Understanding and Calculating Rate Laws in Chemical Kinetics
Chemical kinetics is the study of reaction rates and mechanisms. A fundamental concept in this field is the rate law, which expresses how the rate of a chemical reaction depends on the concentration of the reactants.
What is a Rate Law?
For a general reaction: \(aA + bB \rightarrow cC + dD\)
The rate law is typically expressed in the form:
Rate = \(k[A]^m[B]^n\)
Where:
- Rate is the speed at which the reaction proceeds (usually in units of concentration per unit time, e.g., M/s).
- \(k\) is the rate constant, a proportionality constant specific to the reaction at a given temperature. Its units depend on the order of the reaction.
- [A] and [B] are the molar concentrations of reactants A and B, respectively.
- \(m\) and \(n\) are the reaction orders with respect to reactants A and B. These are typically small integers (0, 1, 2) or simple fractions, and they must be determined experimentally. They are NOT necessarily equal to the stoichiometric coefficients \(a\) and \(b\).
The overall reaction order is the sum of the individual orders: \(m + n\).
How to Determine Rate Laws Experimentally
Rate laws cannot be determined from the stoichiometry of a balanced chemical equation alone. They must be found through experiments. The most common method is the method of initial rates.
The Method of Initial Rates
This method involves running several experiments where the initial concentrations of reactants are systematically varied, and the initial rate of the reaction is measured for each set of conditions. By comparing how the initial rate changes when the concentration of one reactant is changed while others are held constant, we can determine the order with respect to that reactant.
Example: Determining Reaction Orders
Consider the reaction:
\(2NO(g) + O_2(g) \rightarrow 2NO_2(g)\)Suppose we conduct the following experiments and measure the initial rates:
| Experiment | Initial [NO] (M) | Initial [O2] (M) | Initial Rate (M/s) |
|---|---|---|---|
| 1 | 0.010 | 0.010 | 2.5 x 10-3 |
| 2 | 0.020 | 0.010 | 1.00 x 10-2 |
| 3 | 0.010 | 0.020 | 5.0 x 10-3 |
Calculating the Rate Law
To find the order with respect to NO (let's call it \(m\)), we compare experiments where [O2] is constant but [NO] changes.
- Compare Experiment 1 and Experiment 2: [O2] is constant (0.010 M). [NO] doubles from 0.010 M to 0.020 M. The rate increases from 2.5 x 10-3 M/s to 1.00 x 10-2 M/s (an increase by a factor of 4).
- Since Rate = \(k[NO]^m[O_2]^n\), when [NO] doubles and [O2] is constant, the rate is proportional to \((2)^m\).
- So, \(4 = 2^m\), which means \(m = 2\). The reaction is second order with respect to NO.
To find the order with respect to O2 (let's call it \(n\)), we compare experiments where [NO] is constant but [O2] changes.
- Compare Experiment 1 and Experiment 3: [NO] is constant (0.010 M). [O2] doubles from 0.010 M to 0.020 M. The rate doubles from 2.5 x 10-3 M/s to 5.0 x 10-3 M/s (an increase by a factor of 2).
- Since Rate = \(k[NO]^m[O_2]^n\), when [O2] doubles and [NO] is constant, the rate is proportional to \((2)^n\).
- So, \(2 = 2^n\), which means \(n = 1\). The reaction is first order with respect to O2.
The overall rate law is: Rate = \(k[NO]^2[O_2]^1\). The overall reaction order is \(2 + 1 = 3\).
Calculating the Rate Constant (\(k\))
Now that we have the rate law, we can use the data from any experiment to calculate the rate constant \(k\). Using data from Experiment 1:
Rate = \(k[NO]^2[O_2]^1\)
\(2.5 \times 10^{-3} \, M/s = k (0.010 \, M)^2 (0.010 \, M)^1\)
\(2.5 \times 10^{-3} \, M/s = k (0.00010 \, M^2) (0.010 \, M)\)
\(2.5 \times 10^{-3} \, M/s = k (1.0 \times 10^{-6} \, M^3)\)
\(k = \frac{2.5 \times 10^{-3} \, M/s}{1.0 \times 10^{-6} \, M^3}\)
\(k = 2.5 \times 10^3 \, M^{-2}s^{-1}\)
This calculator helps you determine the reaction orders and the rate constant using the method of initial rates, provided you have experimental data.
Rate Law Calculator (Method of Initial Rates)
Enter data from at least two experiments where one reactant concentration is changed while others are held constant. You will compare pairs of experiments to find the orders.
Experiment Pair 1 (For Reactant A):
M/sM
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M/s
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M
Note: For Experiment Pair 1, make sure [Reactant B] is constant between Exp 1 and Exp 2.
Experiment Pair 2 (For Reactant B):
M/sM
M
Note: For Experiment Pair 2, make sure [Reactant A] is constant between Exp 1 (or another suitable experiment) and Exp 3.
Calculate Rate Constant (k)
Enter the determined orders for Reactants A and B, and data from ONE experiment.
M/s
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M