C8863 Free Energy Calculations -1C8863 Free Energy Calculations Petr KulhΓ‘nek 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 2 Chemical Potential C8863 Free Energy Calculations -2- Overview C8863 Free Energy Calculations -3Chemical Mixture chemical potential The internal energy of closed system without change in chemical composition (recapitulation): π‘ˆ(𝑆, 𝑉) π‘‘π‘ˆ = πœ•π‘ˆ πœ•π‘† 𝑉 𝑑𝑆 + πœ•π‘ˆ πœ•π‘‰ 𝑆 𝑑𝑉 The internal energy of closed system with change in chemical composition needs an extension: π‘ˆ(𝑆, 𝑉, 𝑛𝑖) π‘‘π‘ˆ = πœ•π‘ˆ πœ•π‘† 𝑉,𝑛 𝑖 𝑑𝑆 + πœ•π‘ˆ πœ•π‘‰ 𝑆,𝑛 𝑖 𝑑𝑉 + ෍ 𝑖 πœ•π‘ˆ πœ•π‘›π‘– 𝑆,𝑉,𝑛 𝑗≠𝑛 𝑖 𝑑𝑛𝑖 π‘‘π‘ˆ = 𝑇𝑑𝑆 βˆ’ 𝑝𝑑𝑉 state variables composition of chemical mixture, molar amounts total differential total differential fundamental thermodynamic relation πœ•π‘ˆ πœ•π‘›π‘– 𝑆,𝑉,𝑛 𝑗≠𝑛 𝑖 = πœ‡π‘– C8863 Free Energy Calculations -4Chemical Mixture, cont. π‘‘π‘ˆ = 𝑇𝑑𝑆 βˆ’ 𝑝𝑑𝑉 + ෍ 𝑖 πœ‡π‘– 𝑑𝑛𝑖 fundamental thermodynamic relations The Legendre transform provides other fundamental relations: 𝑑𝐻 = 𝑇𝑑𝑆 + 𝑉𝑑𝑝 + ෍ 𝑖 πœ‡π‘– 𝑑𝑛𝑖 𝑑𝐴 = βˆ’π‘†π‘‘π‘‡ βˆ’ 𝑝𝑑𝑉 + ෍ 𝑖 πœ‡π‘– 𝑑𝑛𝑖 𝑑𝐺 = βˆ’π‘†π‘‘π‘‡ + 𝑉𝑑𝑝 + ෍ 𝑖 πœ‡π‘– 𝑑𝑛𝑖 (total) chemical potential πœ•π‘ˆ πœ•π‘›π‘– 𝑆,𝑉,𝑛 𝑗≠𝑛 𝑖 = πœ‡π‘– πœ•π» πœ•π‘›π‘– 𝑆,𝑝,𝑛 𝑗≠𝑛 𝑖 = πœ‡π‘– πœ•π΄ πœ•π‘›π‘– 𝑇,𝑉,𝑛 𝑗≠𝑛 𝑖 = πœ‡π‘– πœ•πΊ πœ•π‘›π‘– 𝑇,𝑝,𝑛 𝑗≠𝑛 𝑖 = πœ‡π‘– equivalent definitions of (total) chemical potential not convenient definition due to constant entropy condition C8863 Free Energy Calculations -5Total Chemical Potential The abstract definition of chemical potential given previously, total change in free energy per extra mole of substance, is more specifically called total chemical potential. If two locations have different total chemical potentials for a species, some of it may be due to potentials associated with "external" force fields (electric potential energy, gravitational potential energy, etc.), while the rest would be due to "internal" factors (density, temperature, etc.). Therefore, the total chemical potential can be split into internal chemical potential and external chemical potential: πœ‡π‘– = πœ•πΊ πœ•π‘›π‘– 𝑇,𝑝,𝑛 𝑗≠𝑛 𝑖 = πœ‡π‘–,𝑖𝑛𝑑 + πœ‡π‘–,𝑒π‘₯𝑑 https://en.wikipedia.org/wiki/Chemical_potential the external potential is the sum of electric potential, gravitational potential, etc. In this course, we will consider only internal chemical potential. C8863 Free Energy Calculations -6Chemical Potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g., in a chemical reaction or phase transition. πœ‡π‘– = πœ•πΊ πœ•π‘›π‘– 𝑇,𝑝,𝑛 𝑗≠𝑛 𝑖 Chemical potential expresses the effort of the substance: β€’ to react with another substance β€’ to change its status β€’ to change its spatial distribution Value of chemical potential: β€’ is related to the very nature of the substance β€’ is related to the environment (temperature, pressure, concentration, ...) β€’ however, it is not related to the nature of the substances with which it reacts or is transformed to C8863 Free Energy Calculations -7Chemical Potential of Ideal Gas 𝑑𝐺 = βˆ’π‘†π‘‘π‘‡ + 𝑉𝑑𝑝 Consider the closed system with one component behaving as an ideal gas: At constant temperature, the Gibbs energy change is: 𝑑𝐺 = 𝑉𝑑𝑝 Consider the pressure change (𝑝0 β†’ 𝑝), then the Gibbs energy change is: 𝐺 = 𝐺0 + ΰΆ± 𝑝0 𝑝 𝑉𝑑𝑝 = 𝐺0 + ΰΆ± 𝑝0 𝑝 𝑛𝑅𝑇 𝑝 𝑑𝑝 = 𝐺0 + 𝑛𝑅𝑇𝑙𝑛 𝑝 𝑝0 𝑝𝑉 = 𝑛𝑅𝑇 Finally, differentiating with respect to molar amount, the following expression for the chemical potential is obtained: πœ‡ = πœ•πΊ πœ•π‘› 𝑇,𝑝 = πœ•πΊ0 πœ•π‘› + 𝑅𝑇𝑙𝑛 𝑝 𝑝0 = πœ‡0 + 𝑅𝑇𝑙𝑛 𝑝 𝑝0 C8863 Free Energy Calculations -8Ideal and Real Gas Mixtures πœ‡π‘– = πœ‡π‘– 0 + 𝑅𝑇𝑙𝑛 𝑝𝑖 𝑝0 Ideal gas: Real gas: 𝑓 - fugacity - effective pressure of the gas substance Unification of two situations: π‘Ž - activity of substance RT i ii ea 0  βˆ’ = πœ‡π‘– = πœ‡π‘– 0 + 𝑅𝑇𝑙𝑛 𝑓𝑖 𝑝0 πœ‡π‘– = πœ‡π‘– 0 + 𝑅𝑇𝑙𝑛(π‘Žπ‘–) 00 p p p f a ii i ο‚»= In chemical thermodynamics, an activity is a measure of the "effective concentration/ pressure" of a species in a mixture, in the sense that the species' chemical potential depends on the activity of a real solution in the same way that it would depend on concentration for an ideal solution. https://en.wikipedia.org/wiki/Thermodynamic_activity C8863 Free Energy Calculations -9Ideal and Real Solutions πœ‡π‘– = πœ‡π‘– 0 + 𝑅𝑇𝑙𝑛(π‘Žπ‘–) 00 c c c c a ii ii ο‚»=  The chemical potential for a substance in ideal and real solutions can be described similarly as gas mixtures: However, the activity is now expressed by concentrations: activity coefficient An activity coefficient is a factor used in thermodynamics to account for deviations from ideal behavior in a mixture of chemical substances. https://en.wikipedia.org/wiki/Activity_coefficient C8863 Free Energy Calculations -10Pure Liquids and Solids The pure liquids and solids have the chemical potential, which is a constant at constant temperature and pressure: πœ‡π‘– = πœ‡π‘– 0 Then, the activity of pure solids and liquids is one. πœ‡π‘– = πœ‡π‘– 0 + 𝑅𝑇𝑙𝑛(π‘Žπ‘–) π‘Žπ‘– = 1 πœ‡π‘– = πœ‡π‘– 0 + 0 𝑙𝑛 1 = 0 Solvent in an ideal diluted solution can be approximated as a pure liquid. Thus, the solvent has activity one. C8863 Free Energy Calculations -11Reference State ➒ To work with the chemical potential, we need to define the reference state. ➒ The choice of the reference state is arbitrary. ➒ But once selected, it must be the same for all compounds considered in the reaction mixture or phase transition. IUPAC recommends a conventional set of reference states, which are called standard states for general use. Standard state (IUPAC): p0 = 100 kPa c0 = 1 mol dm-3 = 1 M πœ‡π‘– = πœ‡π‘– 0 + 𝑅𝑇𝑙𝑛 𝑓𝑖 𝑝0 πœ‡π‘– = πœ‡π‘– 0 + 𝑅𝑇𝑙𝑛 𝛾𝑖 𝑐𝑖 𝑐0 ? ? we need some suitable choice of zero to properly reference all compounds. C8863 Free Energy Calculations -12Standard Chemical Potential Standard chemical potential is the change in Gibbs energy that is associated with the formation of one mole of compound in the standard state. The change of the Gibbs energy is most often expressed in the form of the standard formation Gibbs energies. 0 , 0 ifi G= Standard formation Gibbs energy is the change of Gibbs energy that corresponds to the formation of one mole of matter from chemical elements in the standard state. Chemical elements in the standard state have zero formation Gibbs energy (this is the definition of the reference state). Standard state (IUPAC): p0 = 100 kPa c0 = 1 mol dm-3 = 1 M https://en.wikipedia.org/wiki/Standard_state C8863 Free Energy Calculations -13- Summary πœ‡π‘– = πœ‡π‘– 0 + 𝑅𝑇𝑙𝑛(π‘Žπ‘–) ➒ The chemical potential is the property of the compound. ➒ It express the ability of the compound to react with another substance, to change its status, or to change its spatial distribution. ➒ The standard chemical potential is the change in the Gibbs energy that is associated with the formation of one mole of compound in the standard state. ➒ At different conditions, the chemical potential is proportional to the standard chemical potential and the activity of the compound. ➒ The activity then express an effective amount in comparison to the standard state. C8863 Free Energy Calculations -14Recommended Literature β€’ Atkins, P. W. Physical Chemistry, 5. ed., repr. (with correct.).; Oxford Univ. Press: Oxford, 1994. β€’ BokshteΔ­n, B. S.; Mendelev, M. I.; Srolovitz, D. J. Thermodynamics and Kinetics in Materials Science: A Short Course; Oxford University Press: New York, 2005. β€’ Dill, K. A.; Bromberg, S. Molecular Driving Forces: Statistical Thermodynamics in Biology, Chemistry, Physics, and Nanoscience, 2nd ed.; Garland Science: London ; New York, 2011.