Heterogeneous equilibrium in solutions

A phase distinct from the solution is also present. We will now consider two simple examples of (i) a solution coexisting with a solid phase and (ii) a solution coexisting with a solid phase at a given pressure of a gas that dissolves and dissociates in the solution. 

What is then the actual distribution of dissolved species and the amount of calcium carbonate precipitated from the solution  

Let us add the moles of precipitated calcium carbonate as a sixth variable x6, modify the calcium and carbonate conservation equations f2(x) and /3(x) in order to account for solid phase contribution, and use the expression of the solubility product as a sixth equation. The six equations to be solved read 

The matrix of partial derivatives is obtained by taking the derivatives of the previous equations 

Let us keep the same initial guess as in the previous example and assume that no solid is present 

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