# 4.8: Rates and Concentrations


$$rate = k$$ (a horizontal line)


rate
|      /
|     /rate = k [A]
|    /
|- -/- - - - - rate = k
|  /
| /
|/_________________
0   1   2   3   4  [A]


For a first order reaction, the plot is a straight line (linear), as shown above, because

$$rate = k \ce{[A]}$$ (a straight line)

Note that $$rate = k$$ when $$\mathrm{[A] = 1}$$.

For a second order reaction, the plot is a branch of a parabola, because

$$rate = k \ce{[A]}^2$$

  rate
|      .
|       rate = k [A]2
|     .   (a branch of
|          a parabola)
|    .
| - . - - - - - -
|  .
|._________________
0   1   2   3   4  [A]


For a reaction with an infinite order, the plot is a step function. The rate is small, almost zero, when $$\ce{[A]}$$ is less than 1. When $$\ce{[A]}$$ is greater than or equal to 1, then the reaction rate is very large. This model applies to nuclear explosion, except that $$\mathrm{[A] = 1}$$ is actually the critical mass of the fission material.

$$rate = k \ce{[A]}^{\infty}$$


rate
|       (order = infinity)
|   |   rate = k [A]00
|   |   (a vertical line)
|   |
|   |
|   |
|   |
|...|_________________
0   1   2   3   4  [A]


Is there a chemical process like this? Well, we all know that one of the key conditions in an atomic bomb is to have a critical mass of the fission material, $$\ce{^235U}$$ or $$\ce{^239Pu}$$. When such a mass is put together, the reaction rate increases dramatically, leading to an explosion. Thus, this model seems to apply; however, the mechanism for the fission reaction is not described by the order of the fission material.