C8863 Free Energy Calculations -1-4. Chemical Equilibrium - Exp. methods
C8863 Free Energy Calculations
Petr Kulhánek
kulhanek@chemi.muni.cz
National Centre for Biomolecular Research, Faculty of Science
Masaryk University, Kamenice 5, CZ-62500 Brno
JS/2022 Present Form of Teaching: Rev1
Lesson 4
Chemical Equilibrium - Experimental Methods
C8863 Free Energy Calculations -2-4. Chemical Equilibrium - Exp. methods
Overview
C8863 Free Energy Calculations -3-4. Chemical Equilibrium - Exp. methods
Revisions
▪ At the given temperature and definition of the standard state, the equilibrium constant
is determined only by the standard reaction Gibbs energy:
▪ The equilibrium constant K is proportional to activities of all compounds in the
equilibrium.
▪ For ideal (diluted) solutions, activities can be approximated by molar concentrations:
Δ𝐺𝑟
0
= −𝑅𝑇 ln 𝐾
𝐾 = ෑ
𝑖=1
𝑁
𝑎 𝑟,𝑖
𝜐 𝑖
𝐾 ≈ ෑ
𝑖=1
𝑁
[𝑋𝑖] 𝑟
𝜐 𝑖
Sign convention for stochiometric coefficients ni
products (end state) - positive value
reactants (initial state) - negative value
C8863 Free Energy Calculations -4-4. Chemical Equilibrium - Exp. methods
Equilibrium
multiple chemical processes
C8863 Free Energy Calculations -5-4. Chemical Equilibrium - Exp. methods
Complex Chemical Mixtures
Composition of the chemical system with multiple reactions is determined by a system of
equations. These equations include
➢ each equilibrium process
➢ balance of all reacting compounds
A + 2B AB2
2A + C A2C
𝐾1 =
𝐴𝐵2
𝐴 𝐵 2
𝐾2 =
𝐴2 𝐶
𝐴 2 𝐶
Example:
C8863 Free Energy Calculations -6-4. Chemical Equilibrium - Exp. methods
Complex Chemical Mixtures
Composition of the chemical system with multiple reactions is determined by a system of
equations. These equations include
➢ each equilibrium process
➢ balance of all reacting compounds
A + 2B AB2
2A + C A2C
𝐾1 =
𝐴𝐵2
𝐴 𝐵 2
𝐾2 =
𝐴2 𝐶
𝐴 2 𝐶
Example:
𝑐0,𝐴 = 𝐴 + 𝐴𝐵2 + 2 𝐴2 𝐶
𝑐0,𝐵 = 𝐵 + 2 𝐴𝐵2
𝐴 , 𝐵 , 𝐶 , 𝐴𝐵2 , 𝐴2 𝐶
Unknowns:
→ 5 equations 𝑐0,𝐶 = 𝐶 + 𝐴2 𝐶
balance
equilibria
initial amount
C8863 Free Energy Calculations -7-4. Chemical Equilibrium - Exp. methods
Numerical Solution I
A + 2B AB2
2A + C A2C
𝐾1 =
𝐴𝐵2
𝐴 𝐵 2
𝐾2 =
𝐴2 𝐶
𝐴 2 𝐶
Example:
𝑐0,𝐴 = 𝐴 + 𝐴𝐵2 + 2 𝐴2 𝐶
𝑐0,𝐵 = 𝐵 + 2 𝐴𝐵2
𝐴 , 𝐵 , 𝐶 , 𝐴𝐵2 , 𝐴2 𝐶
Unknowns:
→ 5 equations 𝑐0,𝐶 = 𝐶 + 𝐴2 𝐶
balance
equilibria
initial amount
Only two components are independent:
• five components
• three balances
C8863 Free Energy Calculations -8-4. Chemical Equilibrium - Exp. methods
𝐴2 𝐶 =
1
2
𝑐0,𝐴 −
1
2
𝐴 −
1
2
𝐴𝐵2
Numerical Solution I, cont.
𝐾1 =
𝐴𝐵2
𝐴 𝐵 2
𝐾2 =
𝐴2 𝐶
𝐴 2 𝐶
𝑐0,𝐴 = 𝐴 + 𝐴𝐵2 + 2 𝐴2 𝐶
𝑐0,𝐵 = 𝐵 + 2 𝐴𝐵2
𝑐0,𝐶 = 𝐶 + 𝐴2 𝐶
0 = log 𝐴𝐵2 − log 𝐴 − 2 log 𝐵 − log(𝐾1)
0 = log 𝐴2 𝐶 − 2log 𝐴 − log 𝐶 − log(𝐾2)
Find [A] and [B] such that the last two equations are satisfied:
𝐴𝐵2 =
1
2
𝑐0,𝐵 −
1
2
𝐵
𝐶 = 𝑐0,𝐶 − 𝐴2 𝐶
1. Determine dependent parameters:
2. Solve system of independent equations:
Octave, Matlab: lsqnonlin
𝑓 𝑿 = 𝟎
C8863 Free Energy Calculations -9-4. Chemical Equilibrium - Exp. methods
Numerical Solution II
Find concentration of all components such that all equations are satisfied:
1. Solve system of equations:
𝐾1 =
𝐴𝐵2
𝐴 𝐵 2
𝐾2 =
𝐴2 𝐶
𝐴 2 𝐶
0 = log 𝐴𝐵2 − log 𝐴 − 2 log 𝐵 − log(𝐾1)
0 = log 𝐴2 𝐶 − 2log 𝐴 − log 𝐶 − log(𝐾2)
𝐴 , 𝐵 , 𝐶 , 𝐴𝐵2 , 𝐴2 𝐶
𝑐0,𝐴 = 𝐴 + 𝐴𝐵2 + 2 𝐴2 𝐶
𝑐0,𝐵 = 𝐵 + 2 𝐴𝐵2
𝑐0,𝐶 = 𝐶 + 𝐴2 𝐶
0 = 𝐴 + 𝐴𝐵2 + 2 𝐴2 𝐶 − 𝑐0,𝐴
0 = 𝐵 + 2 𝐴𝐵2 − 𝑐0,𝐵
0 = 𝐶 + 𝐴2 𝐶 − 𝑐0,𝐶
𝑓 𝑿 = 𝟎
this might be numerically less stable
C8863 Free Energy Calculations -10-4. Chemical Equilibrium - Exp. methods
Problems
C8863 Free Energy Calculations -11-4. Chemical Equilibrium - Exp. methods
Host with two binding sites
+
H + G HG
+
+
HG + G HG2
𝐾1
𝐾2
𝐾1 𝐾2
𝐾 = 𝐾1 𝐾2
Note: binding sites are chemically equivalent
host (H) guest (G)
C8863 Free Energy Calculations -12-4. Chemical Equilibrium - Exp. methods
Host with two binding sites, tasks
1. Are 𝐾1 and 𝐾2 equal?
2. Determine the composition of the reaction mixture for 𝑐0,𝐻 = 1 𝑚𝑀 titrated by guest
up to 6 molar equivalents for:
• 𝐾1 = 102
• 𝐾1 = 105
3. Determine Job Plots for 𝑐0,𝐻 = 1 𝑚𝑀 and
• 𝐾1 = 101
• 𝐾1 = 102
• 𝐾1 = 103
• 𝐾1 = 104
C8863 Free Energy Calculations -13-4. Chemical Equilibrium - Exp. methods
Host Dimerization
KD
+
KD
2H H2
• What is 𝐾 𝐷 for dimerization process of the host? Selected 1H NMR signal (fast exchange)
undergoes the following change during the sample dilution.
TBA
C8863 Free Energy Calculations -14-4. Chemical Equilibrium - Exp. methods
References
• Hibbert, D. B.; Thordarson, P. The Death of the Job Plot, Transparency, Open Science
and Online Tools, Uncertainty Estimation Methods and Other Developments in
Supramolecular Chemistry Data Analysis. Chemical Communications 2016, 52 (87),
12792–12805. https://doi.org/10.1039/C6CC03888C.
• Gilson, M. K.; Irikura, K. K. Symmetry Numbers for Rigid, Flexible, and Fluxional
Molecules: Theory and Applications. J. Phys. Chem. B 2010, 114 (49), 16304–16317.
https://doi.org/10.1021/jp110434s.
• Duboué-Dijon, E.; Hénin, J. Building Intuition for Binding Free Energy Calculations:
Bound State Definition, Restraints, and Symmetry. J. Chem. Phys. 2021, 154 (20),
204101. https://doi.org/10.1063/5.0046853.
https://arxiv.org/abs/2102.06089