Library / Solution chemistry, equilibria and kinetics / SCH-40
Solution chemistry, equilibria and kinetics
Product fraction for parallel first-order reactions
Fraction of the products that is B when A goes to B and to C by competing irreversible first-order routes, with rate constants kB and kC.
- Ref
- SCH-40
- Rev
- v1
- Field
- Solution chemistry, equilibria and kinetics
- Result
- fB
When A forms B and C by competing irreversible first-order reactions, both disappear processes follow the same exponential, so the ratio of products is fixed by the rate constants: the fraction of the product that is B is kB/(kB + kC), at all times. It does not depend on the initial concentration or on time. If some product was present initially, subtract it first. The overall decay of A is first-order with constant kB + kC.
Fraction of the products that is B when A goes to B and to C by competing irreversible first-order routes, with rate constants kB and kC. It does not depend on time, concentration or the order of the reactions in A.
Calculator
- Fraction as C, kC/(kB + kC)0.25
- Selectivity ratio, kB/kC3
- Overall decay constant of A, kB + kC0.4 1/s
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.
- 75 % of the A that reacts becomes B and 25 % becomes C, in a ratio of 3 to 1, at any time.
- Because both routes are first-order, changing the concentration does not change the product ratio; only the temperature, which changes kB and kC differently, can.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| kB Rate constant to B | 1e-4 to 100 1/s | 9.99e-4 | 0.999 | 0.998 | +10% | +2.33 % |
| kC Rate constant to C | 1e-4 to 100 1/s | 0.002991 | 0.9997 | 0.9967 | +10% | −2.44 % |
Sweep of Rate constant to B
| Rate constant to B (1/s) | Fraction as B |
|---|---|
| 1e-4 | 9.99e-4 |
| 5.623e-4 | 0.005592 |
| 0.003162 | 0.03065 |
| 0.01778 | 0.151 |
| 0.1 | 0.5 |
| 0.5623 | 0.849 |
| 3.162 | 0.9693 |
| 17.78 | 0.9944 |
| 100 | 0.999 |
Sweep one variable
Variables and how each one contributes
Fraction as B
What the equation gives you. Everything else feeds it.
Rate constant to B
The rate constant for the route to B, in 1/s. A larger kB raises the share of B. Raising it by 10 % (to 0.33) moves the result by +2.33 %, from 0.75 to 0.7674.
Rate constant to C
The rate constant for the route to C, in 1/s. A larger kC lowers the share of B. Raising it by 10 % (to 0.11) moves the result by −2.44 %, from 0.75 to 0.7317.
Worked example with the values above
- Total rate constant: kB + kC = 0.3 + 0.1 = 0.4 1/s.
- Share of B: kB divided by that.
- fB = 0.3 / 0.4 = 0.75.
Assumptions and limits
- Constant volume, a well-mixed closed batch at fixed temperature.
- The simple rate law stated, with the same mechanism over the range studied.
Common mistakes
- Using this for reversible reactions, where the final ratio comes from the equilibrium.
- Mixing up the rate constants of the two routes.
- Using it when the reaction orders in A differ between the two routes.
Statistics and uncertainty
Rate constants are normally known to 5 % to 10 % and change by about 7 % per kelvin near room temperature for a typical activation energy, so temperature control matters.
Where it is used
- Finding reaction orders and rate constants from experiments.
- Predicting selectivity and relaxation to equilibrium.
Maths you need
- Rate law: rate = k[A]^m[B]^n
- Stoichiometric coefficient: ν, negative for reactants
- First-order decay: c = c₀ e^(−kt)