Rate Constant (k) Calculator
Understanding the Rate Constant (k)
In chemical kinetics, the rate constant, denoted by k, is a proportionality constant that relates the rate of a chemical reaction to the concentration of the reactants. It is a crucial parameter that quantics the speed at which a reaction proceeds at a given temperature. The units of the rate constant depend on the overall order of the reaction.
Rate Laws and Reaction Order
The rate law for a reaction expresses the relationship between the reaction rate and the concentrations of reactants. For a general reaction:
aA + bB → Products
The rate law is often expressed as:
Rate = k[A]m[B]n
Where:
- Rate is the speed of the reaction (e.g., in M/s).
- k is the rate constant.
- [A] and [B] are the molar concentrations of reactants A and B.
- m and n are the partial orders of the reaction with respect to reactants A and B. The overall reaction order is the sum of the partial orders (m + n).
The order of a reaction (zero, first, second, etc.) dictates how the rate changes with the concentration of reactants.
Integrated Rate Laws and Calculating k
While the rate law relates instantaneous rate to concentration, integrated rate laws relate concentration to time. These integrated rate laws can be used to determine the rate constant k if we know the initial concentration, the concentration at a later time, and the time elapsed.
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Zero-Order Reaction: [A]t = –kt + [A]0
Rearranging for k: k = ([A]0 – [A]t) / t -
First-Order Reaction: ln[A]t = –kt + ln[A]0
Rearranging for k: k = (ln[A]0 – ln[A]t) / t -
Second-Order Reaction: 1/[A]t = kt + 1/[A]0
Rearranging for k: k = (1/[A]t – 1/[A]0) / t
Where:
- [A]0 is the initial concentration of reactant A.
- [A]t is the concentration of reactant A at time t.
- t is the time elapsed.
How the Calculator Works
This calculator allows you to input the initial and final concentrations of a reactant, the time elapsed, and the order of the reaction. Based on these values and the appropriate integrated rate law, it calculates and displays the rate constant k along with its correct units.
Example Calculation
Let's consider a second-order reaction where the initial concentration of a reactant is 0.50 M. After 60 seconds, the concentration has decreased to 0.25 M.
- Initial Concentration ([A]0): 0.50 M
- Final Concentration ([A]t): 0.25 M
- Time Elapsed (t): 60 s
- Reaction Order: Second Order
Using the integrated rate law for a second-order reaction: k = (1/[A]t – 1/[A]0) / t
k = (1/0.25 M – 1/0.50 M) / 60 s
k = (4.0 M-1 – 2.0 M-1) / 60 s
k = 2.0 M-1 / 60 s
k = 0.033 M-1s-1
The rate constant for this reaction is 0.033 M-1s-1.