Week 2 Liquid/Solid Equilibrium in multicomponent systems: defining solubility#
Let us begin by writing the solid-liquid equilibrium conditions at temperature \(T\) and pressure \(P\), between a pure crystalline phase and a multicomponent liquid phase characterised by composition \(\vec{x}\). Equilibrium is stated by the equality of chemical potentials:
which is equivalent to the equality of the fugacity in the two phases:
where \(f_i^s(T,P)\) is the fugacity of a pure crystalline solid at \(T\) and \(P\), while \(\hat{f}_i^\ell(T,P,\vec{x})\), is the fugacity of a \emph{real} mixture at the same temperature and pressure, at composition \(\vec{x}\).
Fugacity of a real solution#
Focusing on the right hand of Eq.\ref{eq:fugacity}, the fugacity in the \textbf{real solution} phase can be rewritten as a function of the fugacity of the \textbf{pure liquid phase} as follows:
where \(x_i\) is the molar fraction that realizes the equality between fugacities, i.e. the solubility of component \(i\), \(f_i^\ell(T,P)x_i\) is the fugacity of an \emph{ideal} solution at molar fraction \(x_i\), \(\gamma_i(T,P,\vec{x})\) is the activity coefficient, capturing the non-idealities of the liquid phase in equilibrium with the solid. By definition of an ideal solution \(f_i^\ell(T,P)\) is the fugacity of the \textbf{pure solute in its liquid phase} at temperature T and pressure P.
Fugacity of a pure crystalline solid#
Let us now focus on the left-hand side of Eq.\ref{eq:fugacity}. Here, we are only concerned with the \textbf{physical properties of the pure solute}. The fugacity of the pure solid phase can also be written as a function of the fugacity of the pure liquid by introducing the definition of fugacity and integrating between the liquid and solid states \emph{\(\ell\)}\(\rightarrow\)\emph{s}:
which, integrated, leads to:
and rearranged gives:
where notably \(\Delta{\mu}_{i,\ell\rightarrow{s}}\), is the change in chemical potential associated with the liquid to solid transition of the \emph{pure solid}. We note that the change in chemical potential for a pure substance corresponds to the the partial molar change in free energy \(\Delta{g_i(T,P)}\):
Now we express the molar enthalpy change \textbf{for the pure solute} by considering a thermodynamic cycle, that goes through the melting temperature \(T_f\) at pressure P. This enables the computation of \(\Delta{h}_i(T,P)\) and \(\Delta{s}_i(T,P)\) via an arbitrary but convenient path, taking advantage of the fact that all terms in the free energy expression are \emph{state functions}.
Enthalpy of phase transition at T and P for pure solute#
The enthalpy change associated with the solid-to-liquid transition becomes:
where:
\(\int_T^{T_f}Cp^{\ell}dT\) is the enthalpy change associated with raising the temperature of the pure solute liquid from \(T\) to \(T_m\)
\(-\int_T^{T_f}Cp^{s}dT\) is the enthalpy change associated with decreasing the temperature of the pure solid from \(T_m\) to \(T\)
\({-\Delta{h_{fus}\left(T_f\right)}}\) is the change in enthalpy associated with the liquid to solid transition of the pure solute at the melting temperature
For heat capacities independent of T, the enthalpic contribution reduces to:
Entropy of phase transition at T and P for pure solute#
Analogously to the enthalpy term, we can write the molar change in entropy via a the same thermodynamic cycle:
where:
\(+\int_T^{T_f}\frac{Cp^{\ell}}{T}dT\) is the entropy change associated with raising the temperature of the pure solute liquid from \(T\) to \(T_m\)
\(-\int_T^{T_f}\frac{Cp^{s}}{T}dT \) is the enthalpy change associated with decreasing the temperature of the pure solid from \(T_m\) to \(T\)
\(-\frac{\Delta{h_{fus}\left(T_f\right)}}{T_f}\) is the change in entropy associated with the liquid to the solid transition of the pure solute at the melting temperature
Again, for heat capacities independent of T, the entropy change reduces to:
Free energy of phase transition at T and P for pure solute#
Defining \(\Delta{Cp}_{\ell\rightarrow{s}}=\left(Cp^{\ell}-Cp^{s}\right)\) we can now recover the expression for \(\Delta{g}_i^s(T,P)\), where the only assumption is that \(Cp^{\ell}\) and \(Cp^{s}\) are independent from the temperature T.
Solubility#
Inserting now this expression in Eq. \ref{eq:fugacitysolid} yields:
which then can be inserted into the equality of fugacities defining equilibrium Eq.\ref{eq:fugacity} giving:
which can be rearranged to yield an expression of the solubility \(x_i\) where the only approximation introduced is in considering the heat capacities of both the crystal and the liquid constant in T.
In this expression, the fugacity of the pure liquid solute, \(f_i^\ell(T,P)\) here cancels out, and the term \(-\Delta{h_{fus}}\) is not approximating a different quantity (like the enthalpy of dissolution). It is actually introduced rigorously by the convenient choice of an appropriate thermodynamic cycle for the calculation of \(\Delta\mu_{i,\ell\rightarrow{s}}\).
The Schroeder-Van Laar Equation#
The Schroeder-Van Laar equation for the solubility is derived from \ref{eq:solubility} by assuming that \(\Delta{Cp}_{\ell\rightarrow{s}}\) is negligible.
This simplification yields:
Thus leading to the Schroeder-Van Laar solubility equation for \(x_i\):