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Concentration of reactants is one of the factors that can affect the reaction rate. Collision theory can be used to show that the higher the concentration of reactants the more collisions they undergo. Consequently, the number of inelastic collisions between these particles increases, and results in a higher reaction rate.
The relationship between the concentration of reactants and the reaction rate is not always directly proportional: for example, doubling the concentration does not necessarily double the reaction rate. It depends on the type of reaction and the nature of the reactants.
This relationship can be expressed mathematically using the rate law.
Rate law is a mathematical relationship between the concentration of reactants in a chemical reaction and its rate. It can be used to calculate the reaction rate at a given temperature.
For a hypothetical chemical reaction |a\ \text{A} + b\ \text{B} \rightarrow c\ \text{C} + d\ \text{D},| the rate law is:
||v = k[\text{A}]^m[\text{B}]^n||
where
|r\!:| reaction rate in |\text{mol/L}{\cdot\text{s}}|
|k\!:| rate constant in varying units
|[\text{A},] [\text{B}]\!:| concentration of reactants in |\text{mol/L}|
|m, n\!:| reaction order with respect to the corresponding reactant
Only concentrations of reactants in solution (e.g. in the gaseous state or in the aqueous state) are included in the rate law. Reactants in the liquid state or solid state are not included in the equation as they are not in solution.
For example, in the reaction |{\text{H}_{2\color{#3A9A38}{\text{(g)}}} + \text{I}_{2\color{#3A9A38}{\text{(g)}}} → 2\ \text{HI}_{\text{(g)}},}| both reactants are gases, so the rate law is: ||r = k[\text{H}_2]^m[\text{I}_2]^n||
However, in the reaction |\text{MgCO}_{3\color{#EC0000}{\text{(s)}}} + 2\ \text{HCl}_\color{#3A9A38}{\text{(aq)}} \rightarrow \text{MgCl}_{2\text{(aq)}} + \text{CO}_{2\text{(g)}} + \text{H}_2\text{O}_{\text{(g)}},| only one of the reactants is in aqueous state, so the rate law is: ||r = k[\text{HCl}]^m||
Before applying the rate law, the reaction order and the rate constant must be determined experimentally.
The reaction order is an experimentally determined value that indicates how the reaction rate changes when the concentration of a given reactant increases or decreases.
The higher the reaction order, the more significant the effect of concentration variation on the reaction rate.
Since the rate of a reaction varies as it progresses, it is often the initial rates that are measured experimentally in order to determine the order of the reaction.
To determine the reaction order with respect to a given reactant, the same chemical reaction is run with two different initial concentrations of the reactant of interest, while all other parameters are kept constant. The reaction rates at each initial concentration are measured and recorded. Then, the ratio of concentrations is compared to the ratio of reaction rates.
In the following equation, the exponent |m| corresponds to the reaction order with respect to Reactant A:
||\left(\dfrac{[\text{A}]_2}{[\text{A}]_1}\right)^m=\dfrac{r_2}{r_1}||
Consider a chemical reaction where the concentration of an aqueous or gaseous Reactant A is doubled: the concentration ratio is |\dfrac{2}{1}| or simply |2.| In response to this change, the reaction rate can remain constant, double, quadruple or octuple, as shown in the following table. The exponent on the base |2| corresponds to the reaction order with respect to Reactant A |(m).|
|
Initial concentration ratio |\left(\dfrac{[\text{A}]_2}{[\text{A}]_1}\right)| |
Initial rate ratio |\left(\dfrac{r_2}{r_1}\right)| |
Comparison between the concentration and rate ratio |\left(\dfrac{[\text{A}]_2}{[\text{A}]_1}\right)^m = \dfrac{r_2}{r_1}| |
Reaction order with respect to Reactant A |(m)| |
|---|---|---|---|
| |\dfrac{2}{1} = 2| |
The rate remains constant: |\dfrac{1}{1} = 1| |
|2^m = 1| |
|\begin{align} 2^m &= 1\\ |
|
The rate doubles: |\dfrac{2}{1} = 2| |
|2^m = 2| |
|\begin{align} 2^m &= 2\\ |
|
|
The rate quadruples: |\dfrac{4}{1} = 4| |
|2^m = 4| |
|\begin{align} 2^m &= 4\\ |
|
|
The rate octuples: |\dfrac{8}{1} = 8| |
|2^m = 8| |
|\begin{align} 2^m &= 8\\ |
The reaction order is usually a positive integer, although reactions with negative or fractional orders exist as well.
Once the reaction order with respect to each reactant is known, we can calculate the impact of a variation in reactant concentration on the initial reaction rate. To do this, we compare the ratio of the rates |(\dfrac{r_2}{r_1})| of the two reactions to the ratio of the expression of their rate law |(\dfrac{[\text{A}]_2^m\times[\text{B}]_2^n}{[\text{A}]_1^m\times[\text{B}]_1^n}).|
||\text{A}_{(aq)}+\text{B}_{(aq)}\rightarrow\text{C}_{(aq)}+\text{D}_{(aq)}||
||r_1 = k[\text{A}]_1^m[\text{B}]_1^n\ \text{and}\ r_2 = k[\text{A}]_2^m[\text{B}]_2^n\\
\dfrac{r_2}{r_1}=\dfrac{\cancel{k}[\text{A}]_2^m\times[\text{B}]_2^n}{\cancel{k}[\text{A}]_1^m\times[\text{B}]_1^n}\\ \dfrac{r_2}{r_1}=\left(\dfrac{[\text{A}]_2}{[\text{A}]_1}\right)^m \times \left(\dfrac{[\text{B}]_2}{[\text{B}]_1}\right)^n||
The reaction order with respect to a given reactant is not necessarily equal to the stoichiometric coefficient found in front of that reactant in the chemical equation. The reaction order has to be determined experimentally.
Only in elementary reactions, can it be assumed that the reaction order is equal to the stoichiometric coefficient.
The synthesis reaction between nitrogen dioxide gas |(\text{NO}_2)| and carbon monoxide gas |(\text{CO})| is conducted in a lab several times:
||\text{NO}_{2\text{(g)}} + \text{CO}_{\text{(g)}} → \text{NO}_{\text{(g)}} + \text{CO}_{2\text{(g)}}||
The initial concentrations of the reactants are recorded along with the initial reaction rates:
|
Experiment |
Initial |[\text{NO}_2], (\text{mol/L})| |
Initial |[\text{CO}], (\text{mol/L})| |
Initial |r, (\text{mol/L}{\cdot\text{s}})| |
|---|---|---|---|
|
Average values from Experiment 1 |
|1.00| |
|2.00| |
|0.0002| |
|
Average values from Experiment 2 |
|2.00| |
|2.00| |
|0.0008| |
|
Average values from Experiment 3 |
|2.00| |
|1.00| |
|0.0004| |
What is the reaction order with respect to each reactant?
The overall reaction order is the sum of individual orders determined relative to each aqueous or gaseous reactant.
In the previous example, it was determined that the order of the following reaction is |2| with respect to |\text{NO}_2| and |1| with respect to |\text{CO}.|
|\text{NO}_{2\text{(g)}} + \text{CO}_{\text{(g)}} → \text{NO}_{\text{(g)}} + \text{CO}_{2\text{(g)}}|
What is the overall reaction order?
The overall reaction order can be used to predict the effect of varying the concentrations of all reactants on the reaction rate. In the previous example, the overall reaction order is |3.| This means that when the concentration of all reactants doubles, the reaction rate increases by a factor of 8, as shown in the following procedure:
In the following formula, the ratio of initial concentration of reactants and the ratio of initial reaction rates are compared using the exponent |x,| which corresponds to the overall reaction order.
|\dfrac{r_2}{r_1}=\left(\dfrac{[Reactants]_2}{[Reactants]_1}\right)^x|
In this example, the known values are:
|\begin{align}\dfrac{[Reactants]_2}{[Reactants]_1} &= \dfrac{2}{1}=2\\x &= 3\end{align}|
Plug in the values and determine the ratio of reaction rates:
|\begin{align}\dfrac{r_2}{r_1}&=2^3\\\\\dfrac{r_2}{r_1} &= \dfrac{8}{1} = 8\end{align}|
The overall reaction order can be used to determine the units of the rate constant |(k),| as shown in this table.
The rate constant |(k)| is a proportionality constant between the reaction rate and the concentration of reactants for a given chemical reaction at a given temperature.
The rate constant is calculated based on the experimentally obtained values.
The rate constant is calculated by isolating it from the rate law.
|k = \dfrac{r}{[\text{A}]^m[\text{B}]^n}|
where
|k\!:| rate constant in varying units
|r\!:| reaction rate in |\text{mol/L}{\cdot\text{s}}|
|[\text{A},] [\text{B}]\!:| concentration of reactants in |\text{mol/L}|
|m, n\!:| reaction order with respect to the corresponding reactant
There are two ways to determine the units of the rate constant:
By isolating |k| from the rate law and simplifying the units.
By determining the overall reaction order and finding the corresponding units in the following table.
|
Overall reaction order |
Rate constant |(k)| units |
|---|---|
|
|0| |
|\text{mol/L}{\cdot\text{s}}| |
|
|1| |
|\text{s}^{-1}| |
|
|2| |
|\text{L/mol}{\cdot\text{s}}| |
|
|3| |
|\text{L}^2\text{/mol}^2{\cdot\text{s}}| |
The synthesis reaction between nitrogen dioxide gas |(\text{NO}_2)| and carbon monoxide gas |(\text{CO})| is conducted in a lab several times at a specific temperature:
||\text{NO}_{2\text{(g)}} + \text{CO}_{\text{(g)}} → \text{NO}_{\text{(g)}} + \text{CO}_{2\text{(g)}}.||
The initial concentrations of the reactants are recorded along with the initial reaction rates:
|
Experiment |
Initial |[\text{NO}_2], (\text{mol/L})| |
Initial |[\text{CO}], (\text{mol/L})| |
Initial |r, (\text{mol/L}{\cdot\text{s}})| |
|---|---|---|---|
|
Average values from Experiment 1 |
|1.00| |
|2.00| |
|0.0002| |
|
Average values from Experiment 2 |
|2.00| |
|2.00| |
|0.0008| |
|
Average values from Experiment 3 |
|2.00| |
|1.00| |
|0.0004| |
In previous examples, it was determined that the order of the following reaction is |2| with respect to |\text{NO}_2| and |1| with respect to |\text{CO}.| The overall reaction order is 3.
Determine the value of the rate constant |(k)| at this temperature.
The decomposition of |\text{N}_2\text{O}_5| is a first-order reaction:
||2\ \text{N}_2\text{O}_{5\text{(g)}}\rightarrow 4\ \text{N}\text{O}_{2\text{(g)}} + \text{O}_{2\text{(g)}}||
At a given temperature, the concentration of |\text{N}_2\text{O}_5| is measured at |0.020\ \text{mol/L}| and the reaction rate is |1.4 \times 10^{-4}\ \text{mol/L}{\cdot\text{s}}.| Determine the rate constant for this reaction at the given temperature.
The synthesis reaction between fluorine gas |(\text{F}_2)| and chlorine dioxide gas |(\text{ClO}_2)| is conducted in a lab several times at the same temperature:
||\text{F}_{2\text{(g)}} + \text{ClO}_{2\text{(g)}} → 2\ \text{FClO}_{2\text{(g)}}||
The initial concentrations of the reactants are recorded along with the initial reaction rates:
|
Experiment |
|[\text{F}_2], (\text{mol/L})| |
|[\text{ClO}_2], (\text{mol/L})| |
|r, (\text{mol/L}{\cdot\text{s}})| |
|---|---|---|---|
|
Average values from Experiment 1 |
|0.070| |
|0.080| |
|6.72\times 10^{-3}| |
|
Average values from Experiment 2 |
|0.070| |
|0.160| |
|13.44\times 10^{-3}| |
|
Average values from Experiment 3 |
|0.140| |
|0.080| |
|13.44\times 10^{-3}| |
Determine the value of the rate constant for this reaction.
The reaction from the previous example is conducted again at the same temperature. The concentration of |\text{F}_2| is |0.900\ \text{mol/L}| and the concentration of |\text{ClO}_2| is |1.20\ \text{mol/L}.| Determine the rate of the reaction.
A second-order chemical reaction with a single aqueous reactant occurs at a rate of |6.76\times10^{-4}\ \text{mol/L}{\cdot\text{s}}| when the rate constant is |0.040\ \text{L/mol}{\cdot\text{s}}.|
||\text{A}_{(aq)}+\text{B}_{(s)}\rightarrow\text{C}_{(aq)}+\text{D}_{(aq)}||
Determine the concentration of Reactant A.
Consider a chemical reaction with the following rate law: |r = k[\text{A}]^2[\text{B}]^2|
If the concentration of Reactant A is doubled and the concentration of Reactant B is tripled, how will the reaction rate change? Assume the temperature remains constant.