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HW7 - 14:440:407 Section 02 Fall 2010 SOLUTION OF HOMEWORK...

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14:440:407 Section 02 Fall 2010 SOLUTION OF HOMEWORK 07 Question 10.2: (a) Rewrite the expression for the total free energy change for nucleation (Equation 10.1) for the case of a cubic nucleus of edge length a (instead of a sphere of radius r). Now differentiate this expression with respect to a (per Equation 10.2) and solve for both the critical cube edge length, a*, and also Δ G*. (b) Is Δ G* greater for a cube or a sphere? Why? Solution: (a) This problem first asks that we rewrite the expression for the total free energy change for nucleation (analogous to Equation 10.1) for the case of a cubic nucleus of edge length a . The volume of such a cubic radius is a 3 , whereas the total surface area is 6 a 2 (since there are six faces each of which has an area of a 2 ). Thus, the expression for Δ G is as follows: Δ G = a 3 Δ G v + 6 a 2 γ Differentiation of this expression with respect to a is as d Δ G da = d ( a 3 Δ G v ) da + d ( 6 a 2 γ ) da = 3 a 2 Δ G v + 12 a γ If we set this expression equal to zero as 3 a 2 Δ G v + 12 a γ = 0 and then solve for a (= a *), we have a * = 4 γ Δ G v Substitution of this expression for a in the above expression for Δ G yields an equation for Δ G * as Δ G * = ( a *) 3 Δ G v + 6( a* ) 2 γ
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