Mass conservation and elemental fractionation - Geochemistry: An Introduction - page 34

The Rayleigh distillation equation

Let us consider a liquid system at high temperature and sufficiently well stirred for its compo-
sition to remain homogeneous. As the liquid cools, a solid phase appears and we are going
to look at how the concentration of an element i changes as all of the liquid in the reservoir
progressively crystallizes. We designate the properties of the newly crystallized solid by the
subscript sol, those of the residual liquid by the subscript liq, and those of the liquid at the
beginning of crystallization by the subscript 0. We write the mass of each phase (solid or
liquid) as M and the mass of element i accommodated by a phase as m i . We apply two con-
ditions: that of the closed system (transformation of liquid to solid, without external inputs
or outputs) and that of equilibrium fractionation of element i between the solid and liquid.
The closed-system condition is written in its incremental form:

d M liq + d M sol = 0 (2.23)

d m i d m i

liq + sol =

0 (2.24)

The equilibrium condition between the last solid formed and the residual liquid is written
using the partition coefficient D i
s/l

in the form:

m i
liq
M liq

d m i
sol

D i s/l

d M sol =

(2.25)

Let us replace the properties of the liquid by those of the solid from (2.23) and (2.24) :
d m liq
d M liq =

m liq
M liq

D i s/l (2.26)

or

d m liq
m liq

d M liq
M liq

D i
s/l

= (2.27)

Taking the logarithm of concentration C liq = m liq / M liq of element i in the liquid and

differentiating gives:

d C liq
C liq

d m liq
m liq

d M liq
M liq

= − (2.28)

By combining (2.27) and (2.28 ) , and introducing the fraction of residual liquid F = M liq / M 0 ,

we obtain:

d C liq
C liq

D i s/l −

1 d F
F

= (2.29)

which is the form sought, applied to progressive crystallization of a liquid, and which leads
us to (2.20) .