Module 8.2: Integrated Rate Laws

I Integrated rate law: An expression that gives the concentration of a species as a function of time

1. The integrated rate law is useful when...

   1. predicting the concentration of a species at any time
   2. finding the rate constant and order of a reaction

II For zeroth-order reactions, the concentration of species is independent of the rate

1. the rate of a zeroth-order reaction is always constant

2. The concentration of A after a time t is defined as: $[\textrm{A}]_{t} = [\textrm{A}]_{0} - \textrm{k}_{r} \textrm{t}$ (Eq. 132)

8.2.1 First-order reactions

III For first-order reactions, the integrated rate law is: $\textrm{ln}\frac{[\textrm{A}]_{t}}{[\textrm{A}]_{0}} = -\textrm{k}_{r} \textrm{t}$ (Eq. 133)

1. first-order reactions undergo exponential decay.

2. there are many useful forms of this integrated rate law.

   1. $[\textrm{A}]_{t} = [\textrm{A}]_{0} \textrm{e}^{-\textrm{k}_{r}\textrm{t}}$ emphasizes exponential decay. (Figure 78)

   2. $\textrm{ln}[\textrm{A}]_{t} = \textrm{ln}[\textrm{A}]_{0} - \textrm{k}_{r} \textrm{t}$ is written in the form of a linear equation. (Figure 77)

3. The larger the rate constant, the faster the species decays. (Figure 78)

4. All first-order reactions must give a linear plot as seen in Figure 77.

5. half-life ($\textrm{t}_{\frac{1}{2}}$): the time it takes for half of the reactants to decay

$\textrm{t}_{\frac{1}{2}} = \frac{\textrm{ln}(2)}{\textrm{k}_{r}}$ (Eq. 134)