Statistical Thermodynamics and Rate Theories/Derivation of thermodynamic functions and variables from partition functions

=Variables of the Canonical Ensemble= The Canonical Ensemble partition function depends on variables including the composition (N), volume (V) and temperature (T) of a given system, where the above partition function equation is still valid with $$Q_{N,V,T} = Q(N,V,T)$$.

The partition function of the canonical ensemble is defined as the sum over all states of a particular system involving each states respective energies, and represented by the following equation,

$$Q = \sum_{j}\exp \left( \frac{-E_j }{k_B T} \right)$$

In this equation, $$k_B$$ represents the Boltzmann Constant with a value of $$k_B = 1.3806503 \times 10^{-23} JK^{-1} $$, T represents the temperature in kelvin and $$E_j$$ is the energy at state j.

Internal Energy
The partition function can also be related to all state functions from classical thermodynamics, such as U, A, G and S. The ensemble average of the internal energy in a given system is the thermodynamic equivalent to internal energy, as stated by the Gibbs postulate, and defined by,

$$U = \langle E \rangle =\frac{\sum_{j} E_j \exp \left( \frac{-E_j }{k_B T} \right)}{Q} $$

Where the variable Q here represents the partition function term. For a canonical ensemble, the internal energy can be derived from the above equation by considering the derivative of Q with respect to T,


 * $$ \begin{alignat}{7}\frac{ \partial{}}{\partial{T}} Q = \frac{ \partial{}}{\partial{T}} \sum_{j} \exp \left( \frac{-E_j }{k_B T} \right) & \\

= \sum_{j} \frac{ \partial{}}{\partial{T}} \exp \left( \frac{-E_j }{k_B T} \right) & \\ = \sum_{j} \exp \left( \frac{-E_j }{k_B T} \right) \frac{ \partial{}}{\partial{T}} \left( \frac{-E_j }{k_B T} \right) & \\ = \frac{E_j}{k_B} \frac{1}{T^2} \exp \left( \frac{-E_j }{k_B T} \right) & \\ \end{alignat} $$


 * $$ \begin{alignat}{7}\frac{ \partial{}}{\partial{T}} Q = \sum_{j} \frac{E_j}{k_B} \frac{1}{T^2} \exp \left( \frac{-E_j }{k_B T} \right)

= \frac{ 1}{k_B T^2} \sum_{j} E_j \exp \left( \frac{-E_j}{k_B T} \right) & \\ = \frac{ Q}{Q} \frac{ 1}{k_B T^2} \sum_{j} E_j \exp \left( \frac{-E_j }{k_B T} \right) & \\ = Q \frac{1}{k_B T^2} \frac{\sum_{j} E_j \exp \left( \frac{-E_j}{k_B T} \right)}{Q} & \\ = \frac{Q}{k_B T^2} \frac{\sum_{j} E_j \exp \left( \frac{-E_j}{k_B T} \right)}{Q} = \frac{Q}{k_B T^2} \langle E \rangle \end{alignat} $$

Rearranging this resulting equation for $$\langle E \rangle$$ yields,

$$ \langle E \rangle = k_B T^2 \frac{1}{Q} \left( \frac{ \partial{Q}}{\partial{T}} \right)_{N,V} $$

and from the Gibbs Postulate,

$$U = k_B T^2 \left( \frac{\partial\textrm{ln} Q }{\partial T} \right)_{N,V}$$

Helmholtz Energy
The Helmholtz Energy, or Helmholtz Free Energy, of a Canonical Ensemble represents the amount of work and energy obtainable by a certain closed system under constant concentration, volume and temperature. The expression for A utilizes both the internal energy and entropy of the system, and is derived by,

$$ A = U - TS $$

Where A is the Helmholtz Energy term, U represents the internal energy and S represents the entropy of the system being studied. Substituting the per-determined values for the canonical ensemble,

$$ A = \langle E \rangle - T \left( \frac{\langle E \rangle}{T} + k_B \ln{Q} \right) $$

$$ A = \langle E \rangle - \langle E \rangle - k_B T \ln{Q} $$

$$A=-k_B T \ln \left( Q \right)$$

Q represents the partition function for this particular system, and is proportional to the absolute free energy. Thus, an increase in $$\ln (Q)$$ will occur with an increase in the number of total accessible states, and a more negative free energy.

Entropy
The Gibbs definition of Entropy is described by,

$$ S = -k_B \sum_{j} P_j \ln{P_j} $$

where $$P_j$$ represents the Probability of being in state i and described by the weight of the state divided by the sum of all possible weights, such as,

$$ P_i = \frac{\exp \left( \frac{-E_i}{k_B T} \right)}{ \sum_{j} \exp \left( \frac{-E_j}{k_B T} \right)} $$

and thus large probabilities of multiple states correlate with large values of S. By using the above Gibbs definition, the entropy of a Canonical Ensemble (NVT) can be derived, in terms of Q, to yield the following expression,

$$ S = \frac{ \langle E \rangle}{T} + k_B \ln{Q} $$ or $$ S = k_B T \left( \frac{ \partial{\ln{Q}}}{\partial{T}} \right)_{N,V} + k_B \ln{Q} $$

The value of Q in this equation can also be represented as $$ Q = \frac{q^n}{N!} $$, where q is equivalent to the partition function of the molecules, with N representing the number of molecules, in the system.

The partition function is also represented by the denominator of the probability term for a certain state, given by the following,

$$ P_i = \frac{\exp \left( \frac{ -E_j}{k_B T}\right)}{Q} $$

Chemical potential
The Chemical Potential, μ, represents the change in the Helmholtz free energy when an additional particle is added to the canonical ensemble system. A heterogeneous system represents a system containing more than one type of gas and the expression for the chemical potential here is μi, where i represents each differing species. This term is derived from Helmholtz free energy, A, with respect to number of molecules for a certain species i. Thus,

$$ \mu_i = \left( \frac{ \partial{A}}{ \partial{N_i}} \right)_{T,V,N} $$

And utilizing the above Helmholtz expression, the following can be derived,

$$  = \left( \frac{ \partial{} -k_B T \ln{Q}}{\partial{N_i}} \right)_{T,V,N} $$

$$  = -k_B T \left( \frac{ \partial{ \ln{ \frac{q_i \left(V, T \right)^{N_i}}{N_i !}}}}{ \partial{N_i}} \right)_{T,V,N} $$

$$  = -k_B T \left( \frac{ \partial{ \ln{q_i \left(V, T \right)^{N_i}}} - \ln{N_i !}}{\partial{N_i}} \right)_{T,V,N} $$

$$  = -k_B T \left( \frac{ \partial{ N_i \ln{q_i \left(V, T \right)}} - N_i \ln{N_i} + N_i}{\partial{N_i}} \right)_{T,V,N} $$

$$  = -k_B T \left( \ln{q_i} \left(V, T \right) - \ln{N_i} - 1 + 1 \right)_{T,V,N} $$

$$ \mu = -k_B T \ln{}\left( \frac{q_i \left(V, T \right)}{N_i} \right) $$

This definition helps describe the change in the Helmholtz free energy with a changing system composition, such as a reaction forming and/or consuming certain molecules in a system.

Heat Capacity
In this section, we will complete the deviation of heat capacity in terms of the partition function Q.

$$Q = {\displaystyle \sum_j \exp(-E_j/k_B T)}$$

According to the Gibbs postulate, the ensemble average is equal to the average internal energy, U

$$U =\langle E \rangle = \frac{\displaystyle \sum_j E_j \exp(-E_j/k_B T)} {Q}$$

Substitute equation for partition function into ensemble average.

$$ \langle E \rangle = \frac{\displaystyle \sum_j E_j \exp(-E_j/k_B T)} {\displaystyle \sum_j \exp(-E_j/k_B T)}$$

$$\beta= \frac {1} {k_B T}$$, which is the Boltzmann's Constant.

Now substituting $$\beta$$ you get,

$$ \langle E \rangle = \frac{\displaystyle \sum_j E_j \exp(-\beta E_j)} {\displaystyle \sum_j \exp(-\beta E_j)}$$

$$C_v = \left( \frac{d\langle E \rangle} {dT}\right)$$, which is the heat capacity.

To find the equation of heat capacity using Statistical Mechanics we first have to differentiate the ensemble average with respect to $$\beta $$.

$$\left( \frac{d\langle E \rangle} {dT}\right)_{N,V} = \frac{\displaystyle \sum_j E_j \exp(-\beta E_j)} {\displaystyle \sum_j \exp(-\beta E_j)}$$ $$ = \frac{-1}{Q} \left( \frac{dQ} {d\beta}\right)_{N,V}$$

Therefore, the equation of heat capacity with respect to the partition function Q is,

$$C_v = \frac{-1}{Q} \left( \frac{dQ} {d\beta}\right)_{N,V}$$