Library / Solution chemistry, equilibria and kinetics / SCH-32
Solution chemistry, equilibria and kinetics
Complex concentration when both reactants are depleted
Complex concentration from the total metal Mt, total ligand Lt and dissociation constant Kd when binding uses up a significant part of both.
- Ref
- SCH-32
- Rev
- v1
- Field
- Solution chemistry, equilibria and kinetics
- Result
- [ML] [mol/L]
When binding uses up a significant part of both reactants, solve the mass balances Mt = M + ML and Lt = L + ML with Kd = ML-free product ratio, Kd = [M][L]/[ML]. The quadratic has the physical root [ML] = 2MtLt/(B + √((Mt − Lt)² + 2Kd(Mt + Lt) + Kd²)), with B = Mt + Lt + Kd. It reduces to the free-ligand shortcut when one reactant is in large excess.
Complex concentration from the total metal Mt, total ligand Lt and dissociation constant Kd when binding uses up a significant part of both. Here B = Mt + Lt + Kd. This is the physical root of the quadratic mass balance.
Calculator
- Free metal, Mt − [ML]0.001 mol/L
- Free ligand, Lt − [ML]0.002 mol/L
- Dissociation quotient [M][L]/[ML] (equals Kd)0.002 mol/L
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.
- The complex reaches 0.001 mol/L, leaving 0.001 mol/L of free metal and 0.002 mol/L of free ligand, which reproduces Kd.
- 50 % of the limiting reactant is bound; the free-ligand shortcut would overestimate the complex because it ignores the depletion of ligand.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mol/L) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Mt Total metal concentration | 1e-6 to 1 mol/L | 6e-7 | 0.002994 | 0.002993 | +10% | +7.81 % |
| Lt Total ligand concentration | 1e-6 to 1 mol/L | 4.999e-7 | 0.001996 | 0.001996 | +10% | +5.72 % |
| Kd Dissociation constant | 1e-9 to 1 mol/L | 5.97e-6 | 0.002 | 0.001994 | +10% | −3.82 % |
Sweep of Dissociation constant
| Dissociation constant (mol/L) | Complex concentration (mol/L) |
|---|---|
| 1e-9 | 0.002 |
| 1.334e-8 | 0.002 |
| 1.778e-7 | 0.002 |
| 2.371e-6 | 0.001995 |
| 3.162e-5 | 0.001942 |
| 4.217e-4 | 0.00155 |
| 0.005623 | 5.985e-4 |
| 0.07499 | 7.508e-5 |
| 1 | 5.97e-6 |
Sweep one variable
Variables and how each one contributes
Complex concentration
What the equation gives you. Everything else feeds it.
Total metal concentration
The total metal concentration, in mol/L. Raises the complex concentration, with saturation as the ligand runs out. Raising it by 10 % (to 0.0022) moves the result by +7.81 %, from 0.001 to 0.001078 mol/L.
Total ligand concentration
The total ligand concentration, in mol/L. Raises the complex concentration, with saturation as the metal runs out. Raising it by 10 % (to 0.0033) moves the result by +5.72 %, from 0.001 to 0.001057 mol/L.
Dissociation constant
The dissociation constant, in mol/L. A larger Kd means weaker binding, so less complex forms. Raising it by 10 % (to 0.0022) moves the result by −3.82 %, from 0.001 to 9.618×10−4 mol/L.
Worked example with the values above
- B = Mt + Lt + Kd = 0.007; root term = 0.005.
- Numerator: 2 Mt Lt = 1.2×10−5; denominator: 0.012.
- [ML] = 0.001 mol/L.
Assumptions and limits
- A 1:1 complex, or a single neutral species partitioning, with constants in concentration units.
- Equilibrium reached in each step, and no side reactions unless the conditional constant includes them.
Common mistakes
- Using the free-ligand shortcut when binding depletes the ligand.
- Using the association constant in place of the dissociation constant.
- Choosing the wrong root of the quadratic, which gives a negative or too large complex concentration.
Statistics and uncertainty
Formation constants depend on the medium and the temperature, and a quoted constant for a different ionic strength can be off by a factor of two or more. Extraction results are sensitive to pH near the pKa.
Where it is used
- Designing complexometric titrations and metal-binding experiments.
- Planning solvent extractions and pH-controlled separations.
Maths you need
- Formation constant: Kf = [ML]/([M][L])
- Partition coefficient: the ratio of a species in two phases
- Mass balance: total equals free plus bound