To complete the analysis we need one more fact or approximation.
We have assumed that the density of the melt differs little from that
of the solid. The Birch-Murnaghan equations (*Birch*, 1938; 1952;
*Anderson*, 1989)
show that the bulk moduli of solid systems change
in a predictable way as a function of the changing density.
A similar result for simple liquids known as Rao's rule
(*Rao*, 1941) also shows that the bulk modulus of many
pure (*i.e.*, single constituent) liquids is also a simple function
of the density.
Based on these results, if the density of the melt is the same
as that of the surrounding solid, then we expect the bulk modulus
of the melt to differ very little (on the order of a few per cent)
even though the shear modulus
has dropped from a finite value to zero.
We have shown in the preceding paragraph that,
if the fluid inclusions have the same bulk modulus as the
solid, then *K*_{sat} = *K*_{g} and *c _{1}* = 0. We expect
this approximation to have the same level of validity as
the approximation that the density is constant (which is to say that
we think of it as a reasonable first approximation). With this
approximation substituted into (dlnvp2), we find that

d v_sd v_p 3.
This is correct as long as . But the shear modulus will
necessarily decrease as the melt fraction increases, since,
for fixed overall shear distortion, less shear
energy can be stored in the system, and this implies that *c _{2}* > 0.
Computing the actual value
of

10/25/1999