Calculator Inputs
Example Data Table
Example dataset for a reactant with A₀ = 1.0000 mol/L and k = 0.2300 h⁻¹.
| Time (h) | Concentration (mol/L) | ln(Concentration) | Percent Remaining |
|---|---|---|---|
| 0 | 1.0000 | 0.0000 | 100.00% |
| 1 | 0.7945 | -0.2300 | 79.45% |
| 2 | 0.6313 | -0.4600 | 63.13% |
| 3 | 0.5016 | -0.6900 | 50.16% |
| 4 | 0.3985 | -0.9200 | 39.85% |
Formula Used
Integrated first-order law: Aₜ = A₀e-kt
Linearized form: ln(Aₜ) = ln(A₀) - kt
Rate constant from one point: k = ln(A₀ / Aₜ) / t
Time to reach a target concentration: t = ln(A₀ / Aₜ) / k
Half-life: t½ = ln(2) / k
Fraction remaining: Aₜ / A₀ = e-kt
Fraction converted: 1 - e-kt
First-order kinetics means the reaction rate is directly proportional to the current reactant concentration. A constant fraction disappears in equal time intervals, so the concentration decays exponentially while the natural log of concentration decreases linearly with time.
How to Use This Calculator
- Select the calculation mode that matches your known variables.
- Choose the time unit that matches your rate constant data.
- Enter concentration values using one consistent unit label.
- Provide the rate constant, time, or concentration inputs required by your chosen mode.
- Set your preferred precision and graph resolution.
- Click Calculate Kinetics to display the result section above the form.
- Review the summary table, decay graph, and derived kinetic quantities.
- Use the CSV or PDF buttons to export the calculated results or the example dataset.
Frequently Asked Questions
1. What defines a first-order reaction?
A first-order reaction has a rate proportional to one reactant concentration. If concentration halves, the instantaneous rate also halves. The integrated plot of ln(concentration) versus time becomes a straight line with slope -k.
2. Why is the graph curved for concentration?
Concentration decays exponentially in first-order kinetics, so the raw concentration curve bends downward. The same data become linear after taking the natural logarithm, which is why kinetic analysis often uses ln(concentration) versus time.
3. Why is half-life constant in first-order kinetics?
The half-life depends only on the rate constant, not on the starting concentration. That is why every successive halving takes the same amount of time, a distinctive feature of first-order decay.
4. Which units should I use for k?
Use inverse time units matching the time unit in your calculation. If time is in hours, k should be h⁻¹. If time is in seconds, k should be s⁻¹.
5. Can I use any concentration unit?
Yes, as long as you stay consistent. The formulas rely on concentration ratios, so mol/L, mg/L, mmol/L, or ppm all work if A₀ and Aₜ share the same unit.
6. What happens if my target concentration exceeds the initial one?
For a simple decay process, the target concentration must be less than the initial concentration. A larger value would imply growth, external input, measurement error, or a different kinetic model.
7. What does fraction converted mean?
Fraction converted is the portion of reactant that has disappeared by time t. It equals 1 minus the fraction remaining. Multiply by 100 to express conversion as a percentage.
8. When should I use the initial concentration mode?
Use it when you know a later concentration, the elapsed time, and the rate constant, but the starting concentration was not recorded. The calculator reconstructs the original value using exponential decay reversal.