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# Concentration and Chemical Reaction Rate

### Skills to develop

• Explain chemical reaction rates
• Explain the concentration effects on reaction rate
• Define the order of a reaction with respect to a reactant
• Define the overall order of a chemical reaction
• Define the rate constant of a chemical reaction
• Determine the order of a reaction by experiment
• # Concentration and Chemical Reaction Rate

In the introduction to chemical kinetics, we have already defined chemical reaction rates. Rates of chemical reactions depend on the nature of the reactants, the temperature, the presence of a catalyst, and concentration. This page discusses how the concentration affect the chemical reaction rates. Concentration effect is important because chemical reactions are usually carried out in solutions.

## Chemical Reaction Rates

The reaction rates of chemical reactions are the amounts of a reactant reacted or the amount of a product formed per unit time, (moles per second). Often, the amount can be expressed in terms of concentrations.
```             amount reacted
or produced
Rate = --------------- units: g/s, mol/s, or %/s
time interval
```
At certain conditions, the rates are functions of concentrations. Depending on the time interval between measurements, the rates are called
average rate: rate measured between long time interval
instantaneous rate: rate measured between very short interval
initial rate: instantaneous rate at the beginning of an experiment

However, a more realistic representation for a reaction rate is the change in concentration per unit time, either the decrease of concentration per unit time of a reactant or the increase of concentration per unit time of a product. In this case, the rate is expressed in Mol/(L sec).

```            concentration change of
a reactant or product
Rate = ----------------------- units: g/(M s), M/s, ppm/s etc
time interval
```

## Measuring Reaction Rate

To measure a reaction rate, we usually monitor either a product or a reactant for its change. Any physical characteristic related to the quantity or concentration of a product or reactant can be monitored. Some of the characteristics to be monitored are:
change in pressure,
change in color (spectroscopic measurement),
temperature for exothermic or endothermic reaction, and
presence of certain key substance,

The change can be plotted on a graph, and from the graph, we can get the average rate or the instantaneous rate by either graphical method or using computer for the data analysis.

## Rate Constants and the Orders

Usually, the rate of a reaction is a function of the concentrations of reactants. For example, the rate of the reaction 2 NO + O2 = 2 NO2
has the form:
Rate = k [O2] [NO]2
The rate is proportional to the concentration of O2, usually written as [O2] and is proportional to the square of [NO], or [NO]2. The orders of 1 and 2 for [O2] and [NO] respectively has been determined by experiment, NOT from the chemical equation. The total order of this reaction is 3 (=2+1).

Note the rates and order in the following example reactions:

H2 + I2 = 2 HI,
Rate = k [H2] [I2],
Total order 2.

H2 + Br2 = 2 HBr,
Rate = k [H2] [Br2]1/2,
Total order 1.5.

In particular, note that orders are NOT determined from the stoichiometry of the reaction equation.

## Rates as Functions of Reactant Concentrations

The order with respect to (wrt) a reactant are determined experimentally by keeping the concentration of other reactants constant, but vary the concentration of one of the reactant, say A in a general reaction a A + b B + c C = products If concentrations of B and C are kept constant, you can measure the reaction rate of A at various concentrations. You can then plot the rate as a function of [A]. For a zeroth order reaction, you will get a horizontal line, because 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 [A]     (a straight line) Note that rate = k when [A] = 1.

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

rate = k [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 [A] less than 1. When [A] is greater than or equal to 1, then the reaction rate is very large. This model applies to nuclear explosion, except that [A] = 1 is actually the critical mass of the fission material.

rate = k [A]00
```

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 bombs is to have a critical mass of the fission material, 235U or 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 discribed by the order of the fission material.

## Variation of Rate, Rate Constant, and Order of a Reaction

If only [A] is varied in experiments, and the order wrt [A] is n, then the rate has the general expression, rate = k [A]n In this expression, k is the specific rate constant, or the rate when [A] = 1.00. Again, the order n is not necessarily an integer, but its most common values are 0.5 (1/2), 1, 2, or 3. Cases in which n is a negative number are rare.

Mathematical models for the effect of concentration on rates are interesting. In general, the rate of a reaction of order n with respect to A can be represented by the equation:

y = k xn,       (n = various values including 0.5, 1, 2, 3, ...)
Plots of equations for various values of n illustrate the dependence of rate on concentration for various orders.

## Evaluation of order by Experiments

For a chemical reaction, we often determine the order with respect to a reagent by determine the initial rate. When more than one reactants are invovled, we vary the concentrations in a systematic way so that the effect of concentration of one of the reactants can be measured.

For example, if a reaction involving three reactants, A, B, and C, we vary [A] from experiment 1 to experiment 2 and find out how the rate varies. Similarly, we vary concentrations of B or C in other experiments, keeping others constant, and investigate its effect. The example below illustrates the strategy for such an approach.

Example

Derive the rate law for the reaction A + B + C => products from the following data, where rate is measured as soon as the reactants are mixed.
Experiment1234
[A]o0.1000.2000.2000.100
[B]o0.1000.1000.3000.100
[C]o0.1000.1000.1000.400
rate0.1000.8007.2000.400

Solution
Assume the orders to be x, y, and z respectively for A, B, and C, we have

rate = k [A]x [B]y [C]z From experiment 1 and 2, we have:
```0.800   k 0.2x  0.1y  0.1z
----- = ---------------
0.100   k 0.1x  0.1y  0.1z
```
Thus, 8 = 2x; and x = ln8 / ln2 = 3

By similar procedures, we get y = 2 and z = 1. Thus, the rate law is:

rate = k [A]3 [B]2 [C]

Discussion
Note the following relationships:

x = yz
ln x = z ln y
z
= ln x / ln y

### Summary

The variation of reaction rates as functions of order and concentrations are summarized in the form of a Table below.

 Differential rate law Plot of rate vs[A] - d[A]/dt = k horizontal line - d[A]/dt = k[A] straight line with slope = k - d[A]/dt = [A]2 a branch of parabola - d[A]/dt = k [A]oo rate = 0 when [A] < 1 rate = infinite when [A] > 1 a vertical line at [A] = 1

### Confidence Building Questions

• If the reaction rate doubles as the concentration of the reactant A increase by a factor of 2, what is the order of the reaction with respect to A?

Skill -
Recognize the order when rate is linear dependent on [A]. Only when n = 1 does the rate depend linearly on the concentration.

• If the reaction rate is a constant as the concentration of a reactant A varies, what is the order of the reaction with respect to A?

Skill -
Recognize the order when rate = k when n = 0.

• For a reaction that is second order with respect to a reactant A, how many times does the rate increase as [A] increases by a factor of 2.

Skill -
Predicts rate increases as the concentration doubles for reactions of various orders.

• What is the reaction rate when the concentrations of all reactants are unity, ([A] = [B] = 1) for a second order reaction?

Discussion -
Rate = k, when [A] = [B] = 1 regardless of the order of the reaction. Note the difference between rate and rate constant, k.

• For a reaction that is 2nd order with respect to A, you can keep concentrations of other reactants constant, and vary the concentration of A. You can measure the initial rate in the experiments, and then plot rate as a function of [A]. What type of curve or line is such a plot?

• In general, the rates increase as the concentrations increase for any reaction with a positive order, true or false?

Skill -
What if n is a negative value?