Gibbs free energy is a thermodynamic potential defined as the (non-stadard) Legendre transform of internal energy regarding volume and entropy:
\[ G(p,T) = U + pV - TS \]
For a system with constant temperature and pressure at chemical equilibrium,
\[ \delta G = 0 \]
A chemical species is molecular entities that the measurement cannot distinguish the set of molecular energy levels populated by.
Each chemical species in a mixture has its own chemical potential. The chemical potential, aka partial molar Gibbs free energy, of a species in a mixture can be defined as the slope of the free energy of the system with respect to a change in the number of moles of just that species.
\[ \mu_i=\left(\frac{\partial G}{\partial N_i}\right)_{T,P,N_j} \]
In the presence of external force fields such as electromagnetic and gravity fields, the above definition serves as the total chemical potential, with external chemical potential being the sum of external force field induced potential and internal chemical potential being the remaining potential.
The standard state of a system is defined as a reference point for state properties such as Gibbs free energy. Typical standard values are 1 bar for pressure, 1 mol/kg for molality (质量摩尔浓度), and 1 mol/L for (amount) concentration (体积摩尔浓度). Temperature is not part of the definition of a standard state.
In chemical thermodynamics, given some standard state \(\ominus\), a species' activity \(a_i\) (solution) or fugacity \(f_i\) (gaseous state) is defined as a cumulative factor that alters the chemical potential such that:
\[ \mu_i = \mu_i^{\ominus} + RT \ln a_i \]
Activity coefficient \(\gamma\) is the ratio of activity to amount fraction, mass fraction; molality, amount concentration or mass concentration (relative to standard values). It indicates deviation from ideal behavior, which equals 1 for ideal gas/solution.
\[ a_i = \gamma_{x,i} x_i\ = \gamma_{w,i} w_i\ = \gamma_{b,i} \frac{b_i}{b^{\ominus}}\ = \gamma_{c,i} \frac{c_i}{c^{\ominus}}\, = \gamma_{\rho,i} \frac{\rho_i}{\rho^{\ominus}} \]
Activity coefficients, as a function of amount fractions, serve as the (additional) constitutive relations of chemical thermodynamic processes. It can be determined either experimentally or theoretically. Theoretical methods include Debye–Hückel theory and Specific ion Interaction Theory for electrolyte solutions, correlative methods for non-electrolyte solutions, salting-out model for uncharged species, and specific models for solvents.
Gibbs energy change
\[ \Delta_r G = \Delta_r G^{\ominus} + R T \ln Q_r \]
Particles always tend naturally to go from a higher chemical potential to a lower one.
In chemical processes, the Second Law of Thermodynamics can be formulated as:
At constant temperature and pressure, free energy of a system never increase in spontaneous chemical reaction.
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For a chemical reaction occurring at constant volume, without an appreciable build-up of reaction intermediates, the reaction rate is defined as:
\[ v = \frac{1}{\nu_i} \frac{\mathrm{d} [i]}{\mathrm{d} t} \]
Here \([i]\) denotes the amount concentration of reagent i. For reactant, \(-\frac{\mathrm{d} [i]}{\mathrm{d} t}\) is called its rate of disappearance. For product, \(\frac{\mathrm{d} [i]}{\mathrm{d} t}\) is called its rate of appearance.
For a reaction occurring in a closed system of varying volume, rate of conversion can be used:
\[ \dot{\xi} = \frac{1}{\nu_i} \frac{\mathrm{d} n_i}{\mathrm{d} t} \]
Here the extent of reaction \(\xi\) is the amount of chemical transformations occurred, indicating the progress of a chemical reaction.
An elementary reaction is a chemical reaction with a single step and with a single transition state. Using collision theory, elementary reactions should be either unimolecular or bimolecular, with rate equations, correspondingly:
\[ \frac{\mathrm{d}[A]}{\mathrm{d}t} = -k [A] \quad \text{or} \quad \frac{\mathrm{d}[A]}{\mathrm{d}t} = \frac{\mathrm{d}[B]}{\mathrm{d}t} = -k [A] [B] \]
For non-elementary reactions, their rate equations must be determined experimentally. It is typically a monomial with rate coefficients \(k\) and partial reaction orders \(n\) which may not be equal to the stoichiometric coefficients:
\[ v = k \prod_i [i]^{n_i} \]
More complex rate laws are called mixed order, in that they may be approximated by the simper forms at different concentrations. Typical forms include polynomial and rational function.
With an assumed multi-step mechanism, the rate equation of a reaction can often be theoretically derived from the underlying elementary reactions using quasi-steady state assumptions.
When the sums of chemical potential of reactants and products are equal, the system is at equilibrium? (multiple equilibria)
At chemical equilibrium or phase equilibrium, the free energy is at a minimum, thus the total sum of chemical potentials is zero.
\[ \sum_{i=1}^I \mu_i \mathrm{d} N_i = 0 \]
In a (reversible) chemical reaction, reaction quotient is the product of the activities of the reagents to the power of their stoichiometric coefficients (化学计量系数) (positive for products, negative for reactants).
\[ Q_r = \frac{\prod_j a_j^{\nu_j}}{\prod_i a_i^{\nu_i}} \]
At chemical equilibrium, the reaction quotient is independent of the analytical concentrations of the reactant and product species in a mixture, thus referred to as equilibrium constant \(K_{eq}\).
\[ \Delta_r G^{\ominus} = -RT \ln K_{eq} \]
From the fundamental thermodynamic relation and Gibbs free energy:
\[ \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p \]
This Gibbs–Duhem equation shows the dependency among chemical potentials for components in a system at thermodynamic equilibrium. At constant temperature and pressure, the changes in chemical potentials of the two participants in a binary mixture are related by:
\[ \mathrm{d}\mu_1 = -\frac{N_2}{N_1} \mathrm{d}\mu_2 \]
Clapeyron equation can be derived from the Gibbs–Duhem equation. So can Raoult's law and Henry's law (ideal gas and solution).
Maxwell relations shows the temperature and pressure variation of chemical potential.
For an ideal gas, the Euler integral of Gibbs energy is given by \[ G = G^{\ominus} + k_B NT \ln \frac{p}{{p^{\ominus}}} \]
Or expressed in chemical potential: \[ \mu = \mu^{\ominus} + k_B T \ln \frac{p}{{p^{\ominus}}} \]
For a pure substance, \( G^\star = \mu^\star n \), where star denotes property of pure substance.