Library / Solution chemistry, equilibria and kinetics / SCH-41
Solution chemistry, equilibria and kinetics
Intermediate in consecutive first-order reactions
Concentration of the intermediate B in A → B → C at time t, starting from pure A.
- Ref
- SCH-41
- Rev
- v1
- Field
- Solution chemistry, equilibria and kinetics
- Result
- B(t) [mol/L]
In A → B → C with first-order steps and only A at the start, the intermediate first rises and then falls: B(t) = A₀ k₁ [e^(−k₁t) − e^(−k₂t)]/(k₂ − k₁). If k₁ = k₂ the formula becomes A₀ k₁ t e^(−k₁t). The calculator factors out the slower exponential, so it stays accurate when the two rate constants are close to each other.
Concentration of the intermediate B in A → B → C at time t, starting from pure A. When the two rate constants are equal the result is A₀k₁t·exp(−k₁t). The calculation is arranged to stay accurate when the two rates are nearly equal.
Calculator
- Concentration of A, A₀ e^(−k₁t)0.1213 mol/L
- Concentration of C, A₀ − A − B0.03096 mol/L
- Time of the maximum of B6.931 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.
- At t = 5 s the mixture holds 0.1213 mol/L of A, 0.04773 mol/L of B and 0.03096 mol/L of C.
- The three amounts always sum to A₀ = 0.2 mol/L, a check on the calculation.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mol/L) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A0 Initial concentration of A | 0.001 to 5 mol/L | 2.387e-4 | 1.193 | 1.193 | +10% | +10 % |
| k1 Rate constant A → B | 1e-4 to 100 1/s | 6.319e-5 | 0.09523 | 0.09517 | +10% | +7.07 % |
| k2 Rate constant B → C | 1e-4 to 100 1/s | 1.214e-4 | 0.07867 | 0.07855 | +10% | −4.44 % |
| t Time | 0 to 1,000 s | 0 | 0.04773 | 0.04773 | +10% | +2.27 % |
Sweep of Time
| Time (s) | Intermediate concentration (mol/L) |
|---|---|
| 0 | 0 |
| 125 | 7.453e-7 |
| 250 | 2.778e-12 |
| 375 | 1.035e-17 |
| 500 | 3.857e-23 |
| 625 | 1.438e-28 |
| 750 | 5.357e-34 |
| 875 | 1.996e-39 |
| 1,000 | 7.44e-45 |
Sweep one variable
Variables and how each one contributes
Intermediate concentration
What the equation gives you. Everything else feeds it.
Initial concentration of A
The initial concentration of A, in mol/L. Raises the intermediate in direct proportion. Raising it by 10 % (to 0.22) moves the result by +10 %, from 0.04773 to 0.0525 mol/L.
Rate constant A → B
The rate constant of A → B, in 1/s. A larger k₁ forms the intermediate faster, so it rises at early times. Raising it by 10 % (to 0.11) moves the result by +7.07 %, from 0.04773 to 0.05111 mol/L.
Rate constant B → C
The rate constant of B → C, in 1/s. A larger k₂ drains the intermediate faster, so its peak is lower. Raising it by 10 % (to 0.22) moves the result by −4.44 %, from 0.04773 to 0.04561 mol/L.
Time
The time, in s. The intermediate rises, peaks and then falls with time. Raising it by 10 % (to 5.5) moves the result by +2.27 %, from 0.04773 to 0.04882 mol/L.
Worked example with the values above
- Exponentials: e^(−k₁t) = 0.60653, e^(−k₂t) = 0.36788.
- Prefactor: A₀ k₁/(k₂ − k₁) = 0.2.
- B(t) = 0.04773 mol/L.
Assumptions and limits
- Irreversible first-order steps for the consecutive scheme, and a single substrate with a steady-state intermediate for the enzyme.
- Constant temperature, pH and volume, with negligible product and substrate depletion for initial rates.
Common mistakes
- Swapping k₁ and k₂ in the formula.
- Using the formula for reversible steps.
- Using it when the intermediate is present at the start.
Statistics and uncertainty
Fitted enzyme constants, Km and Vmax, carry uncertainties of 10 % to 30 % from the data scatter, and a wide range of substrate concentrations is needed to define both.
Where it is used
- Following intermediates in sequential reactions and decay chains.
- Describing enzyme saturation and inhibition.
Maths you need
- Integrated rate law: concentration as a function of time
- Steady state: the intermediate’s concentration stays constant
- Michaelis constant: Km, the substrate concentration at half of Vmax