At the heart of Statistical Mechanics is the partition function (see Derivation of the partition function):
Q = ∑ exp(-E_{i}/kT) (1)
Where k is Boltzmann's constant, T is the temperature and E_{i} reflects each possible energetic state of the system. The partition function provides a measure of the total number of energetic states available to the system at a given temperature. Similarly,
exp(-E_{i}/kt) (2)
provides a measure of the number of energetic states of a particular energy that are likely to be occupied at a given temperautre.
Dividing Equation 2 by Equation 1 gives the probability of finding the system in a particular energetic state, i:
p_{i} = exp(-E_{i}/kT)/Q (3)
This probability can be used to find the average value, which corresponds to the macroscopic value, of any property, J, that depends on the energetic state of the system by using the formula:
<J> = ∑p_{i}J_{i} = ∑J_{i}exp(-E_{i}/kT)/Q (4)
Where <J> is the average value of property J.
Equation 4 can be applied to the internal energy, U, and pressure, P:
U = ∑E_{i}exp(-E_{i}/kT)/Q (5)
P = ∑P_{i}exp(-E_{i}/kT)/Q (6)
Subsequently, Equations 5 and 6 can be combined with known thermodynamic relationships between U and P to arrive at an expression for P in terms of only temperature, volume and the partition function. Similar relationships in terms of the partition function can be derived for other thermodynamic properties as shown in the following table.
Table 1 | |
Helmholtz free energy: | A = -kT ln Q |
Internal Energy: | U = kT^{2}(dlnQ/dT)_{N,V} |
Pressure: | P = kT(dlnQ/dV)_{N,T} |
Entropy: | S = klnQ + U/T |
Gibbs free energy: | G = -kT ln Q + kTV(dlnQ/dV)_{N,T} |
Enthalpy: | H = U + PV |
Constant Volume Heat Capacity: | C_{V} = (dU/dT)_{N,V} |
Constant Pressure Heat Capacity: | C_{P} = (dH/dT)_{N,P} |
Chemical Potential: | μ_{i} = -kT(dlnQ/dN_{i})_{T,V,N} |
It is often useful to consider the energy of a given molecule to be distributed among a number of modes. For example, translational energy refers to that portion of energy associated with the motion of the center of mass of the molecule. Configurational energy refers to that portion of energy associated with the various attractive and repulsive forces between molecules in a system. The other modes are all considered to be internal to each molecule. They include rotational, vibrational, electronic and nuclear modes. If we assume that each mode is independent the total energy can be expressed as the sum of each of the components:
E = E_{t} + E_{c} + E_{n} + E_{e} + E_{r} + E_{v} (2)
Where the subscripts t, c, n, e, r, and v correspond to translational, configurational, nuclear, electronic, rotational and vibrational modes, respectively. The relationship in Equation 2 can be substituted into Equation 1 to give:
Q = ∑ exp(-(E_{ti}+E_{ti}+E_{ci}+E_{ni}+E_{ei}+E_{ri}+E_{vi}) /kT)
= ∑ exp(-E_{ti}/kT)*exp(-E_{ci}/KT)*exp(-E_{ni}/kT)*exp(-E_{ei}/kT)*exp(-E_{ri}/kT)*exp(-E_{vi}/kT)
= Q_{t}Q_{c}Q_{n}Q_{e}Q_{r}Q_{v}
Thus a partition function can be defined for each mode. Simple expressions have been derived relating each of the various modes to various measurable molecular properties, such as the characteristic rotational or vibrational frequencies.
Expressions for the various molecular partition functions are shown in Table 2:
Table 2 | |
Nuclear | Q_{n} = 1 (T<10^{8} K) |
Electronic | Q_{e} = W_{0}exp(kT D_{e} + W_{1}exp(-θ_{e1}/T) + ... |
vibrational | Q_{v} = ∏exp(-θ_{vj}/2T)/(1-exp(-θ_{vj}/T)) |
rotational (linear) | Q_{r} = T/σθ_{r} |
rotational (non-linear) | Q_{r} = √π/σ*(T^{3}/(θ_{A}θ_{B}θ_{C})^{1/2} |
Translational | Q_{t} = (2πmkT)^{3/2}/h^{3} |
Configurational (ideal gas) | Q_{c} = V |
The equations in Table 2 can be combined with those in Table 1 to determine the contribution of a particular energy mode to a thermodynamic property. For example the "rotational pressure" could be determined in this manner. The total pressure could be found by summing the pressure contributions from all of the individual modes, ie:
P = P_{t} + P_{c} + P_{n} + P_{e} + P_{r} + P_{v}