Chemical Thermodynamics

Concept

Chemical Thermodynamics — Potentials, Equilibria, and Electrochemistry

Scope: rigorous derivation of chemical equilibrium and reaction energetics from first principles, including both macroscopic thermodynamic and microscopic/statistical bases. Extends to electrochemical systems, temperature and pressure dependence, and nonequilibrium reaction thermodynamics.

1. Chemical Reactions and Thermodynamic Potentials

For a reacting system of N species and R independent reactions, each reaction r can be represented as: ∑iνirAi=0,\sum_i \nu_{ir} A_i = 0, where νir\nu_{ir} are stoichiometric coefficients (positive for products, negative for reactants).

Extent of reaction: dni=νirdξrdn_i = \nu_{ir} d\xi_r. At constant T and P, total Gibbs energy differential: dG=∑iμidni=∑r(∑iνirμi)dξr.dG = \sum_i \mu_i dn_i = \sum_r \left(\sum_i \nu_{ir} \mu_i\right) d\xi_r. Define the chemical affinity ArA_r: Ar=−∑iνirμi.A_r = -\sum_i \nu_{ir} \mu_i. At equilibrium: Ar=0.A_r = 0.

Thus, equilibrium occurs when the Gibbs free energy is at a minimum with respect to ξr\xi_r at constant T, P.

2. Chemical Potential and Standard States

Definition: μi(T,P,xi)=μi∘(T,P∘)+RTln⁡ai,\mu_i(T,P,x_i) = \mu_i^\circ(T,P^\circ) + RT \ln a_i, where aia_i is the activity (dimensionless measure of escaping tendency). Standard state conventions:

  • Ideal gas: ai=fi/P∘a_i = f_i / P^\circ.
  • Ideal solution: ai=xi.a_i = x_i.
  • Electrolyte: ai=γimi/m∘a_i = \gamma_i m_i / m^\circ.

3. Equilibrium Constant and the Gibbs Function

For a general reaction ∑iνiAi=0\sum_i \nu_i A_i = 0: ΔrG=∑iνiμi=ΔrG∘+RTln⁡Q,\Delta_r G = \sum_i \nu_i \mu_i = \Delta_r G^\circ + RT \ln Q, where Q=∏iaiνiQ = \prod_i a_i^{\nu_i} is the reaction quotient. At equilibrium ΔrG=0\Delta_r G = 0: ΔrG∘=−RTln⁡K.\boxed{\Delta_r G^\circ = -RT \ln K}.

Therefore: K(T,P)=exp⁡(−ΔrG∘/RT).K(T,P) = \exp(-\Delta_r G^\circ / RT).

This relation defines the thermodynamic equilibrium constant, valid for any reaction form.

4. Temperature and Pressure Dependence (Van’t Hoff and Le Chatelier)

4.1 Van’t Hoff Equation

Differentiate ln⁡K=−ΔrG∘/RT\ln K = -\Delta_r G^\circ / RT: d(ln⁡K)dT=ΔrH∘RT2.\frac{d(\ln K)}{dT} = \frac{\Delta_r H^\circ}{RT^2}.

Thus, exothermic reactions (ΔH∘<0\Delta H^\circ < 0) have K decreasing with T; endothermic reactions increase with T. Integration yields: ln⁡K2K1=−ΔrH∘R(1T2−1T1).\ln\frac{K_2}{K_1} = -\frac{\Delta_r H^\circ}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right).

4.2 Le Chatelier’s Principle (Quantitative Form)

At constant T, increasing P favors the side with smaller molar volume: (∂ln⁡K∂P)T=−ΔrV∘RT.\left(\frac{\partial \ln K}{\partial P}\right)_T = -\frac{\Delta_r V^\circ}{RT}.

5. Statistical-Mechanical Basis of Chemical Equilibrium

5.1 Partition Functions and Chemical Potential

For an ideal gas: μi=−kBTln⁡(qiNi)+kBTln⁡(PiP∘).\mu_i = -k_B T \ln\left(\frac{q_i}{N_i}\right) + k_B T \ln\left(\frac{P_i}{P^\circ}\right). where qiq_i is the molecular partition function: qi=qtransqrotqvibqelec.q_i = q_{trans} q_{rot} q_{vib} q_{elec}.

5.2 Reaction Constant from Partition Functions

For ∑iνiAi=0\sum_i \nu_i A_i = 0: Kp(T)=∏i(qiV)νi(kBT)∑iνiexp⁡(−ΔrE0kBT).K_p(T) = \prod_i\left(\frac{q_i}{V}\right)^{\nu_i} (k_B T)^{\sum_i \nu_i} \exp\left(-\frac{\Delta_r E_0}{k_B T}\right). This microscopic derivation connects quantum states to macroscopic equilibrium constants.

6. Thermochemistry — Enthalpy and Heat of Reaction

Reaction enthalpy: ΔrH∘=∑iνihi∘.\Delta_r H^\circ = \sum_i \nu_i h_i^\circ. Temperature dependence via Kirchhoff’s Law: d(ΔrH∘)dT=ΔrCp∘,thus ΔrH∘(T2)=ΔrH∘(T1)+∫T1T2ΔrCp∘ dT.\frac{d(\Delta_r H^\circ)}{dT} = \Delta_r C_p^\circ, \quad \text{thus } \Delta_r H^\circ(T_2) = \Delta_r H^\circ(T_1) + \int_{T_1}^{T_2} \Delta_r C_p^\circ\, dT.

Entropy of reaction: ΔrS∘=∑iνisi∘.\Delta_r S^\circ = \sum_i \nu_i s_i^\circ.

Gibbs free energy of reaction: ΔrG∘=ΔrH∘−TΔrS∘.\Delta_r G^\circ = \Delta_r H^\circ - T\Delta_r S^\circ.

7. Electrochemical Systems

Electrochemical cells are reactions with separated oxidation and reduction steps producing electrical work.

7.1 EMF and Gibbs Free Energy

For cell reaction: ΔrG=−nFE.\Delta_r G = -nFE. At equilibrium (reversible cell): E=E∘−RTnFln⁡Q.E = E^\circ - \frac{RT}{nF}\ln Q. Thus: E∘=RTnFln⁡K.E^\circ = \frac{RT}{nF} \ln K.

7.2 Temperature Coefficients

(∂E∂T)P=ΔrSnF.\left(\frac{\partial E}{\partial T}\right)_P = \frac{\Delta_r S}{nF}. A positive slope implies endothermic reaction (entropy increase).

7.3 Cell Exergy and Second-Law Efficiency

Exergy of electrical work = nFEnFE. Maximum work = ΔG\Delta G; thermal losses correspond to T0ΔSgenT_0\Delta S_{gen}. Efficiency: η2=nFEΔH.\eta_2 = \frac{nFE}{\Delta H}.

8. Nonideal Reactions and Activities

For real systems: μi=μi∘+RTln⁡(γixi)⇒Q=∏i(γixi)νi.\mu_i = \mu_i^\circ + RT\ln (\gamma_i x_i) \Rightarrow Q = \prod_i (\gamma_i x_i)^{\nu_i}. Equilibrium constant splits: K=KγKx,with Kγ=∏iγiνi.K = K_\gamma K_x, \quad \text{with } K_\gamma = \prod_i \gamma_i^{\nu_i}.

At high pressures, replace aia_i with fugacities fif_i: fi=yiϕiP,μi=μi∘+RTln⁡(fi/P∘).f_i = y_i \phi_i P, \quad \mu_i = \mu_i^\circ + RT \ln(f_i/P^\circ).

9. Reaction Equilibrium by Gibbs Energy Minimization

For multi-reaction, multi-phase systems: G(T,P,ni)=∑iniμi(T,P,xj).G(T,P,{n_i}) = \sum_i n_i \mu_i(T,P,{x_j}). Constraints: An=bA n = b (elemental conservation). Minimize G subject to constraints using Lagrange multipliers λ\lambda: ∂∂ni(G+λT(An−b))=0⇒μi=AiTλ.\frac{\partial}{\partial n_i}\left(G + \lambda^T(A n - b)\right) = 0 \Rightarrow \mu_i = A_i^T \lambda.

This yields equilibrium compositions without explicit K.

10. Nonequilibrium Thermodynamics of Reactions

At finite rates, entropy generation is nonzero: S˙gen=−1T∑rArξ˙r.\dot S_{gen} = -\frac{1}{T} \sum_r A_r \dot\xi_r. Reversible limit: Ar=0.A_r = 0. Linear response near equilibrium (Onsager): ξ˙r=∑sLrsAs/T.\dot\xi_r = \sum_s L_{rs} A_s/T. Coefficients LrsL_{rs} satisfy reciprocal relations Lrs=LsrL_{rs} = L_{sr}. This framework unifies chemical kinetics and thermodynamics.

11. Coupled Reactions and Energy Conversion

In biochemical or catalytic systems, multiple reactions share intermediates. Coupling allows an exergonic reaction (ΔG<0\Delta G < 0) to drive an endergonic one (ΔG>0\Delta G > 0) via common intermediates.

Example: ATP hydrolysis couples to biosynthesis. ΔrGtotal=ΔrG1+ΔrG2.\Delta_r G_{total} = \Delta_r G_1 + \Delta_r G_2. If total ΔG<0\Delta G < 0, the coupled process is spontaneous.

12. Phase and Chemical Equilibrium Combined

For reactive, multiphase systems: μi(α)=μi(β)  (phase eq.),∑iνirμi=0  (chem eq.).\mu_i^{(\alpha)} = \mu_i^{(\beta)} \; \text{(phase eq.)}, \quad \sum_i \nu_{ir}\mu_i = 0 \; \text{(chem eq.)}. Both must be solved simultaneously. This forms the basis for reactive distillation and electrochemical interface models.

13. Example: Hydrogen Combustion

Reaction: H2+12O2→H2O(g)H_2 + \frac{1}{2}O_2 \to H_2O(g)

Property298 KUnits
ΔH∘\Delta H^\circ-241.8kJ/mol
ΔS∘\Delta S^\circ-44.5J/mol·K
ΔG∘\Delta G^\circ-237.1kJ/mol

At 298 K: K=e−ΔG∘/(RT)=e95.5≈3×1041K = e^{-\Delta G^\circ/(RT)} = e^{95.5} \approx 3\times10^{41}. Thus essentially complete conversion at standard conditions.

14. Summary Equations

ConceptEquation
Chemical affinityAr=−∑iνirμiA_r = -\sum_i \nu_{ir} \mu_i
Equilibrium conditionAr=0A_r = 0
Gibbs relationΔrG=ΔrG∘+RTln⁡Q\Delta_r G = \Delta_r G^\circ + RT\ln Q
Equilibrium constantK=e−ΔrG∘/(RT)K = e^{-\Delta_r G^\circ/(RT)}
Van’t Hoffdln⁡K/dT=ΔrH∘/(RT2)d\ln K/dT = \Delta_r H^\circ/(RT^2)
EMF–ΔG\Delta G linkΔG=−nFE\Delta G = -nFE
Nernst equationE=E∘−(RT/nF)ln⁡QE = E^\circ - (RT/nF)\ln Q
Entropy generationS˙gen=−∑r(Arξ˙r)/T\dot S_{gen} = -\sum_r (A_r \dot\xi_r)/T
Nonequilibrium fluxξ˙r=LrrAr/T\dot\xi_r = L_{rr}A_r/T
  • mixtures-phases.md — activity, fugacity, and phase equilibrium foundations.
  • 07_Exergy_and_Irreversibility.md — exergy of chemical reactions.
  • Fluid_Dynamics/09_Reactive_Flows.md — coupling of reaction kinetics and transport phenomena.