Library / Solution chemistry, equilibria and kinetics / SCH-39
Solution chemistry, equilibria and kinetics
Relaxation to equilibrium for reversible first-order conversion
Concentration of A in the closed, constant-volume reaction A ⇌ B at time t, which approaches the equilibrium value Aeq = kr·Ctot/(kf + kr) with rate constant kf + kr.
- Ref
- SCH-39
- Rev
- v1
- Field
- Solution chemistry, equilibria and kinetics
- Result
- A(t) [mol/L]
For A ⇌ B with forward and reverse first-order constants kf and kr, the total Ctot = A + B is fixed, and the equilibrium value is Aeq = kr Ctot/(kf + kr). The deviation from equilibrium decays exponentially with the rate constant kf + kr: A(t) = Aeq + (A₀ − Aeq) e^(−(kf + kr)t). The approach to equilibrium is therefore governed by the sum of the two rate constants, not by either alone.
Concentration of A in the closed, constant-volume reaction A ⇌ B at time t, which approaches the equilibrium value Aeq = kr·Ctot/(kf + kr) with rate constant kf + kr. Ctot = A + B is conserved.
Calculator
- Equilibrium concentration of A, kr Ctot/(kf + kr)0.1 mol/L
- Relaxation time constant, 1/(kf + kr)3.333 s
- Equilibrium constant, kf/kr2
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 5 s A has fallen from 0.3 to 0.1446 mol/L, on its way to 0.1 mol/L at equilibrium.
- The relaxation time constant is 3.333 s, so equilibrium is within 1 % after about 15.4 s; the ratio kf/kr = 2 is the equilibrium constant.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mol/L) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ctot Total concentration A + B | 0.3 to 5 mol/L | 0.1446 | 1.362 | 1.217 | +10% | +5.37 % |
| kf Forward rate constant | 1e-4 to 100 1/s | 2.997e-4 | 0.2999 | 0.2996 | +10% | −6.39 % |
| t Time | 0 to 1,000 s | 0.1 | 0.3 | 0.2 | +10% | −4.3 % |
| kr Reverse rate constant | 1e-4 to 100 1/s | 0.1104 | 0.2994 | 0.189 | +10% | +2.01 % |
| A0 Initial concentration of A | 0 to 0.3 mol/L | 0.07769 | 0.1446 | 0.06694 | +10% | +4.63 % |
Sweep of Time
| Time (s) | Concentration of A (mol/L) |
|---|---|
| 0 | 0.3 |
| 125 | 0.1 |
| 250 | 0.1 |
| 375 | 0.1 |
| 500 | 0.1 |
| 625 | 0.1 |
| 750 | 0.1 |
| 875 | 0.1 |
| 1,000 | 0.1 |
Sweep one variable
Variables and how each one contributes
Concentration of A
What the equation gives you. Everything else feeds it.
Forward rate constant
The forward rate constant, in 1/s. Speeds the relaxation and lowers the equilibrium amount of A. Raising it by 10 % (to 0.22) moves the result by −6.39 %, from 0.1446 to 0.1354 mol/L.
Reverse rate constant
The reverse rate constant, in 1/s. Speeds the relaxation and raises the equilibrium amount of A. Raising it by 10 % (to 0.11) moves the result by +2.01 %, from 0.1446 to 0.1475 mol/L.
Total concentration A + B
The total concentration of A + B, in mol/L. Raises the equilibrium and the instantaneous concentration in direct proportion. Raising it by 10 % (to 0.33) moves the result by +5.37 %, from 0.1446 to 0.1524 mol/L.
Initial concentration of A
The initial concentration of A, in mol/L (up to Ctot). Raises the starting point, so the whole curve rises while the early part stays above equilibrium. Raising it by 10 % (to 0.33) moves the result by +4.63 %, from 0.1446 to 0.1513 mol/L.
Time
The time, in s. A longer time brings A closer to its equilibrium value. Raising it by 10 % (to 5.5) moves the result by −4.3 %, from 0.1446 to 0.1384 mol/L.
Worked example with the values above
- Equilibrium: Aeq = kr Ctot/(kf + kr) = 0.1 × 0.3 / 0.3 = 0.1 mol/L.
- Decay factor: exp(−(kf + kr) t) = exp(−0.3 × 5) = 0.22313.
- A(t) = 0.1 + (0.3 − 0.1) × 0.22313 = 0.14463 mol/L.
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 the wrong sign of the stoichiometric coefficient for reactants and products.
- Forgetting that a pseudo-first-order constant includes the excess reactant’s concentration.
- Using rates from different times in a method of initial rates.
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)