HW 9-Sol

HW 9-Sol - HW9 solution The two stages involved in the...

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HW9 solution The two stages involved in the formation of particles of a new phase are nucleation and growth . The nucleation process involves the formation of normally very small particles of the new phase(s) which are stable and capable of continued growth. The growth stage is simply the increase in size of the new phase particles. The total free energy change is equal to the sum of free energy of nucleation and free energy of surface of nucleation. That is SL V r G r G γ π π 2 3 4 3 4 + Δ = Δ Since * r and * G Δ appear at the maximum on the free energy-versus-radius curve of Figure 10.2 b , derivation of expressions for these two parameters is a simple matter. For * r , we differentiate the G Δ equation with respect to r , set the resulting expression equal to zero, and then solve for ) ( * r r = . That is, ( ) 0 ) 2 ( 4 3 3 4 2 = + Δ = Δ r r r G dr G d SL V γ π π which leads to the result V SL G r Δ = γ 2 * Now, substitution of this expression for * r into the equation of total free energy change yields the following expression for * G Δ Δ = Δ = Δ 2 2 2 3 2 3 * 1 3 16 ) ( 3 16 T L T G G V M SL V SL πγ πγ with m V V T T L G Δ = Δ The total free energy change is equal to the sum of free energy of nucleation and free energy of surface of nucleation for the case of a cubic nucleus of edge length a . That is SL V a G a G γ 2 3 6 + Δ = Δ Since * r and * G Δ appear at the maximum on the free energy-versus-radius curve of Figure 10.2 b , derivation of expressions for these two parameters is a simple matter. For * r , we differentiate the G Δ equation with respect to r , set the resulting expression equal to zero, and then solve for ) ( * r r = . That is, ( ) 0 12 3 2 = + Δ = Δ a r a G da G d SL V γ which leads to the result V SL G r Δ = γ 4 *

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