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,

${\begin{alignedat}{7}{\frac {\partial {}}{\partial {T}}}Q={\frac {\partial {}}{\partial {T}}}\sum _{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)&\\=\sum _{j}{\frac {\partial {}}{\partial {T}}}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)&\\=\sum _{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right){\frac {\partial {}}{\partial {T}}}\left({\frac {-E_{j}}{k_{B}T}}\right)&\\={\frac {E_{j}}{k_{B}}}{\frac {1}{T^{2}}}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)&\\\end{alignedat}}$

${\begin{alignedat}{7}{\frac {\partial {}}{\partial {T}}}Q=\sum _{j}{\frac {E_{j}}{k_{B}}}{\frac {1}{T^{2}}}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)={\frac {1}{k_{B}T^{2}}}\sum _{j}E_{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)&\\={\frac {Q}{Q}}{\frac {1}{k_{B}T^{2}}}\sum _{j}E_{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)&\\=Q{\frac {1}{k_{B}T^{2}}}{\frac {\sum _{j}E_{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)}{Q}}&\\={\frac {Q}{k_{B}T^{2}}}{\frac {\sum _{j}E_{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)}{Q}}={\frac {Q}{k_{B}T^{2}}}\langle E\rangle \end{alignedat}}$

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,

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