The equilibrium coefficients developed in the previous post are used in reaction rate expressions, but they aren't very helpful for exploring the overall equilibrium of the system. To find the equilibrium of a given set of compounds, we need to minimize the Gibbs free energy. It will take a few posts to develop the equations and conduct the minimization.
The Gibbs free energy can be partitioned into two contributions...(1) the free energies of the compounds, and (2) the impact of mixtures and pressure. I will develop the first contribution in this post.
Derivation of F(T)
The first equation below is the Gibbs-Helmholtz equation, with F as the free energy of a compound. Integration of this equation will produce F(T).
I tried using the form of the last equation with F(T) and H(T), letting Mathcad conduct the integration numerically. It worked, but not reliably. Trials with some temperatures resulted in "could not converge". Therefore, I manually integrate the equation below and arrive at a vector-matrix solution that doesn't require the numerical integration.
Continue by substituting for the enthalpy, H, in the integral term.
The DHT term can be easily integrated. I used the symbolic math capability below.
The above result will be assigned to a new function, DFT(t). In order to make the first element dimensionless, I divide t by K, the default temperature unit.
After integration of all of the parts, the following function is obtained:
The above function computes a vector of the free energies of the compounds at temperature T and 1 atm. Recall that cp is a matrix of the coefficients for the cubic polynomials for the component heat capacities.
An example result for the reforming system is shown below
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