First-Order Reaction Rate Constant Calculator
Result:
The calculated rate constant (k) will appear here.
Understanding First-Order Reactions and Rate Constants
In chemical kinetics, reactions are classified based on their order, which describes how the rate of the reaction depends on the concentration of the reactants. A first-order reaction is one where the reaction rate is directly proportional to the concentration of a single reactant raised to the power of one.
The Rate Law for a First-Order Reaction
For a general first-order reaction A → Products, the rate law is expressed as:
Rate = k[A]1
Where:
- 'Rate' is the speed at which the reaction proceeds.
- 'k' is the rate constant, a proportionality constant specific to the reaction at a given temperature.
- '[A]' is the molar concentration of reactant A.
Integrated Rate Law
To determine the rate constant 'k' from experimental data, we use the integrated rate law for a first-order reaction. This equation relates the concentration of the reactant at any given time to its initial concentration and the rate constant. It can be derived by integrating the differential rate law and is commonly expressed in two forms:
Natural Logarithm Form:
ln([A]t) = -kt + ln([A]0)
Where:
- [A]t is the concentration of reactant A at time 't'.
- [A]0 is the initial concentration of reactant A at time t=0.
- 'k' is the rate constant.
- 't' is the time elapsed.
Rearranging this equation to solve for 'k', we get:
k = (ln([A]0) – ln([A]t)) / t
Or, using the properties of logarithms:
k = ln([A]0 / [A]t) / t
Graphical Form:
The integrated rate law can also be written as:
ln([A]t) = -kt + ln([A]0)
This equation is in the form of a straight line (y = mx + c), where:
- y = ln([A]t)
- x = t
- m = -k (the slope of the line)
- c = ln([A]0) (the y-intercept)
By plotting ln([A]t) versus time 't', a first-order reaction will yield a straight line with a slope equal to -k.
Units of the Rate Constant (k)
For a first-order reaction, the rate constant 'k' has units of time-1. Common units include s-1, min-1, or hr-1, depending on the units used for time in the calculation.
Example Calculation
Consider a hypothetical first-order decomposition reaction where 0.1 M of a reactant is reduced to 0.05 M in 300 seconds. We want to calculate the rate constant (k).
- Initial Concentration ([A]0) = 0.1 M
- Final Concentration ([A]t) = 0.05 M
- Time Elapsed (t) = 300 s
Using the formula k = ln([A]0 / [A]t) / t:
k = ln(0.1 M / 0.05 M) / 300 s
k = ln(2) / 300 s
k ≈ 0.6931 / 300 s
k ≈ 0.00231 s-1
Therefore, the rate constant for this reaction under these conditions is approximately 0.00231 s-1.
This calculator helps you quickly determine the rate constant 'k' for a first-order reaction given initial concentration, final concentration, and the time elapsed.