First, the general relationship between (1) and (5) is trivial, because it is well-known that the harmonic mean is always less than or equal to the mean (Abramowitz, 1972). Thus, I find
Then, for a binary mixture with x1 = x and x2 = 1-x, consider the difference
<1V>^2 - 1V_W^2 = 2x(1-x)[(_1_2K_2K_1)^12 - 12(_1K_2+_2K_1)] 0. The fact that the right hand side is less than or equal to zero follows from the fact that the geometric mean is always less than or equal to the mean (Abramowitz, 1972). This result establishes the general relation for binary mixtures that
Finally, to establish the result (8) for multicomponent mixtures, I consider
<1V>^2 = [x_i(_iK_i )^12]^2 = [(x_i_i)^12 (x_i/K_i)^12]^2 x_i_i x_i/K_i = 1V_W^2, where the inequality in (9) follows from Cauchy's inequality (Abramowitz, 1972) for sums . Thus, (9) shows that (8) is a general result for multicomponent mixtures.
Combining these results, I obtain the set of inequalities
V_W <1/V>^-1 <V>. These results have been derived for arbitrary multicomponent fluid mixtures, including but not restricted to binary mixtures.