Change in concentration of a reactant or product per unit time.
$$ \textrm{Rate}=\frac{\textrm{conc. of A at }t_2 - \textrm{conc. of A at }t_1}{t_2 - t_1} $$
$$ \textrm{Rate}=\frac{\Delta [\textrm{A}]}{\Delta t} $$
\( [\mathrm{A}] \) means concentration of A in mol/L
A is the reactant or product being considered
The Decomposition of Nitrogen Dioxide
\( \chem{2NO_2(g) \rightarrow 2NO(g) + O_2(g)} \) (at \( 300^\circ \textrm{C}\))
Time (sec)
\( [\mathrm{NO}_2] \) (mol/L)
\( [\mathrm{NO}] \) (mol/L)
\( [\mathrm{O}_2] \) (mol/L)
0
0.0100
0
0
50
0.0079
0.0021
0.0011
100
0.0065
0.0035
0.0018
150
0.0055
0.0045
0.0023
200
0.0048
0.0052
0.0026
250
0.0043
0.0057
0.0029
350
0.0034
0.0066
0.0033
400
0.0031
0.0069
0.0035
The Decomposition of Nitrogen Dioxide
Instantaneous Rate
Value of the rate at a particular time.
Can be obtained by computing the slope of a line tangent to the curve at that point.
Rate Law
Shows how the rate depends on the concentrations of reactants.
For the decomposition of nitrogen dioxide:
$$ \chem{2NO_2(g)\rightarrow 2NO(g)+O_2(g)} $$
$$ \mathrm{Rate}=k\conc{NO_2}^2 $$
\(k\) is the rate constant
\(n\) is the order of the reactant
The concentrations of the products do not appear in the rate law because the reaction rate is being studied under conditions where the reverse reaction does not contribute to the overall rate.
The value of the exponent \(n\) must be determined by experiment; it cannot be written from the balanced equation.
Types of Rate Laws
Differential Rate Law (rate law) - shows how the rate of a reaction depends on concentrations.
Integrated Rate Law - shows how the concentrations of species in the reaction depend on time.
Rate Laws: A Summary
Because we typically consider reactions only under conditions where the reverse reaction is unimportant, our rate laws will involve only concentrations of reactants.
Because the differential and integrated rate laws for a given reaction are related in a well–defined way, the experimental determination of either of the rate laws is sufficient.
Experimental convenience usually dictates which type of rate law is determined experimentally.
Knowing the rate law for a reaction is important mainly because we can usually infer the individual steps involved in the reaction from the specific form of the rate law.
What does it mean to determine the form of the rate law?
Determine experimentally the power to which each reactant concentration must be raised in the rate law.
Method of Initial Rates
The value of the initial rate is determined for each experiment at the same value of t as close to t = 0 as possible.
Several experiments are carried out using different initial concentrations of each of the reactants, and the initial rate is determined for each run.
The results are then compared to see how the initial rate depends on the initial concentrations of each of the reactants.
Overall Reaction Order
The sum of the exponents in the reaction rate equation.
$$ \mathrm{Rate}=k\conc{A}^n\conc{B}^m $$
$$\text{Overall reaction order} = n+m $$
\(k\) is the rate constant
\(\conc{A}\) is the concentration of reactant A
\(\conc{B}\) is the concentration of reactant B
First-Order
\(\mathrm{Rate}=k\conc{A}\)
Integrated:
$$ \ln\conc{A}=-kt+\ln\conc{A}_0 $$
\(\conc{A}\) is the concentration of A at time t
\(k\) is the rate constant
\(t\) is the time
\(\conc{A}_0\) is the initial concentration of A
Plot of \(\ln\conc{N_2O_5}\) versus Time
\(\ln\conc{N_2O_5}\)
Time(s)
-2.303
0
-2.649
50
-2.996
100
-3.689
200
-4.382
300
-5.075
400
Half-Life in First-Order Reactions
Time required for a reactant to reach half its original concentration
Half-life:
$$ t_\bfrac{1}{2} = \frac{0.693}{k} $$
where \(k\) is the rate constant
Half-life does not depend on the concentration of reactants.
Example Problem
A first order reaction is 35% complete at teh end of 55 minutes. What is the value of \(k\)?