Library / Foundations: Chemistry / FCH-34
Foundations: Chemistry
Rate constant at a second temperature (Arrhenius)
Rate constant at one temperature from its value at another, for the same mechanism, assuming the activation energy and the pre-exponential factor stay constant over the range.
- Ref
- FCH-34
- Rev
- v1
- Field
- Foundations: Chemistry
- Result
- k2
The Arrhenius equation k = A exp(−Ea/RT) links the rate constant to temperature. Taking the ratio at two temperatures cancels the pre-exponential factor A: k₂ = k₁ exp[(Ea/R)(1/T₁ − 1/T₂)], with R = 8.314 J/(mol K). Temperatures are absolute (kelvin) and Ea is in kJ/mol. A higher temperature raises the rate constant, more steeply when Ea is large.
Rate constant at one temperature from its value at another, for the same mechanism, assuming the activation energy and the pre-exponential factor stay constant over the range.
Calculator
- Rate ratio, k₂/k₁1.926
- Temperature rise, T₂ − T₁10 K
- Activation energy that would just double the rate over this interval52.9 kJ/mol
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.
- Warming from 298 K to 308 K multiplies the rate constant by 1.926 for Ea = 50 kJ/mol, so k rises from 0.01 to 0.01926.
- A rule of thumb says a 10 K rise doubles the rate; here a 10 K rise gives a factor of 1.93, which works for activation energies near 50 kJ/mol at room temperature but is not general: it depends on Ea and on T.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| T2 Second temperature | 200 to 1,000 K | 5.077e-7 | 14,200 | 14,200 | +10% | +490 % |
| T1 First temperature | 200 to 1,000 K | 1.356e-8 | 379.2 | 379.2 | +10% | −84 % |
| k1 Rate constant at the first temperature | 1e-6 to 100 | 1.926e-6 | 192.6 | 192.6 | +10% | +10 % |
| Ea Activation energy | 5 to 300 kJ/mol | 0.01068 | 0.5097 | 0.499 | +10% | +6.77 % |
Sweep of Second temperature
| Second temperature (K) | Rate constant at T₂ |
|---|---|
| 200 | 5.077e-7 |
| 300 | 0.01144 |
| 400 | 1.717 |
| 500 | 34.73 |
| 600 | 257.8 |
| 700 | 1,079 |
| 800 | 3,158 |
| 900 | 7,281 |
| 1,000 | 14,200 |
Sweep one variable
Variables and how each one contributes
Rate constant at T₂
What the equation gives you. Everything else feeds it.
Rate constant at the first temperature
The rate constant at the first temperature, in any unit; the result has the same unit. Raises k₂ in direct proportion, because k₂ is a multiple of k₁. Raising it by 0.005 (to 0.015) moves the result by +50 %, from 0.01926 to 0.02888.
Activation energy
The activation energy, in kJ/mol. Raises k₂ steeply when T₂ is above T₁, because the factor grows exponentially with Ea. Raising it by 5 kJ/mol (to 55 kJ/mol) moves the result by +6.77 %, from 0.01926 to 0.02056.
First temperature
The first temperature, in kelvin, at which k₁ was measured. Lowers k₂: with k₁ fixed, a warmer starting point leaves a smaller gap to T₂. Raising it by 2 K (to 300 K) moves the result by −12.6 %, from 0.01926 to 0.01683.
Second temperature
The second temperature, in kelvin, at which k₂ is wanted. Raises k₂: a higher temperature makes the reaction faster. Raising it by 2 K (to 310 K) moves the result by +13.4 %, from 0.01926 to 0.02184.
Worked example with the values above
- Exponent: (Ea/R)(1/T₁ − 1/T₂) = (50,000 / 8.3145) × (1/298 − 1/308) = 0.6552.
- Factor: e0.6552 = 1.926.
- k₂ = 0.01 × 1.926 = 0.01926.
Assumptions and limits
- The rate constant follows the Arrhenius law with Ea constant over the temperature range.
- Absolute temperatures in kelvin and the same reaction mechanism at both.
Common mistakes
- Using degrees Celsius in place of kelvin.
- Entering Ea in J/mol when kJ/mol is expected, or vice versa.
- Extrapolating far beyond the temperature range where Ea was measured.
Statistics and uncertainty
The temperature appears in an exponent, so an error of 1 K at 300 K in a temperature difference changes the factor by about Ea/(RT²) per kelvin, about 7 % per K for 50 kJ/mol near 300 K. Activation energies from only two temperatures carry large uncertainty; a straight-line fit of ln k against 1/T over several temperatures is much better.
Where it is used
- Predicting how fast a reaction, ageing process or degradation proceeds at a different temperature.
- Estimating the shelf life of propellants and energetic materials at storage temperature.
Maths you need
- Activation energy: the energy barrier that must be overcome
- Absolute temperature: in kelvin, starting at absolute zero