## Chemical Kinetics: Reaction Rates

Attention: Print this page in 'landscape' mode because the table is too wide to fit when printed in 'portrait' mode.
Reaction Order Differential Rate Law Integrated Rate Law Linear Form Characteristic
Kinetic Plot
Half-life
Expression
Units of
Rate Constant
Zero
 - Δ[A] Δt = rate = k
[A] = [A]0 - k t [A] = -k t + [A]0 [A] vs t
 t½ = [A]0 2k
mole L-1 sec-1
First
 - Δ[A] Δt = rate = k [A]
[A] = [A]0 e- k t ln[A] = -kt + ln[A]0 ln [A] vs t
 t½ = ln(2) k
sec-1
Second
 - Δ[A] Δt = rate = k [A]2
 [A] = [A]0 1 + k t [A]0
 1 [A] = k t + 1 [A]0
1/[A] vs t
 t½ = 1 k[A]0
L mole-1 sec-1

The series of three graphs shown below illustrate the use of the characteristic kinetic plots. The graph on the left shows [A] vs t plots for a zero-order (red line), first-order (green line), and second-order (blue line) reaction. The graph in the middle shows ln [A] vs t plots and the graph on the right shows 1/[A] vs t plots for each reaction order.  page break

## Kinetics Using Calculus

### Zero Order

```            –d[A]
(1)  rate = ————— = k
dt
```
Differential Rate Law
How rate depends on concentration.
```(2)     d[A] = –kdt
```
 ```(3) ``` [A] t ∫ d[A] =   –k ∫ dt [A]0 0
```(4)   [A] – [A]0 = –kt
```
```(5)   [A] = –kt + [A]0
```
Integrated rate law (linear form)
How concentration depends on time.

 Remember:```  ``` ∫ dx =  x

### First Order

```            –d[A]
(1)  rate = ————— = k[A]
dt
```
Differential Rate Law
How rate depends on concentration.
```       d[A]
(2)   ------ = –kdt
[A]
```
 ```(3) ``` [A] t ∫ d[A] =   –k ∫ dt `————` [A] [A]0 0
```(4)   ln[A] – ln[A]0 = –kt
```
```(5)   ln[A] = –kt + ln[A]0
```
Integrated rate law (linear form)
How concentration depends on time.

 Remember:```  ``` ∫ dx =  ln(x) `————` x

### Second Order

```            –d[A]
(1)  rate = ————— = k[A]2
dt
```
Differential Rate Law
How rate depends on concentration.
```     – d[A]
(2)   ------ = kdt
[A]2
```
 ```(3) ``` [A] t – ∫ d[A] =    k ∫ dt `————` [A]2 [A]0 0
```(4)   –((–1/[A]) – (–1/[A]0)) = kt
```
```(5)   (1/[A]) – (1/[A]0) = kt
```
```(6)   (1/[A]) = kt + (1/[A]0)
```
Integrated rate law (linear form)
How concentration depends on time.

 Remember:```  ``` ∫ dx 1 `————` `  = – ———` x2 x

### Units of the Rate Constant (k)

 Order Sample Rate Law Units 0 rate = k M/s or M s–1 1 rate = k[A]1 1/s or s–1 2 rate = k[A]2 1/(M s) or M–1 s–1
Rule: M–(n–1) s–1 where n = overall order of reaction
The graphs and table on the first page were borrowed from a document on the Davidson College Chemistry Department site but the page no longer exists.
Last updated: Feb 17, 2022       Home