Chemical equilibrium

At chemical equilibrium, a reversible chemical reaction has equal forward and reverse reaction rates, so its macroscopic composition stays constant. Individual molecules can keep changing identity: the reaction has not stopped. This dynamic balance is the definition described by the ^[ref]. Equal reaction rates also need not mean equal concentrations. A small population converting rapidly can balance a large population converting slowly.

This distinction gives two useful questions. Which composition is favored at the specified temperature and other constraints? Thermodynamics answers through chemical potentials and an equilibrium constant. How quickly does a prepared mixture approach that composition? Chemical kinetics answers through reaction rates. Knowing the destination does not determine the journey's duration. A composition that barely changes during observation may reflect extremely slow chemical kinetics, so constancy alone does not establish chemical equilibrium.

Consider the idealized elementary chemical reactionAssume a well-mixed closed system at fixed temperature and volume, with first-order law of mass action rates in both directions and no competing chemical reactions. Write $a=[\mathrm{A}]$ and $b=[\mathrm{B}]$ for concentrations in $\mathrm{M}=\mathrm{mol}/\mathrm{L}$. The total $a+b=C=1\mathrm{M}$ is conserved because each conversion replaces one molecule by one molecule. This conservation is special to the chosen stoichiometry.

The forward and reverse reaction rates, measured as conversions per unit volume per unit time, are $J_f=k_{f}a$ and $J_r=k_{r}b$. HenceSetting this derivative to zero givesWith $a(0)=1\mathrm{M}$ and $b(0)=0$, solving the ordinary differential equation yieldsAt $t=5\mathrm{s}$, these are approximately $0.3515\mathrm{M}$ and $0.6485\mathrm{M}$. Initially $J_f=0.3M/s$ and $J_r=0$. At chemical equilibrium, both are $0.075M/s$, while the net reaction rate $J_f-J_r$ is zero. The persistent exchange is visible in the lower panel even after the upper panel has nearly flattened.

Why does chemical equilibrium also involve a free energy minimum? For a closed system at fixed temperature and pressure, without imposed non-expansion work, stable equilibrium minimizes Gibbs free energy over compositions allowed by conserved elemental inventories. See W. C. Carter, MIT thermodynamics notes, sections 2-1.14 and 2-1.19. This is constrained Gibbs minimization for chemical equilibrium: freely minimizing over unrelated species amounts would change the inventory. At fixed temperature and volume, the appropriate potential is Helmholtz free energy. In either setting, varying an allowed extent of reaction gives the corresponding free energy slope ${\sum }_i{\nu }_i{\mu }_i$, with molar chemical potentials and signed stoichiometric vector entries. An interior equilibrium has zero slope; stability requires a minimum. For A converting to B, this says ${\mu }_{\mathrm{B}}={\mu }_{\mathrm{A}}$.

The beauty of this connection is that molecular exchange, a macroscopic composition, and a geometric minimum describe the same balance. In the ideal concentration model, let $x=b/C$ and choose the standard molar chemical potential difference consistent with $K=3$. Up to a composition-independent constant, the fixed-volume Helmholtz free energy then obeyswhere $n$ is the conserved total amount of substance and $R$ the molar gas constant. Its derivative is $\ln [x/(1-x)]-\ln 3$, vanishing at $x=3/4$. Its second derivative is $1/x+1/(1-x)>0$ inside $0<x<1$, so this balance is the unique minimum. The mixing entropy terms explain why the favored composition still contains A.

For a mineral reaction producing one mole of water, let $H^{\circ },S^{\circ }$ refer to the $P^{\circ }$ standard state and let the solids’ volume change $\Delta V_s$ be constant. If $A_s$ is the product/reactant solid-activity ratio, the Gibbs energy of reaction isFor temperature-independent specified activities and fugacity, equilibrium givesThe gas pressure work is already represented by the fugacity term; adding a second constant water-volume term double counts it. For an ideal gas, $f=P$; a given fugacity coefficient gives $f={\varphi }_{f}P$. If these quantities depend on temperature, the equilibrium equation must instead be solved implicitly. Pressure units in the solid volume correction must match those used for $\Delta V_s$.

For ideal-gas single-particle functions $Z_1^{(i)}=C_{i}V{T}^{3/2}$ and reaction $A\rightleftharpoons 2B$, equilibrium gives $K_N=N_B^2/N_A=(C_B^2/C_A)V{T}^{3/2}$. This particle-number ratio contains a volume factor; the concentration ratio $n_B^2/n_A=K_N/V$ does not.

For $A\rightleftharpoons 2B$, the conserved amount is $M=N_A+N_B/2$. The total particle count $N_A+N_B$ changes during reaction. Defining a dissociated fraction by $\alpha =N_B/(2M)$ gives $N_A=M(1-\alpha )$.

For $A\rightleftharpoons 2B$, ${\alpha }^2/(1-\alpha )=K_N/(4M)$. The left side increases monotonically on $0<\alpha <1$, so increasing the conserved formula-unit density at fixed temperature reduces dissociation in the ideal model.

The product of hydrogen-ion and hydroxide-ion activities at a specified temperature. In dilute aqueous solution at 25 degrees Celsius, its usual concentration approximation is ten to the minus fourteen.

Reversible association of two molecular units to form a dimer.

The linked equilibria among dissolved carbon dioxide, bicarbonate, carbonate and hydrogen ions in seawater. At a fixed alkalinity, added carbon changes speciation and acidity, limiting how much extra carbon can be held at a given atmospheric partial pressure. Exchange and chemical partitioning therefore do not behave like irreversible destruction by a single first-order sink.

Adjustment of ocean and sediment carbonate chemistry through dissolution or burial of carbonate minerals. Following a carbon-dioxide perturbation, additional carbonate dissolution can increase alkalinity and permit further ocean uptake. The adjustment is much slower than surface gas exchange, operating over millennial timescales.

The temperature relation $\mathrm{d}\ln K/\mathrm{d}T={\Delta }_r{H}^{\circ }/(R{T}^2)$ for a dimensionless thermodynamic equilibrium constant. With constant reaction enthalpy, integration gives the difference of reciprocal temperatures.

The stoichiometric sum of standard molar entropies of products minus those of reactants: ${\Delta }_r{S}^{\circ }={\sum }_i{\nu }_i{S}_i^{\circ }$.

The derivative of Gibbs energy with respect to reaction extent at fixed temperature and pressure: ${\Delta }_{r}G={\sum }_i{\nu }_i{\mu }_i$. For a reversible electrochemical cell reaction, ${\Delta }_{r}G=-nFE$.

At fixed temperature, $dG=VdP$ gives the pressure correction for the participating solids. Constant reaction volume reduces the integral to $\Delta V_{\mathrm{solids}}(P-P_0)$. A gas product treated by chemical fugacity instead contributes $\nu RT\ln (f/P_0)$ relative to its reference state. Adding a separate gas-volume correction to that term would double count the same pressure dependence.

The Gibbs energy of reaction for specified standard states. It determines the dimensionless equilibrium constant through ${\Delta }_r{G}^{\circ }=-RT\ln K$.

For a nondegenerate spin-one-half ideal gas with energies $\mp {\mu }_{B}B(\mathbf{r})$, the two component chemical potentials are $k_{B}T\ln (n_{\pm }/n_Q)\mp {\mu }_{B}B$. Diffusive and spin-conversion equilibrium require one spatially constant common chemical potential, provided spin conversion is allowed. Hence the displayed densities give total number density $n=2n_Q{e}^{\beta \mu }\cosh (\beta {\mu }_{B}B)$ and magnetization $M=2{\mu }_B{n}_Q{e}^{\beta \mu }\sinh (\beta {\mu }_{B}B)=n{\mu }_B\tanh (\beta {\mu }_{B}B)$. The high-temperature fixed-density limit is $M\simeq n{\mu }_B^{2}B/(k_{B}T)$, a Curie law. At fixed chemical potential the density itself varies with temperature, so the unnormalized magnetization need not have a pure $1/T$ dependence.

A chemical system is out of local chemical equilibrium when reaction adjustment cannot keep pace with transport, imposed changes or external driving. Comparing the chemical relaxation time with transport and irradiation times identifies plausible departures. Disequilibrium chemistry in an exoplanet atmosphere combines this principle with mixing, horizontal advection and atmospheric photochemistry.

For one particle species in chemical equilibrium under $2\varphi \leftrightarrow 3\varphi$, chemical balance gives $2{\mu }_{\varphi }=3{\mu }_{\varphi }$, hence ${\mu }_{\varphi }=0$. Elastic $2\varphi \leftrightarrow 2\varphi$ imposes no such constraint. In an expanding universe the number-changing reaction must remain fast compared with the Hubble parameter; after chemical freeze-out a nonzero chemical potential can develop even if elastic collisions maintain kinetic equilibrium.

The chemical relaxation time estimates how rapidly a perturbed abundance returns toward chemical equilibrium. Comparing it with the eddy mixing time identifies a chemical quench level.

For the A-to-B model in chemical equilibrium, deviations decay by exponential decay with ${\tau }_{\mathrm{chem}}=1/(k_f+k_r)=2.5\mathrm{s}$. After one such interval, $1/e$ of the initial deviation remains; reaching one percent takes $\ln (100){\tau }_{\mathrm{chem}}\simeq 11.51\mathrm{s}$. Multiplying both rate constants by two preserves their ratio and the equilibrium composition, but halves ${\tau }_{\mathrm{chem}}$ to $1.25\mathrm{s}$. The equilibrium exchange rates double to $0.15M/s$. Faster exchange therefore coexists with the same final composition and the same free energy minimum.

For a specified chemical reaction and standard state convention, the dimensionless equilibrium constant is the equilibrium value of the reaction quotient $Q={\prod }_i{a}_i^{{\nu }_i}$, where $a_i$ denotes thermodynamic activity and ${\nu }_i$ a signed stoichiometric vector entry. Thermodynamic activities incorporate the reference state and departures from ideal behavior. See IUPAC Gold Book, equilibrium constant.

In the ideal concentration model for A converting to B, use $a_{\mathrm{A}}=[\mathrm{A}]/c^{\circ }$ and $a_{\mathrm{B}}=[\mathrm{B}]/c^{\circ }$ with the same reference $c^{\circ }=1\mathrm{M}$. Their ratio gives $Q=[\mathrm{B}]/[\mathrm{A}]$ and $K=0.75/0.25=3$. Equality $K=k_f/k_r$ applies to this elementary first-order pair; it is not a universal ratio of two measured rate constants for an arbitrary multistep chemical reaction.

At a specified constant temperature, with consistent standard states, the standard molar reaction Gibbs free energy isHere ${\Delta }_{r}G$ is the local slope per mole of extent of reaction at fixed temperature and pressure, rather than the finite total Gibbs free energy change between two mixtures. At equilibrium ${\Delta }_{r}G=0$, while ${\Delta }_r{G}^{\circ }$ generally remains nonzero. For $K=3$ at $298\mathrm{K}$, it is about $-2.72kJ/mol$. For positive activities, $Q<K$ favors forward conversion and $Q>K$ favors reverse conversion; neither sign supplies a reaction rate. See MIT 5.111, Lecture 18, reaction free energy and equilibrium.

For ideal concentration activities and signed stoichiometric coefficients ${\nu }_i$, $K_c={\prod }_i(c_i/c^{\circ }{)}^{{\nu }_i}$ at equilibrium. Its numerical value depends on the stated standard concentration when ${\sum }_i{\nu }_i\ne 0$. Number concentrations require a number-concentration reference; molar concentrations require the corresponding molar reference.

For $A+B\rightleftharpoons AB$, a concentration association constant is $K_a=[AB]/([A][B])$. Comparisons require matched solvent, temperature and standard-state conventions.

A standard state is a specified reference state used to define a species' standard chemical potential ${\mu }_i^{\circ }$. The reference convention, including temperature, must be consistent when calculating thermodynamic activities, reaction quotients, and equilibrium constants.

The reference pressure used to define a standard gas activity; a common convention is one bar. Normalizing pressure by this reference keeps a thermodynamic equilibrium constant dimensionless.

The reference concentration $c^{\circ }$ used to make concentration activities dimensionless. Changing this reference changes the numerical dimensionless equilibrium constant when the reaction changes the total number of solute or gas molecules.

The thermodynamic activity is the dimensionless quantity defined by ${\mu }_i={\mu }_i^{\circ }+RT\ln a_i$, where $R$ is the molar gas constant. For an ideal dilute solution with a specified concentration reference, $a_i=c_i/c^{\circ }$; departures from ideality are represented by an activity coefficient multiplying this ratio.

An effective gas pressure defined by $\mu ={\mu }^{\circ }+RT\ln (f/p^{\circ })$ in a specified standard-state convention. It approaches the gas pressure at low density and differs from the dimensionless fugacity.

The dimensionless ratio of a gas species' chemical fugacity to its pressure in the pure-fluid convention. Its thermodynamic activity relative to a gas reference pressure $P^{\circ }$ is $\varphi P/P^{\circ }$. For mixtures the pressure denominator uses the species' partial pressure. The ideal-gas limit has $\varphi =1$.

For a chemical reaction with signed stoichiometric vector entries ${\nu }_i$, positive for products and negative for reactants, the reaction quotient is the product of the species' thermodynamic activities raised to these powers. It equals the equilibrium constant at chemical equilibrium, with all quantities using the same standard state convention.

The chemical potential of photons in thermal equilibrium is zero because their number is not conserved.