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Equations describing chemical equilibria

Chemical equilibrium are best represented by the free-energy minimization. In our implementation, the condition of equilibrium are state in term of the minimisation of the Gibbs free energy.

In the problem of solving equilibrium in combustion process, we could consider an ideal gas and pure condensed phase. So, the equation of state for the mixture is

$\displaystyle PV = nRT$ (1)

where $ P$ is the pressure ($ N/m^2$), V specific volume ($ m^3/kg$), n moles, and T temperature (K).

The variable V and n refer to gases only while the mass is for the entire mixture including condensed species.

The molecular weight of the mixture is $ M$ and is define as

$\displaystyle M = \frac{ \overset{n}{\underset{j=1}{\sum}}{n_j M_j}}{\overset{m}{\underset{j=1}{\sum}}{n_j}}$ (2)

For a mixture of $ n$ species, the Gibbs free energy is given by

$\displaystyle g = \sum_{j=1}^{n}{\mu_j n_j}$ (3)

and the chemical potential may be written

$\displaystyle \mu_j = \begin{cases}\mu_j^0 + RT\ln{(n_j/n)} + RT\ln{P_{atm}}& (j=1,...,m)  \mu_j^0& (j=m+1,...,n) \end{cases}$ (4)

where $ \mu_j^0$ is the chemical potential in the standard state. It is habitually defined as

$\displaystyle \mu^0 = H_T^0 + TS_T^0$ (5)

The equilibrium condition is the minimisation of free energy and is subject to the mass balance constraint:

$\displaystyle b_i - b_i^0 = 0 \quad i = 1,...,l$ (6)

where $ b_i$ is the number of moles of one atom in the mixture and $ b_i^0$ is the number of moles of the same atom of total reactants.

$\displaystyle b_i = \sum_{j=1}^{n}{a_{ij} n_j} \quad i = 1,...,l$ (7)

We could now define a term $ G$

$\displaystyle G = g + \sum_{i=1}^{l}{\lambda_i (b_i - b_i^0)}$ (8)

where $ \lambda_i$ are Lagrangian multipliers. Taking the first derivative, we get

$\displaystyle \delta G = \sum_{j=1}^{n}{\left(\mu_j + \sum_{i=1}^{l}{\lambda_i a_{ij}}\right)\delta n_j} + \sum_{i=1}^{l}{(b_i - b_i^0)\delta \lambda_i}=0$ (9)

The variations $ \delta n_j$ and $ \delta \lambda_i$ are independant and we get two conditions for equilibrium

$\displaystyle \mu_j + \sum_{i=1}^{l}{\lambda_i a_{ij}} = 0 \quad j = 1,...,n$ (10)
$\displaystyle b_i - b_i^0 = 0$ (11)


next up previous
Next: Gibbs iteration equations Up: cpropep Previous: Introduction
Antoine Lefebvre 2000-03-26