An expression to derive the definition of activity

Definition of symbolic variables

Definition of gibbs energy of the olivine solid solution

(1)RT(2yfolog(yfo)+(22yfo)log(1yfo))+Wyfo(1yfo)+gfao(1yfo)+gfooyfo

Definition of the gibss energy of forsterite in the olivine solid solution

(2)2RTlog(yfo)+Wyfo22Wyfo+W+gfoo

Definition of activity

Activity of forsterite is defined as the gibbs energy of the endmember in the solid solution (g_fo_ss gfo) relative to the gibbs energy of the pure endmember forsterite gfoo

(3)afo=exp(gfogfooRT)
(4)yfo2eW(yfo22yfo+1)RT

being awkwards the expression is usually expressed as RTlnafo

(5)RTlog(yfo2eWyfo22Wyfo+W+gfogfooRT)

Just note that

(6)2RTlog(yfo)+Wyfo22Wyfo+W
(7)W(yfo22yfo+1)

and that

(8)W(yfo1)2

so ,

(9)RTln(afo)=2RTln(yfo)+W(yfo1)2