9. 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 conditions is stated by the equality of chemical potentials:

(102)#\[ \mu_i^s(T,P)=\mu_i^\ell(T,P,\vec{x}) \]

which is equivalent to the equality of the fugacity in the two phases:

(103)#\[ f_i^s(T,P)=f_i^\ell(T,P,\vec{x}) \]

Focusing on the right hand of Eq.(103), the fugacity in the liquid phase can be rewritten as a function of the fugacity of the pure liquid phase as follows:

(104)#\[ f_i^\ell(T,P,\vec{x})=f_i^\ell(T,P)x_i\gamma_i(T,P,\vec{x}) \]

where \(x_i\) is the molar fraction of the solute and \(\gamma_i(T,P,\vec{x})\) is the activity coefficient of that specie in the liquid phase.

Let us now focus on the left hand side of Eq. (103). The fugacity of the pure solid phase can also be written as a function of the fugacity of the pure liquid:

(105)#\[ \int^{T,P,s}_{T,P,\ell}d\mu_i=\int^{T,P,s}_{T,P,\ell}RTd\ln{f_i} \]
(106)#\[ \Delta{\mu}_{i,\ell\rightarrow{s}}=RTln\left(\frac{f_i^s(T,P)}{f_i^\ell(T,P)}\right) \]
(107)#\[ f_i^s(T,P)=f_i^\ell(T,P)e^{\frac{\Delta{\mu}_{i,\ell\rightarrow{s}}}{RT}} \]

The change in chemical potential can be written for a pure substance as the partial molar change in free energy \(\Delta{g_i(T,P)}\):

(108)#\[ \Delta\mu_{i,\ell\rightarrow{s}}=\Delta{g}_i^s(T,P)=\Delta{h}_i(T,P)-T\Delta{s}_i(T,P) \]

The molar enthalpy change is:

(109)#\[ \Delta{h}_i(T,P)=-\Delta{h_{fus}\left(T_f\right)}+\int_T^{T_f}Cp^{\ell}dT+\int_T^{T_f}Cp^{s}dT\simeq-\Delta{h_{fus}\left(T_f\right)}+\left(Cp^{\ell}-Cp^{s}\right)\left(T_f-T\right) \]

while the molar entropy change is written as:

(110)#\[ \Delta{s}_i(T,P)=-\frac{\Delta{h_{fus}\left(T_f\right)}}{T_f}+\int_T^{T_f}\frac{Cp^{\ell}}{T}dT+\int_T^{T_f}\frac{Cp^{s}}{T}dT \simeq -\frac{\Delta{h_{fus}}}{T_f}+\left(Cp^{\ell}-Cp^{s}\right)\ln{\left(\frac{T_f}{T}\right)} \]

Defining \(\Delta{Cp}_{\ell\rightarrow{s}}=\left(Cp^{\ell}-Cp^{s}\right)\)

(111)#\[ \Delta{g}_i^s(T,P)=-\Delta{h_{fus}\left(1-\frac{T}{T_f}\right)}+\Delta{Cp}_{\ell\rightarrow{s}}\left(T_f-T-T\ln{\left(\frac{T_f}{T}\right)}\right) \]

Since \(\Delta{Cp}_{\ell\rightarrow{s}}\) is usually small compared the above expression is typically simplified to:

(112)#\[ \Delta{g}_i^s(T,P)\simeq-\Delta{h_{fus}\left(1-\frac{T}{T_f}\right)} \]

Inserting this equation into the equality of fugacities yields:

(113)#\[ f_i^\ell(T,P)x_i\gamma_i(T,P,\vec{x})=f_i^\ell(T,P)e^{\frac{\Delta{\mu}_{i,\ell\rightarrow{s}}}{RT}} \]

hence the solubility of component \(i\), \(x_i\) is:

(114)#\[ x_i=\frac{1}{\gamma_i(T,P,\vec{x})}e^{-\frac{\Delta{h_{fus}}}{R}\left(\frac{1}{T}-\frac{1}{T_f}\right)} \]