Derivation of the Scheil equation

Equating the amount of solute in the shaded areas gives:

( C_L - C>_S ) df = ( 1 - f ) dC_L

Substituting for C_S, using:

C_S = k C₀

where k is the partition coefficient gives:

C_L ( 1- k ) df = ( 1 - f ) dC_L

We then integrate, from zero to f_S, the fraction solidified, and from C₀ (since C_L = C₀ when f_S = 0) to C_L:

\[\int_0^{{f_S}} {\frac{{df}}{{\left( {1 - f} \right)}} = \int_{{C_0}}^{{C_{\rm{L}}}} {\frac{{d{C_{\rm{L}}}}}{{{C_{\rm{L}}}\left( {1 - k} \right)}}} } \]

\[ - \ln \left( {1 - {f_{\rm{s}}}} \right) = \frac{1}{{1 - k}}\ln \frac{{{C_{\rm{L}}}}}{{{C_0}}}\]

Taking exponentials of both sides, and rearranging gives:

C_L = C₀ ( 1 - f_S )^k-1

We are more interested in the concentration of the solid, so this is usually written as:

C_S = k C₀ ( 1 - f_S )^k-1

This is known as the Scheil equation.