Equation |
Rate-Determining Mechanism |
Differential Form 
|
Integral Form 
|
EqnNo |
Diffusion Models |
|
|
|
|
Parabola Law |
One-Dimensional Diffusion |
1/2x |
x2 |
D1 |
Valensi Eqn. |
Two-Dimensional Diffusion |
-[ln(1-x)]-1 |
(1-x)ln(1- x)+ x |
D2 |
Jander Eqn. |
Three-Dimensional Diffusion |
(3/2)(1-x)2/3[1-(1- x)1/3]-1 |
[1-(1-x)1/3]2 |
D3 |
Ginstling-Brounshtein Eqn. |
Three-Dimensional Diffusion |
(3/2)[(1-x)-1/3-1]-1 |
1-2x/3-(1-x)2/3 |
D4 |
Zhuravlevl, Lesokin, Tempelman Eqn. |
Three-Dimensional Diffusion |
(3/2)(1-x)4/3[(1- x)-1/3-1]-1 |
[(1-x)1/3-1]2 |
D5 |
Phase Boundary Reaction |
|
|
|
|
Power Law |
Contracting Cylinder |
2(1-x)1/2 |
1-(1-x)1/2 |
P1 |
Power Law |
Contracting Sphere |
3(1-x)2/3 |
1-(1-x)1/3 |
P2 |
Reaction-Order Models |
|
|
|
|
First-order |
Chemical Reaction |
(1-x) |
X |
R1 |
Second-order |
Chemical Reaction |
(1-x)2 |
[1/(1-x)]-1 |
R2 |
Third-order |
Chemical Reaction |
(1-x)3 |
(1/2)[(1-x)-2-1] |
R3 |
Nucleation Models |
|
|
|
|
Mampel power law |
Nucleation |
(2/3)x-1/2 |
x3/2 |
N1 |
Mampel power law |
Nucleation |
2x1/2 |
x1/2 |
N2 |
Mampel power law |
Nucleation |
3x2/3 |
x1/3 |
N3 |
Mampel power law |
Nucleation |
4x3/4 |
x1/4 |
N4 |
Exponential law |
Nucleation |
X |
ln x |
N5 |
Avrami-Erofeev Eqn. |
Random Nucleation and Subsequent Growth |
2(1-x)[-ln(1-x)]1/2 |
[-ln(1-x)]1/2 |
N6 |
Erofeev Eqn. |
Random Nucleation and Subsequent Growth |
3(1-x)[-ln(1-x)]2/3 |
[-ln(1-x)]1/3 |
N7 |
Prout-Tomkins Ewn. |
Branching Nuclei |
x(1-x) |
ln[x/(1-x)] |
N8 |