  In thermodynamically characterizing a mixture comprising a finite number of
molecules, we consider two kinds of protocol for producing a mixture from a
pure substance. The first is a single alchemical operation, whereas the second
is a series of processes with feedback control in information thermodynamics
and conventional mixing with semipermeable membranes. A comparison of the two
numerically determined free-energy changes provides a combinatorial factor that
indicates the indistinguishability of the molecules and an alternative
Jarzynski equality. The comparison also uncovers a work relation for
determining the mixing free energy without using semipermeable membranes. We
demonstrate a numerical calculation of applying the work relation to a mixture
of argon and krypton. The mixing free energy clearly shows the characteristics
of liquid--vapor transition.
