Library / Chemical / CHE-04
Chemical
First-order batch kinetics
Concentration remaining after time for a first-order reaction.
- Ref
- CHE-04
- Rev
- v1
- Field
- Chemical
- Result
- CA [mol/m³]
CA concentration at time t, CA0 the initial concentration and k the first-order rate constant in 1/s. Constant volume and temperature; irreversible first-order consumption.
In a first-order reaction the rate is proportional to what is left, so the concentration falls by the same fraction in every equal time interval.
Calculator
- Half-life ln2 / k69.31 s
- Fraction converted69.88 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- After one half-life (ln 2 / k) half remains; after 3.3 half-lives about 10%.
- The half-life of a first-order reaction does not depend on the starting concentration.
- Because k depends strongly on temperature, a small temperature change shifts the result a lot (see Arrhenius).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mol/m³) | Local step | Change in result |
|---|---|---|---|---|---|---|
| CA0 Initial concentration | 1 to 1,000 mol/m³ | 0.3012 | 301.2 | 300.9 | +10% | +10 % |
| k Rate constant | 5e-4 to 0.1 1/s | 6.144e-4 | 94.18 | 94.18 | +10% | −11.3 % |
| t Time | 10 to 3,600 s | 2.32e-14 | 90.48 | 90.48 | +10% | −11.3 % |
Sweep of Time
| Time (s) | Concentration (mol/m³) |
|---|---|
| 10 | 90.48 |
| 20.87 | 81.16 |
| 43.56 | 64.69 |
| 90.91 | 40.29 |
| 189.7 | 15 |
| 396 | 1.906 |
| 826.5 | 0.02574 |
| 1,725 | 3.227e-6 |
| 3,600 | 2.32e-14 |
Sweep one variable
Variables and how each one contributes
Concentration
What the equation gives you. Everything else feeds it.
Initial concentration
Concentration at the start. Directly proportional.
Rate constant
First-order rate constant (changes with temperature). Raises the decay rate.
Time
Elapsed time. Decay is exponential in time.
Worked example with the values above
- Exponent: kt = 0.01 × 120 = 1.2.
- Remaining fraction: e−1.2 = 0.3012.
- CA = 100 × 0.3012 = 30.12 mol/m³.
Assumptions and limits
- First-order, irreversible consumption.
- Constant volume and temperature.
- Well-mixed batch.
Common mistakes
- Using this for second-order reactions.
- Mixing minutes and seconds between k and t.
- Forgetting temperature dependence of k.
Statistics and uncertainty
Rate constants from experiments commonly carry ±5–10% error, and k appears in the exponent. For kt = 1.2, a 10% error in k changes the remaining concentration by about 13% (multiplied by e0.12).
Where it is used
- Shelf-life and degradation of propellants and explosives.
- Decontamination and agent decay.
- Batch reactor design.
Maths you need
- Exponentials: e^(−x)
- Rates: rate of change of concentration