We now apply Eq. (5.61) to a volume containing solvent
molecules and solute particles . The sum over *i* then runs over solvent molecules and
solute particles. The averaging may be performed by a conditional average
given some particle configuration, followed by an average over all particle
configurations.

We first perform the sum over solvent molecules. Dividing the space occupied
by liquid into small cubes, large enough to be called a many body system,
but small enough for gradients of cube properties to be small, we may write
for the contribution of the solvent molecules

The averages are over particle configurations. The integrals in the first two terms are over the space occupied by liquid. The sum in the last term is over solute particles, and is a vectorial surface element of particle

We now calculate the contribution from the solute particles. Averaging over
the solvent configurations turns
*m*_{i}*dv*_{i}/*dt* into
*m*_{i}*DV*_{i}/*Dt*given in Eq. (4.39). The contribution to the stress tensor by
the solute particles then reads

where is the joint probability function for the solute particles to be at positions . Combining the results we find

The sum of the second term in Eq. (5.62) and the first term in Eq. (5.63) is equal to the total mass weighed velocity fluctuation divided by the volume, and has been put equal to zero, which is reasonable for macroscopic volumes of fluid.

The first term in Eq. (5.64) reads

= | |||

= | |||

= | (5.65) |

In the second step we have used the fact that within the solute particle velocity gradients are zero. In the third step we have used , for any function .