The Method of Intercepts: An Alternative Derivation Boyd L. Earl
University of Nevada, Las Vegas, Las Vegas, NV 89154
Partial molar quantities are discussed, a t least hriefly, in most physical chemistry texts, and the method of intercepts for their determination is presented in several. Using volume as an example, the total volume of a binary solution obeys the equation V = n,V,
+ n,V,
(1)
where Vl = (aVlanl),,TF and V2 = (aVIan2)n,,TFare the partial molar volumes of components 1and 2, respectively. Dividing through by nl + nz yields the equation for the total molar volume, V.
v = x'V' + r,V,
(2)
where X I and xz are the mole fractions. When P is plotted against one of the mole fractions, as in the figure, the methods of intercepts provides values for Vl and V2a t any composition xlO,as the intercepts of the line tangent t o the curve at the point [xlO, V(xlo)]. The common derivation justifying this method follows the pattern of that in the text by Moore' or Atkins2. The result for the equation of the tangent line is
This result shows that the intercept a t X I = 0 is Vz(xlo),and hy reversing the roles of the variables (switching the indices), it is trivial to conclude that the intercept a t rl = 1is V,(x,O). .. . . Presented here is an alternative derivation that is at once less elegant and more direct and that has one additional property that is advantageous: the derived equation explicitly gives the results for both intercepts simultaneously3. We calculate the equation of the tangent line by "brute force", by finding the slope of the curve a t x10 by direct differentiation and using the point-slope form of the straight-line equation. The slope is the derivative of eq 2 with respect to xl dVi &2 d V, V1+rl-+-V,+T,(4) dr1 &I dx1 &I Although total derivatives are shown, we recognize that temperature and pressure are constant. This equation is simplified by noting that dx&l = -1, and that, by the GihbsDuhem equation dtr -=
so the second and fourth terms cancel. The result for the slope a t x10is
Using the point-slope form of the equation for a straight line, we obtain for the equation of the tangent line 58
Journal of Chemical Education
A representative graph showlng the evaluation of partial molar volumes for a
hypothetical System by the method of intercepts.
v-
tr(x,O)= [V,(x,O)- Yz(x,O)l(~, - x,')
(7)
Now, by eq 2 V(r,O) = z,OVl(.r,O)+ (1- z,O)Vz(x,O) (8) Substituting this into the left side of eq 7 and simplifying yields the desired equation for the tangent line (9) V = &(x,0) + [P,(x,O)- V2(r,0)]x, from which it is readily seen that when xl = 1, V = V,(xlo) and when X I = 0, V = t;(xlo). Two points might he noted: (1) Equation 2 can be written V = V2 (V' - VZ)X~. Thus this equation has the peculiar characteristic that the equation of the curve and of the tangent line are of thesame form, with that of the line having constant coefficients in place of the variable coefficients in that of the curve. (2) Given this deceptive similarity of form, one might he tempted to conclude by inspection that the slope a t any point is obviously Vl - Vz. This "quick and dirty" analysis is incorrect for two reasons: it amounts to taking the partial derivative of the right side with respect to xl, which is inadmissible (XI and xz cannot vary independently), and since Vl and Vz are functions of XI, the slope, as given by the total derivative, must include the derivatives of these volumes with respect to XI.
+
Moore, Walter J. Basic Physical Chemistry, Prentice-Hall: Englewood Cliffs. NJ, 1983; p 179. Atkins. P. W. Physical Chembtly. 2nd ed.; Freeman: San Francisco, 1982; pp 248-249. As pointed out by the referee, the text. Klotz, I. M.; Rosenberg, R. M. Chemical Thermodynamics,4th ed.; BenjaminlCummings:Menlo Perk. 1986: DD 373-374, contains aderivation similar to the one here presented, which follows the same steps through eq 6,
