heteronuclear molecules and a value of 2 for homonuclear molecules:

Q_r = 8 ² µ r² k T / ( h² ) .

 

The rotational partition function of non-linear molecules

can be obtained by a generalization of the above formula. The most important point in this generalization is that linear molecules have only two degrees of rotational freedom with identical moments of inertia:

I_x = I_y = (µ r²).
 

In contrast, non-linear molecules have three degrees of rotation with different moments of inertia I_x, I_y, and I_z. (Some of these may be identical due to symmetry restrictions.) The partition function is obtained as:

Q_r = ( 8 ² k T ) (I_x I_y I_z ) / ( h³ ) .
 

The symmetry number is now defined as the number of different orientations, which can be obtained by rotations of the respective molecule: 2 for H₂O, 3 for NH₃, 12 for C₆H₆, and so on. This definition also includes linear molecules.
 

The vibrational partition function

can be obtained using the harmonic approximation. According to this, the allowed energy levels are _i = (i+) h and the partition function for one vibration mode is

Q = _i=0, ^{-(i+) h / kT}.

 
Using the short hand notation x = h / kT, one obtaines

Q = ^-x + ^-x + ... ;
 
Q · ^x = ^x + ^-x + ^-x + ... ;
 
Q · ^x = ^x + Q ;
 
Q · (^x - 1) = ^x ;
 
Q = ^x / (^x - 1) .

 
(Expansion of nominator and denominator in Taylor series results in a very simple approximate result for x<<1 - i.e. high temperatures: Q 1 / x .)

Statical Thermodynamics.
Introduction.
 
 Dr. Michael Ramek.