Physical chemistry includes three major subfields:

1. Atomic structure of chemical species;
2. Chemical thermodynamics: Direction (in the space of equilibrium states) and distance (as of free energy change) of a non-equilibrium system to equilibrium;
3. Chemical kinetics: The rate and path to equilibrium;

Josiah Willard Gibbs laid the foundation of chemical thermodynamics in On the Equilibrium of Heterogeneous Substances [@Gibbs1876]. Gibbs' theory unifies physical chemistry by integrating various physicochemical processes from the perspective of energy change, including chemical reaction, phase transition, and dissolution/diffusion. It applies the criteria for thermodynamic equilibrium to determine the direction of spontaneous chemical processes. Statistical mechanics is later applied to derive state variables and equations of state.

Gibbs free energy is a thermodynamic potential defined as the (non-stadard) Legendre transformation of internal energy regarding volume and entropy:

$$G(p, T, \{N_i\}) = U(V, S, \{N_i\}) + pV - TS$$

## Definition

A chemical species are molecular entities whose set of molecular energy levels cannot be distinguished by the measurement. A phase is a form of matter that is homogeneous in chemical composition and state of matter. A component is a chemical species in a specific phase.

For a species, the internal energy of formation is the energy difference between the species and the constituting atoms. It can be calculated as the total bond energies of the species. In a spontaneous chemical reaction, the difference in total internal energy of formation between the products and the reactants will be released as heat and even mechanical work.

The chemical potential, aka partial molar Gibbs free energy, of a species/component in a mixture can be defined as the change of Gibbs free energy or internal energy of the system per unit change in moles of just that species, under difference conditions.

$$\mu_i = \left(\frac{\partial G}{\partial N_i}\right)_{p,T,N_j} = \left(\frac{\partial U}{\partial N_i}\right)_{V,S,N_j}$$

Each chemical species in a mixture has its own chemical potential, so that the energy change of a chemical reaction can be calculated easily.

Using the Euler integral of internal energy:

$$G(p, T, \{N_i\}) = \sum_i \mu_i N_i$$

For a chemical reaction, the extent of reaction $\xi$ is the amount of chemical transformations occurred, and stoichiometric coefficients (化学计量系数) are positive for products and negative for reactants. Chemical potentials of species are preferred to the chemical affinity $\mathbb{A}$ of a reaction, which are related by:

$$-\mathbb{A} \equiv \left( \frac{\partial G}{\partial \xi} \right)_{p, T} = \sum_i \nu_i \mu_i$$

In the presence of external force fields such as electromagnetic or gravity fields, the above definition serves as the total chemical potential, which includes external chemical potential, the sum of external force field induced potential, and internal chemical potential, the remaining chemical potential.

The standard state $\ominus$ 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.

Standard states of matter:

• Gases: pure substance at standard pressure, assuming ideal gas behavior.
• Condensed phases: pure substance at standard pressure.
• Solutes: solute at standard molality, standard concentration or standard pressure, assuming infinitely dilute solution behavior.

In chemical thermodynamics, given some standard state $\ominus$, a species' activity $a_i$ (in solution) or fugacity $f_i$ (in 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.

## Properties

### Direction of Chemical Process

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, Gibbs free energy of a system never increase in spontaneous chemical reaction.

Actions:

• Diffusion/dissolution: For a given temperature, a molecule has a higher chemical potential in a higher-concentration area, and a lower chemical potential in a low concentration area.
• Melting: Above 0°C, an H2O molecule in the liquid phase has a lower chemical potential than one that is in the solid phase; below 0°C, the difference reverts.
• Reaction: Reaction proceeds from the side with higher sum of chemical potential to the other.

### Reaction rates of chemical processes

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:

$$\dot{\xi} = \frac{1}{\nu_i} \frac{\mathrm{d} n_i}{\mathrm{d} t}$$

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:

\begin{align} \frac{\mathrm{d}[A]}{\mathrm{d}t} &= -k [A] \\ \frac{\mathrm{d}[A]}{\mathrm{d}t} = \frac{\mathrm{d}[B]}{\mathrm{d}t} &= -k [A] [B] \end{align}

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 be different from 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. Mixed order rate laws are typically polynomials and rational functions.

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.

### Equilibrium

In a (reversible) chemical reaction, reaction quotient is the product of the activities of the reagents to the power of their stoichiometric coefficients.

$$Q_r = \frac{\prod_j a_j^{\nu_j}}{\prod_i a_i^{\nu_i}}$$

Gibbs energy change of a chemical reaction can be found to be:

$$\Delta_r G = \Delta_r G^{\ominus} + R T \ln Q_r$$

At chemical equilibrium or phase equilibrium, the Gibbs free energy of a closed system is at a minimum, thus the total chemical energy change is zero.

$$\left( \mathrm{d} G \right)_{p, T} = \sum_{i=1}^I \mu_i \mathrm{d} N_i = 0$$

At chemical equilibrium with constant temperature, pressure and ionic strength, the reaction quotient is thus independent of the analytical concentrations of the reactant and product species in a mixture, and referred to as equilibrium constant $K_{eq}$.

$$\Delta_r G^{\ominus} = -RT \ln K_{eq}$$

### Gibbs–Duhem equation

When chemical species and phases are differentiated at equilibrium, a species has a unique chemical potential in each phase, and chemical potentials of components in each phase are related through the Gibbs-Duhem relation.

Because of the Gibbs–Duhem equation, we can say that the chemical potential of a pure substance is a function of temperature and pressure.

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, Raoult's law, and Henry's law (ideal gas and solution) can be derived from the Gibbs–Duhem equation.

### Dependence on temperature and pressure

Maxwell relations shows the temperature and pressure variation of chemical potential.

$$\left( \frac{\partial \mu}{\partial T} \right)_{p,n} = - \left( \frac{\partial S}{\partial n} \right)_{T,p}$$

$$\left( \frac{\partial \mu}{\partial p} \right)_{T,n} = \left( \frac{\partial V}{\partial n} \right)_{T,p}$$

### Euler integral

For pure substances: (star denotes property of pure substance)

$$G^\star = \mu^\star n$$

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: $$\mu = \mu^{\ominus} + k_B T \ln \frac{p}{{p^{\ominus}}}$$