2. The partial molar volumes of the components of a binary solution may be writt

Question

2. The partial molar volumes of the components of a binary solution may be written as Where n1, n2, M1, and M2, are the number of moles and molecular weights of component one (water) and component 2 (methanol), respectively, and p is the density of the solution. Do the following: a. Rewrite the above expression in terms of the mole fractions x1 and X2. Remember that the partial derivatives must also be converted. b. The density of a water-methanol solution in g/cm^3 at 298 K may be expressed in terms of the mole fraction X2 as follows: Substitute this into the expressions obtained in part a to obtain analytical expressions for V1 and V2. c. Evaluate V1, V2, and Vt0t when X2 = 0.1 and the total number of moles is 1. d. How does Vtot compare with the sum of the unmixed volumes?

Explanation / Answer

(dV/dn1)n2 = [(dV/dX1)(dX1/dn1)]n2

X1 = n1/(n1+n2)

X2 = n2/(n1+n2)

(dX1/dn1)n2 = [d{(n1)/(n1+n2)}/dn1]n2

= {(n1+n2)(1) - (n1)(1)}/(n1+n2)2

So, (dX1/dn1)n2 = n2/(n1+n2)2?

Similarly, (dX2/dn2)n1 = n1/(n1+n2)2?.

So,

(dV/dn1)n2 = [(dV/dX1)(dX1/dn1)]n2

= [(dV/dX1){n2/(n1+n2)2?}]n2

(dV/dn2)n1 = [(dV/dX2)(dX2/dn2)]n1

= [(dV/dX2){n1/(n1+n2)2?}]n1

[dp/dn1]n2 = [(dp/dX1).{n2/(n1+n2)2?}]n2

[dp/dn2]n1 = [(dp/dX2).{n1/(n1+n2)2?}]n1

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