Chemical Equilibria

In a chemical reaction, chemical equilibrium is the state in which both reactants and products are present in concentrations which have no further tendency to change with time.^[1] Usually, this state results when the forward reaction proceeds at the same rate as the reverse reaction. The reaction rates of the forward and backward reactions are generally not zero, but equal. Thus, there are no net changes in the concentrations of the reactant(s) and product(s). Such a state is known as dynamic equilibrium.

Historical Introduction

 

 Burette, a common laboratorl apparatus for carrying out titration, an important experimental technique in equilibrium and analytical chemistry.


The concept of chemical equilibrium was developed after Berthollet (1803) found that some chemical reactions are reversible. For any reaction mixture to exist at equilibrium, the rates of the forward and backward (reverse) reactions are equal. In the following chemical equation with arrows pointing both ways to indicate equilibrium, A and B are reactant chemical species, S and T are product species, and α, β, σ, and τ are the stoichiometric coefficients of the respective reactants and products:

α A + β B ⇌ σ S + τ T

The equilibrium concentration position of a reaction is said to lie "far to the right" if, at equilibrium, nearly all the reactants are consumed. Conversely the equilibrium position is said to be "far to the left" if hardly any product is formed from the reactants.
Guldberg and Waage (1865), building on Berthollet’s ideas, proposed the law of mass action:

${\displaystyle {\mbox{forward reaction rate}}=k_{+}\mathrm {A} ^{\alpha }\mathrm {B} ^{\beta }\,\!}$

${\displaystyle {\mbox{backward reaction rate}}=k_{-}\mathrm {S} ^{\sigma }\mathrm {T} ^{\tau }\,\!}$

where A, B, S and T are active masses and k₊ and k₋ are rate constants. Since at equilibrium forward and backward rates are equal:

${\displaystyle k_{+}\left\{\mathrm {A} \right\}^{\alpha }\left\{\mathrm {B} \right\}^{\beta }=k_{-}\left\{\mathrm {S} \right\}^{\sigma }\left\{\mathrm {T} \right\}^{\tau }\,}$

and the ratio of the rate constants is also a constant, now known as an equilibrium constant.

${\displaystyle K_{c}={\frac {k_{+}}{k_{-}}}={\frac {\{\mathrm {S} \}^{\sigma }\{\mathrm {T} \}^{\tau }}{\{\mathrm {A} \}^{\alpha }\{\mathrm {B} \}^{\beta }}}}$

By convention the products form the numerator. However, the law of mass action is valid only for concerted one-step reactions that proceed through a single transition state and is not valid in general because rate equations do not, in general, follow the stoichiometry of the reaction as Guldberg and Waage had proposed (see, for example, nucleophilic aliphatic substitution by S_N1 or reaction of hydrogen and bromine to form hydrogen bromide). Equality of forward and backward reaction rates, however, is a necessary condition for chemical equilibrium, though it is not sufficient to explain why equilibrium occurs.
Despite the failure of this derivation, the equilibrium constant for a reaction is indeed a constant, independent of the activities of the various species involved, though it does depend on temperature as observed by the van 't Hoff equation. Adding a catalyst will affect both the forward reaction and the reverse reaction in the same way and will not have an effect on the equilibrium constant. The catalyst will speed up both reactions thereby increasing the speed at which equilibrium is reached.^[2][4]
Although the macroscopic equilibrium concentrations are constant in time, reactions do occur at the molecular level. For example, in the case of acetic acid dissolved in water and forming acetate and hydronium ions,

CH₃CO₂H + H₂O ⇌ CH
3CO−
2 + H₃O⁺

a proton may hop from one molecule of acetic acid on to a water molecule and then on to an acetate anion to form another molecule of acetic acid and leaving the number of acetic acid molecules unchanged. This is an example of dynamic equilibrium. Equilibria, like the rest of thermodynamics, are statistical phenomena, averages of microscopic behavior.
Le Châtelier's principle (1884) gives an idea of the behavior of an equilibrium system when changes to its reaction conditions occur. If a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to partially reverse the change. For example, adding more S from the outside will cause an excess of products, and the system will try to counteract this by increasing the reverse reaction and pushing the equilibrium point backward (though the equilibrium constant will stay the same).
If mineral acid is added to the acetic acid mixture, increasing the concentration of hydronium ion, the amount of dissociation must decrease as the reaction is driven to the left in accordance with this principle. This can also be deduced from the equilibrium constant expression for the reaction:

${\displaystyle K={\frac {\ce {\{CH3CO2^{-}\}\{H3O+\}}}{\ce {\{CH3CO2H\}}}}}$

If {H₃O⁺} increases {CH₃CO₂H} must increase and CH
3CO−
2 must decrease. The H₂O is left out, as it is the solvent and its concentration remains high and nearly constant.
A quantitative version is given by the reaction quotient.
J. W. Gibbs suggested in 1873 that equilibrium is attained when the Gibbs free energy of the system is at its minimum value (assuming the reaction is carried out at constant temperature and pressure). What this means is that the derivative of the Gibbs energy with respect to reaction coordinate (a measure of the extent of reaction that has occurred, ranging from zero for all reactants to a maximum for all products) vanishes, signalling a stationary point. This derivative is called the reaction Gibbs energy (or energy change) and corresponds to the difference between the chemical potentials of reactants and products at the composition of the reaction mixture.^[1] This criterion is both necessary and sufficient. If a mixture is not at equilibrium, the liberation of the excess Gibbs energy (or Helmholtz energy at constant volume reactions) is the "driving force" for the composition of the mixture to change until equilibrium is reached. The equilibrium constant can be related to the standard Gibbs free energy change for the reaction by the equation

${\displaystyle \Delta _{r}G^{\ominus }=-RT\ln K_{\mathrm {eq} }}$

where R is the universal gas constant and T the temperature.
When the reactants are dissolved in a medium of high ionic strength the quotient of activity coefficients may be taken to be constant. In that case the concentration quotient, K_c,

${\displaystyle K_{\mathrm {c} }={\frac {[\mathrm {S} ]^{\sigma }[\mathrm {T} ]^{\tau }}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }}}}$

where [A] is the concentration of A, etc., is independent of the analytical concentration of the reactants. For this reason, equilibrium constants for solutions are usually determined in media of high ionic strength. K_c varies with ionic strength, temperature and pressure (or volume). Likewise K_p for gases depends on partial pressure. These constants are easier to measure and encountered in high-school chemistry courses.

Thermodynamics

At constant temperature and pressure, one must consider the Gibbs free energy, G, while at constant temperature and volume, one must consider the Helmholtz free energy: A, for the reaction; and at constant internal energy and volume, one must consider the entropy for the reaction: S.
The constant volume case is important in geochemistry and atmospheric chemistry where pressure variations are significant. Note that, if reactants and products were in standard state (completely pure), then there would be no reversibility and no equilibrium. Indeed, they would necessarily occupy disjoint volumes of space. The mixing of the products and reactants contributes a large entropy (known as entropy of mixing) to states containing equal mixture of products and reactants. The standard Gibbs energy change, together with the Gibbs energy of mixing, determine the equilibrium state.^[5][6]
In this article only the constant pressure case is considered. The relation between the Gibbs free energy and the equilibrium constant can be found by considering chemical potentials.^[1]
At constant temperature and pressure, the Gibbs free energy, G, for the reaction depends only on the extent of reaction: ξ (Greek letter xi), and can only decrease according to the second law of thermodynamics. It means that the derivative of G with ξ must be negative if the reaction happens; at the equilibrium the derivative being equal to zero.

${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=0~}$:     equilibrium

In order to meet the thermodynamic condition for equilibrium, the Gibbs energy must be stationary, meaning that the derivative of G with respect to the extent of reaction: ξ, must be zero. It can be shown that in this case, the sum of chemical potentials of the products is equal to the sum of those corresponding to the reactants. Therefore, the sum of the Gibbs energies of the reactants must be the equal to the sum of the Gibbs energies of the products.

${\displaystyle \alpha \mu _{\mathrm {A} }+\beta \mu _{\mathrm {B} }=\sigma \mu _{\mathrm {S} }+\tau \mu _{\mathrm {T} }\,}$

where μ is in this case a partial molar Gibbs energy, a chemical potential. The chemical potential of a reagent A is a function of the activity, {A} of that reagent.

${\displaystyle \mu _{\mathrm {A} }=\mu _{A}^{\ominus }+RT\ln\{\mathrm {A} \}\,}$

(where μo
A is the standard chemical potential).
The definition of the Gibbs energy equation interacts with the fundamental thermodynamic relation to produce

${\displaystyle dG=Vdp-SdT+\sum _{i=1}^{k}\mu _{i}dN_{i}}$.

Inserting dN_i = ν_i dξ into the above equation gives a Stoichiometric coefficient (${\displaystyle \nu _{i}~}$) and a differential that denotes the reaction occurring once (dξ). At constant pressure and temperature the above equations can be written as

${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\sum _{i=1}^{k}\mu _{i}\nu _{i}=\Delta _{\mathrm {r} }G_{T,p}}$ which is the "Gibbs free energy change for the reaction .

This results in:

${\displaystyle \Delta _{\mathrm {r} }G_{T,p}=\sigma \mu _{\mathrm {S} }+\tau \mu _{\mathrm {T} }-\alpha \mu _{\mathrm {A} }-\beta \mu _{\mathrm {B} }\,}$.

By substituting the chemical potentials:

${\displaystyle \Delta _{\mathrm {r} }G_{T,p}=(\sigma \mu _{\mathrm {S} }^{\ominus }+\tau \mu _{\mathrm {T} }^{\ominus })-(\alpha \mu _{\mathrm {A} }^{\ominus }+\beta \mu _{\mathrm {B} }^{\ominus })+(\sigma RT\ln\{\mathrm {S} \}+\tau RT\ln\{\mathrm {T} \})-(\alpha RT\ln\{\mathrm {A} \}+\beta RT\ln\{\mathrm {B} \})}$,

the relationship becomes:

${\displaystyle \Delta _{\mathrm {r} }G_{T,p}=\sum _{i=1}^{k}\mu _{i}^{\ominus }\nu _{i}+RT\ln {\frac {\{\mathrm {S} \}^{\sigma }\{\mathrm {T} \}^{\tau }}{\{\mathrm {A} \}^{\alpha }\{\mathrm {B} \}^{\beta }}}}$

${\displaystyle \sum _{i=1}^{k}\mu _{i}^{\ominus }\nu _{i}=\Delta _{\mathrm {r} }G^{\ominus }}$:

which is the standard Gibbs energy change for the reaction that can be calculated using thermodynamical tables. The reaction quotient is defined as:

${\displaystyle Q_{\mathrm {r} }={\frac {\{\mathrm {S} \}^{\sigma }\{\mathrm {T} \}^{\tau }}{\{\mathrm {A} \}^{\alpha }\{\mathrm {B} \}^{\beta }}}}$

Therefore,

${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\Delta _{\mathrm {r} }G_{T,p}=\Delta _{\mathrm {r} }G^{\ominus }+RT\ln Q_{\mathrm {r} }}$

At equilibrium:

${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\Delta _{\mathrm {r} }G_{T,p}=0}$

leading to:

${\displaystyle 0=\Delta _{\mathrm {r} }G^{\ominus }+RT\ln K_{\mathrm {eq} }}$

and

${\displaystyle \Delta _{\mathrm {r} }G^{\ominus }=-RT\ln K_{\mathrm {eq} }}$

Obtaining the value of the standard Gibbs energy change, allows the calculation of the equilibrium constant.

Addition of reactants or products

For a reactional system at equilibrium: Q_r = K_eq; ξ = ξ_eq.

• If are modified activities of constituents, the value of the reaction quotient changes and becomes different from the equilibrium constant: Q_r ≠ K_eq

${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\Delta _{\mathrm {r} }G^{\ominus }+RT\ln Q_{\mathrm {r} }~}$

and

${\displaystyle \Delta _{\mathrm {r} }G^{\ominus }=-RT\ln K_{eq}~}$

then

${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=RT\ln \left({\frac {Q_{\mathrm {r} }}{K_{\mathrm {eq} }}}\right)~}$

• If activity of a reagent i increases

${\displaystyle Q_{\mathrm {r} }={\frac {\prod (a_{j})^{\nu _{j}}}{\prod (a_{i})^{\nu _{i}}}}~}$, the reaction quotient decreases.

then

${\displaystyle Q_{\mathrm {r} }<K_{\mathrm {eq} }~}$     and     ${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}<0~}$

The reaction will shift to the right (i.e. in the forward direction, and thus more products will form).

• If activity of a product j increases

then

${\displaystyle Q_{\mathrm {r} }>K_{\mathrm {eq} }~}$     and     ${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}>0~}$

The reaction will shift to the left (i.e. in the reverse direction, and thus less products will form).

Note that activities and equilibrium constants are dimensionless numbers.

Treatment of activity

The expression for the equilibrium constant can be rewritten as the product of a concentration quotient, K_c and an activity coefficient quotient, Γ.

${\displaystyle K={\frac {[\mathrm {S} ]^{\sigma }[\mathrm {T} ]^{\tau }...}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }...}}\times {\frac {{\gamma _{\mathrm {S} }}^{\sigma }{\gamma _{\mathrm {T} }}^{\tau }...}{{\gamma _{\mathrm {A} }}^{\alpha }{\gamma _{\mathrm {B} }}^{\beta }...}}=K_{\mathrm {c} }\Gamma }$

[A] is the concentration of reagent A, etc. It is possible in principle to obtain values of the activity coefficients, γ. For solutions, equations such as the Debye–Hückel equation or extensions such as Davies equation^[7] Specific ion interaction theory or Pitzer equations^[8] may be used.^{Software (below)}. However this is not always possible. It is common practice to assume that Γ is a constant, and to use the concentration quotient in place of the thermodynamic equilibrium constant. It is also general practice to use the term equilibrium constant instead of the more accurate concentration quotient. This practice will be followed here.
For reactions in the gas phase partial pressure is used in place of concentration and fugacity coefficient in place of activity coefficient. In the real world, for example, when making ammonia in industry, fugacity coefficients must be taken into account. Fugacity, f, is the product of partial pressure and fugacity coefficient. The chemical potential of a species in the gas phase is given by

${\displaystyle \mu =\mu ^{\ominus }+RT\ln \left({\frac {f}{\mathrm {bar} }}\right)=\mu ^{\ominus }+RT\ln \left({\frac {p}{\mathrm {bar} }}\right)+RT\ln \gamma }$

so the general expression defining an equilibrium constant is valid for both solution and gas phases.

Concentration quotients

In aqueous solution, equilibrium constants are usually determined in the presence of an "inert" electrolyte such as sodium nitrate NaNO₃ or potassium perchlorate KClO₄. The ionic strength of a solution is given by

${\displaystyle I={\frac {1}{2}}\sum _{i=1}^{N}c_{i}z_{i}^{2}}$

where c_i and z_i stand for the concentration and ionic charge of ion type i, and the sum is taken over all the N types of charged species in solution. When the concentration of dissolved salt is much higher than the analytical concentrations of the reagents, the ions originating from the dissolved salt determine the ionic strength, and the ionic strength is effectively constant. Since activity coefficients depend on ionic strength the activity coefficients of the species are effectively independent of concentration. Thus, the assumption that Γ is constant is justified. The concentration quotient is a simple multiple of the equilibrium constant.^[9]

${\displaystyle K_{\mathrm {c} }={\frac {K}{\Gamma }}}$

However, K_c will vary with ionic strength. If it is measured at a series of different ionic strengths the value can be extrapolated to zero ionic strength.^[8] The concentration quotient obtained in this manner is known, paradoxically, as a thermodynamic equilibrium constant.
To use a published value of an equilibrium constant in conditions of ionic strength different from the conditions used in its determination, the value should be adjusted.

Metastable mixtures

A mixture may appear to have no tendency to change, though it is not at equilibrium. For example, a mixture of SO₂ and O₂ is metastable as there is a kinetic barrier to formation of the product, SO₃.

2 SO₂ + O₂ ⇌ 2 SO₃

The barrier can be overcome when a catalyst is also present in the mixture as in the contact process, but the catalyst does not affect the equilibrium concentrations.
Likewise, the formation of bicarbonate from carbon dioxide and water is very slow under normal conditions

CO₂ + 2 H₂O ⇌ HCO−
3 + H₃O⁺

but almost instantaneous in the presence of the catalytic enzyme carbonic anhydrase.

Pure substances

When pure substances (liquids or solids) are involved in equilibria their activities do not appear in the equilibrium constant^[10] because their numerical values are considered one.
Applying the general formula for an equilibrium constant to the specific case of a dilute solution of acetic acid in water one obtains

CH₃CO₂H + H₂O ⇌ CH₃CO₂⁻ + H₃O⁺

${\displaystyle K_{\mathrm {c} }={\frac {\mathrm {[{CH_{3}CO_{2}}^{-}][{H_{3}O}^{+}]} }{\mathrm {[{CH_{3}CO_{2}H}][{H_{2}O}]} }}}$

For all but very concentrated solutions, the water can be considered a "pure" liquid, and therefore it has an activity of one. The equilibrium constant expression is therefore usually written as

${\displaystyle K={\frac {\mathrm {[{CH_{3}CO_{2}}^{-}][{H_{3}O}^{+}]} }{\mathrm {[{CH_{3}CO_{2}H}]} }}=K_{\mathrm {c} }}$.

A particular case is the self-ionization of water itself

2 H₂O ⇌ H₃O⁺ + OH⁻

Because water is the solvent, and has an activity of one, the self-ionization constant of water is defined as

${\displaystyle K_{\mathrm {w} }=\mathrm {[H^{+}][OH^{-}]} }$

It is perfectly legitimate to write [H⁺] for the hydronium ion concentration, since the state of solvation of the proton is constant (in dilute solutions) and so does not affect the equilibrium concentrations. K_w varies with variation in ionic strength and/or temperature.
The concentrations of H⁺ and OH⁻ are not independent quantities. Most commonly [OH⁻] is replaced by K_w[H⁺]⁻¹ in equilibrium constant expressions which would otherwise include hydroxide ion.
Solids also do not appear in the equilibrium constant expression, if they are considered to be pure and thus their activities taken to be one. An example is the Boudouard reaction:^[10]

2 CO ⇌ CO₂ + C

for which the equation (without solid carbon) is written as:

${\displaystyle K_{\mathrm {c} }={\frac {\mathrm {[CO_{2}]} }{\mathrm {[CO]^{2}} }}}$

Multiple equilibria

Consider the case of a dibasic acid H₂A. When dissolved in water, the mixture will contain H₂A, HA⁻ and A²⁻. This equilibrium can be split into two steps in each of which one proton is liberated.

${\displaystyle {\begin{array}{rl}{\ce {H2A<=>{HA^{-}}+{H+}}}:&K_{1}={\frac {\ce {[HA-][H+]}}{\ce {[H2A]}}}\\{\ce {HA-<=>{A^{2-}}+{H+}}}:&K_{2}={\frac {\ce {[A^{2-}][H+]}}{\ce {[HA-]}}}\end{array}}}$

K₁ and K₂ are examples of stepwise equilibrium constants. The overall equilibrium constant, β_D, is product of the stepwise constants.

${\displaystyle {\ce {{H2A}<=>{A^{2-}}+{2H+}}}}$:     ${\displaystyle \beta _{\ce {D}}={\frac {\ce {[A^{2-}][H^{+}]^{2}}}{\ce {[H_{2}A]}}}=K_{1}K_{2}}$

Note that these constants are dissociation constants because the products on the right hand side of the equilibrium expression are dissociation products. In many systems, it is preferable to use association constants.

${\displaystyle {\begin{array}{ll}{\ce {{A^{2-}}+{H+}<=>HA-}}:&\beta _{1}={\frac {\ce {[HA^{-}]}}{\ce {[A^{2-}][H+]}}}\\{\ce {{A^{2-}}+{2H+}<=>H2A}}:&\beta _{2}={\frac {\ce {[H2A]}}{\ce {[A^{2-}][H+]^{2}}}}\end{array}}}$

β₁ and β₂ are examples of association constants. Clearly β₁ = 1/K₂ and β₂ = 1/β_D; log β₁ = pK₂ and log β₂ = pK₂ + pK₁^[11] For multiple equilibrium systems, also see: theory of Response reactions.

Effect of temperature

The effect of changing temperature on an equilibrium constant is given by the van 't Hoff equation

${\displaystyle {\frac {d\ln K}{dT}}={\frac {\Delta H_{\mathrm {m} }^{\ominus }}{RT^{2}}}}$

Thus, for exothermic reactions (ΔH is negative), K decreases with an increase in temperature, but, for endothermic reactions, (ΔH is positive) K increases with an increase temperature. An alternative formulation is

${\displaystyle {\frac {d\ln K}{d(T^{-1})}}=-{\frac {\Delta H_{\mathrm {m} }^{\ominus }}{R}}}$

At first sight this appears to offer a means of obtaining the standard molar enthalpy of the reaction by studying the variation of K with temperature. In practice, however, the method is unreliable because error propagation almost always gives very large errors on the values calculated in this way.

Effect of electric and magnetic fields

The effect of electric field on equilibrium has been studied by Manfred Eigen among others.

Types of equilibrium

Haber–Bosch process

1. N₂ (g) ⇌ N₂ (adsorbed)
2. N₂ (adsorbed) ⇌ 2 N (adsorbed)
3. H₂ (g) ⇌ H₂ (adsorbed)
4. H₂ (adsorbed) ⇌ 2 H (adsorbed)
5. N (adsorbed) + 3 H(adsorbed) ⇌ NH₃ (adsorbed)
6. NH₃ (adsorbed) ⇌ NH₃ (g)

• In the gas phase: rocket engines^[12]
• The industrial synthesis such as ammonia in the Haber–Bosch process (depicted right) takes place through a succession of equilibrium steps including adsorption processes
• Atmospheric chemistry
• Seawater and other natural waters: chemical oceanography
• Distribution between two phases
  • log D distribution coefficient: important for pharmaceuticals where lipophilicity is a significant property of a drug
  • Liquid–liquid extraction, Ion exchange, Chromatography
  • Solubility product
  • Uptake and release of oxygen by haemoglobin in blood
• Acid–base equilibria: acid dissociation constant, hydrolysis, buffer solutions, indicators, acid–base homeostasis
• Metal–ligand complexation: sequestering agents, chelation therapy, MRI contrast reagents, Schlenk equilibrium
• Adduct formation: host–guest chemistry, supramolecular chemistry, molecular recognition, dinitrogen tetroxide
• In certain oscillating reactions, the approach to equilibrium is not asymptotically but in the form of a damped oscillation .^[10]
• The related Nernst equation in electrochemistry gives the difference in electrode potential as a function of redox concentrations.
• When molecules on each side of the equilibrium are able to further react irreversibly in secondary reactions, the final product ratio is determined according to the Curtin–Hammett principle.

In these applications, terms such as stability constant, formation constant, binding constant, affinity constant, association/dissociation constant are used. In biochemistry, it is common to give units for binding constants, which serve to define the concentration units used when the constant’s value was determined.

Composition of a mixture

When the only equilibrium is that of the formation of a 1:1 adduct as the composition of a mixture, there are any number of ways that the composition of a mixture can be calculated. For example, see ICE table for a traditional method of calculating the pH of a solution of a weak acid.
There are three approaches to the general calculation of the composition of a mixture at equilibrium.

1. The most basic approach is to manipulate the various equilibrium constants until the desired concentrations are expressed in terms of measured equilibrium constants (equivalent to measuring chemical potentials) and initial conditions.
2. Minimize the Gibbs energy of the system.^[13][14]
3. Satisfy the equation of mass balance. The equations of mass balance are simply statements that demonstrate that the total concentration of each reactant must be constant by the law of conservation of mass.

Mass-balance equations

In general, the calculations are rather complicated or complex. For instance, in the case of a dibasic acid, H₂A dissolved in water the two reactants can be specified as the conjugate base, A²⁻, and the proton, H⁺. The following equations of mass-balance could apply equally well to a base such as 1,2-diaminoethane, in which case the base itself is designated as the reactant A:

${\displaystyle T_{\mathrm {A} }=\mathrm {[A]+[HA]+[H_{2}A]} \,}$

${\displaystyle T_{\mathrm {H} }=\mathrm {[H]+[HA]+2[H_{2}A]-[OH]} \,}$

With T_A the total concentration of species A. Note that it is customary to omit the ionic charges when writing and using these equations.
When the equilibrium constants are known and the total concentrations are specified there are two equations in two unknown "free concentrations" [A] and [H]. This follows from the fact that [HA] = β₁[A][H], [H₂A] = β₂[A][H]² and [OH] = K_w[H]⁻¹

${\displaystyle T_{\mathrm {A} }=\mathrm {[A]} +\beta _{1}\mathrm {[A][H]} +\beta _{2}\mathrm {[A][H]} ^{2}\,}$

${\displaystyle T_{\mathrm {H} }=\mathrm {[H]} +\beta _{1}\mathrm {[A][H]} +2\beta _{2}\mathrm {[A][H]} ^{2}-K_{w}[\mathrm {H} ]^{-1}\,}$

so the concentrations of the "complexes" are calculated from the free concentrations and the equilibrium constants. General expressions applicable to all systems with two reagents, A and B would be

${\displaystyle T_{\mathrm {A} }=[\mathrm {A} ]+\sum _{i}p_{i}\beta _{i}[\mathrm {A} ]^{p_{i}}[\mathrm {B} ]^{q_{i}}}$

${\displaystyle T_{\mathrm {B} }=[\mathrm {B} ]+\sum _{i}q_{i}\beta _{i}[\mathrm {A} ]^{p_{i}}[\mathrm {B} ]^{q_{i}}}$

It is easy to see how this can be extended to three or more reagents.

Polybasic acids

Species concentrations during hydrolysis of the aluminium.

The composition of solutions containing reactants A and H is easy to calculate as a function of p[H]. When [H] is known, the free concentration [A] is calculated from the mass-balance equation in A.
The diagram alongside, shows an example of the hydrolysis of the aluminium Lewis acid Al³⁺_(aq)^[15] shows the species concentrations for a 5 × 10⁻⁶ M solution of an aluminium salt as a function of pH. Each concentration is shown as a percentage of the total aluminium.

Solution and precipitation

The diagram above illustrates the point that a precipitate that is not one of the main species in the solution equilibrium may be formed. At pH just below 5.5 the main species present in a 5 μM solution of Al³⁺ are aluminium hydroxides Al(OH)²⁺, AlOH+
2 and Al
13(OH)7+
32, but on raising the pH Al(OH)₃ precipitates from the solution. This occurs because Al(OH)₃ has a very large lattice energy. As the pH rises more and more Al(OH)₃ comes out of solution. This is an example of Le Châtelier's principle in action: Increasing the concentration of the hydroxide ion causes more aluminium hydroxide to precipitate, which removes hydroxide from the solution. When the hydroxide concentration becomes sufficiently high the soluble aluminate, Al(OH)−
4, is formed.
Another common instance where precipitation occurs is when a metal cation interacts with an anionic ligand to form an electrically neutral complex. If the complex is hydrophobic, it will precipitate out of water. This occurs with the nickel ion Ni²⁺ and dimethylglyoxime, (dmgH₂): in this case the lattice energy of the solid is not particularly large, but it greatly exceeds the energy of solvation of the molecule Ni(dmgH)₂.

Minimization of Gibbs energy

At equilibrium, at a specified temperature and pressure, the Gibbs energy G is at a minimum:

${\displaystyle dG=\sum _{j=1}^{m}\mu _{j}\,dN_{j}=0}$

For a closed system, no particles may enter or leave, although they may combine in various ways. The total number of atoms of each element will remain constant. This means that the minimization above must be subjected to the constraints:

${\displaystyle \sum _{j=1}^{m}a_{ij}N_{j}=b_{i}^{0}}$

where a_ij is the number of atoms of element i in molecule j and b0
i is the total number of atoms of element i, which is a constant, since the system is closed. If there are a total of k types of atoms in the system, then there will be k such equations. If ions are involved, an additional row is added to the a_ij matrix specifying the respective charge on each molecule which will sum to zero.
This is a standard problem in optimisation, known as constrained minimisation. The most common method of solving it is using the method of Lagrange multipliers, also known as undetermined multipliers (though other methods may be used).
Define:

${\displaystyle {\mathcal {G}}=G+\sum _{i=1}^{k}\lambda _{i}\left(\sum _{j=1}^{m}a_{ij}N_{j}-b_{i}^{0}\right)=0}$

where the λ_i are the Lagrange multipliers, one for each element. This allows each of the N_j and λ_j to be treated independently, and it can be shown using the tools of multivariate calculus that the equilibrium condition is given by

${\displaystyle 0={\frac {\partial {\mathcal {G}}}{\partial N_{j}}}=\mu _{j}+\sum _{i=1}^{k}\lambda _{i}a_{ij}}$

${\displaystyle 0={\frac {\partial {\mathcal {G}}}{\partial \lambda _{i}}}=\sum _{j=1}^{m}a_{ij}N_{j}-b_{i}^{0}}$

(For proof see Lagrange multipliers.) This is a set of (m + k) equations in (m + k) unknowns (the N_j and the λ_i) and may, therefore, be solved for the equilibrium concentrations N_j as long as the chemical potentials are known as functions of the concentrations at the given temperature and pressure. (See Thermodynamic databases for pure substances.) Note that the second equation is just the initial constraints for minimization.
This method of calculating equilibrium chemical concentrations is useful for systems with a large number of different molecules. The use of k atomic element conservation equations for the mass constraint is straightforward, and replaces the use of the stoichiometric coefficient equations.^[12]. The results are consistent with those specified by chemical equations. For example, if equilibrium is specified by a single chemical equation: ^[16],

${\displaystyle \sum _{j=0}^{m}\nu _{j}R_{j}=0}$

where ν_j is the stochiometric coefficient for the j th molecule (negative for reactants, positive for products) and R_j is the symbol for the j th molecule, a properly balanced equation will obey:

${\displaystyle \sum _{j=1}^{m}a_{ij}\nu _{j}=0}$

Multiplying the first equilibrium condition by ν_j yields

${\displaystyle 0=\sum _{j=1}^{m}\nu _{j}\mu _{j}+\sum _{j=1}^{m}\sum _{i=1}^{k}\nu _{j}\lambda _{i}a_{ij}=\sum _{j=1}^{m}\nu _{j}\mu _{j}}$

As above, defining ΔG

${\displaystyle \Delta G=\sum _{j=1}^{m}\nu _{j}\mu _{j}=\sum _{j=1}^{m}\nu _{j}(\mu _{j}^{\ominus }+RT\ln(\{R_{j}\}))=\Delta G^{\ominus }+RT\ln \left(\prod _{j=1}^{m}\{R_{j}\}^{\nu _{j}}\right)=\Delta G^{\ominus }+RT\ln(K_{eq})}$

which will be zero at equilibrium.

#2 Intoduction to Communication

VERBAL COMMUNICATION


We use verbal communication for most purposes. Verbal communication may be oral or written.

a)ORAL COMMUNICATION:


Oral Communication is more natural and immediately available for responding to a comment / statement. In natural and informal situations, we speak readily without hesitation in order to communicate with others; but in a formal and official situation, many persons feel nervous and cannot speak easily. It needs training, practice and skill to speak effectively in a formal situation.
Oral communication requires the presence and simultaneous attention of both the persons. Need for personal presence makes certain demands on the skills of both; each must be able to respond to the body language of the other, and must be able to make immediate response to what the other says.
Oral communication occurs in situations like conversations, telephone talk, interviews, presentations, group discussions, and meetings.


FACE-TO-FACE CONVERSATION:

Oral communication is best when it is face-to-face. A face-to-face setting is possible between two individuals or among a small group of persons at an interview, or in a small meeting, where both the sender and the receiver could see each other and communicate. Communication can flow both ways in these situations. Here, an immediate feedback, which gives clarification is possible. Besides, a face-to-face setting offers a rich communication experience owing to the presence of the living personality whose voice, tone, expressions and movements add significance to the words.


TELEPHONE TALK:


Telephone talk depends entirely on the voice and its quality. It does not have the advantage of physical presence or facial expressions since there is no option to look at others physical appearance at live. Clarity of speech and skillful use of voice are important in this kind of communication. There can be confusion between similar sounding words like “pale” and “bale”, or between “light” and “like”. Names and addresses communicated on the telephone are sometimes wrongly received. It is therefore customary in telephonic conversation to clarify spellings by saying G for God, P for pen etc.


PRESENTATION:


It has a face-to-face setting. It is a formal, well-prepared talk on a specific topic, delivered to knowledgeable and interested audience. It looks odd and slumbers if the presentation is not welcomed by the audience to which it is presented. At times a touch of humour always enriches the presentation. The purpose for such kind of communication is to give / pass on the information rather than making them dull and sleepy.


PUBLIC SPEECH:

 A public speech or lecture also has a face-to-face setting, but here the space between the speaker and audience do matters. This distance increases as the audience gets larger, as in an open air public meeting. This way of communication much depends on the speaker’s skill in using gestures and using the microphone in the correct order.

INTERVIEW:


An interview is a meeting at which one person or panel of persons, who are the interviewers, discuss a matter with another person or ask questions of another person, who is the interviewee. The purpose is, usually to assess, to judge whether it would be worthwhile to enter into a relationship with the other. An interview is of structured question and answer type of communication.



MEETING:

 Usually a meeting involves many persons; there is a chair person or leader who leads and guides the communication and maintains perfect order. There is a fixed agenda, that is, a list of issues to be discussed at the meeting. Meetings are of many types, from the small committee meeting consisting of three or four persons to the large conference or the share holders’ meeting. This type of oral communication is backed up by note-taking and writing up of minutes.



b)WRITTEN COMMUNICATION:


Written communication is used for many purposes. Many types of documents are required for official work. Letters, circulars, memos, notices, reports and minutes are constantly prepared and exchanged in and between organizations. All has a format and layout which is fixed by custom.

Letter:
Letters are the most widely used form of written communication. They are used mostly for external communication. A letter has a complex lay-out which has to be carefully followed.


Memo:
 Memo, short form of memorandum, is an informal message between members of an organization and generally relates to daily work. Information or instructions can be conveyed by a memo. A memo may or may not be signed.


Notice:
 A notice is used in order to communicate the same message within an organization. It is the most common method of mass communication, within an organization. It should be short, its language should be simple and the type should be large and well spaced for easy reading.


Circular:
 A circular is a detailed document giving information, instructions or orders on a specific matter. A circular has a number and date for reference, and is signed by the authorized signatory of the issuing office. They are generally issued by government department and other official bodies like government departments, councils, universities and Head Offices of organizations.


Report:
 A report is a document prepared by an individual or a committee entrusted with the task of collecting information on a given subject. It requires careful research, collection of data and presentation of the findings, conclusions and recommendations. Reports are of varying length and may be anything from two pages to a full book dived into chapters.

Minutes:
 Minutes are the written record of decisions taken at a meeting. Different bodies have their own convention of recordings the discussion and the decisions. Minutes may be written by hand or typed and pasted in minute books, or typed and filled in a minute file. Minutes are a legal document.



NON-VERBAL COMMUNICATION

Non-verbal methods of communication can be consciously created and used with both written and oral communication. Graphics of all kinds can enrich the message presented in a document or in a speech. Pictures, maps, charts, diagrams, sketches, cutouts, models, etc., communicate more effectively quality vice and clarity vice than verbal communication. Apart from these symbols we consciously may convey the meaning by facial expressions, gestures, eye contact, clothing, posture, etc. These are called body language. They do communicate more than verbal communication.
Non-verbal communication occurs even when there is no verbal communication. Going by the road side, on seeing the no parking board, we are not parking our vehicles near it. Rather a NSS volunteer person when suggesting not to leave our vehicle in that place, often we ignore him. Thus we say that non-verbal communication, by way of a picture here, communicates something more than what is communicated through verbal communication. Henceforth, a good understanding of non-verbal communication will entitle a person or persons to communicate more effectively than what is conveyed through verbal communication.

Introduction to Communication

Introduction

 Have you ever felt the messages you convey are not communicated properly or have you ever felt guilty of not conveying the message as it wants to be conveyed? If so it is because of your weakness towards communication skills. Apart from the basic necessities, you need to be equipped with habits for good communication skills, as this is what will make you a happy and successful social being. In order to develop these habits, you need to first acknowledge the fact that communication skills need an improvement from time to time. The only constant in life is change, and the more you accept your strengths and work towards dealing with shortcomings, especially in the area of communication skills, the better will be your interactions and the more your social popularity. Thus the present unit enables you to get a detailed picture of the need and importance of developing communication skills and feel confident and empowering to face any type of situation in life.

1 Communication and Its Process

We use ‘communication’ usually to mean speaking or writing or sending a message to another person. Communication is really much more than that. It involves a number of choices and decisions but being natural and unnoticed in informal situations. In formal situations, our communication needs to be more effective and carefully chosen, that is, we need to plan our communication. Here comes the question what is Communication or how communication could be defined?
Communication may be broadly defined as the process of meaningful interaction among human beings. More specifically, it is the process by which meanings are perceived and understandings are reached among human beings. – D.E. McFarland.
Looked at more closely, what is essential for communication to occur is the cooperation between two parties, one active or at the giving end and the other passive or at the receiving end.
The sender selects appropriate symbols to suit the situation and realizes the meaning through speech or writing depending upon the socially regulated requirements or self-perceived needs. At the receiving end the symbols are identified and identification obviously implies recognition and realization of meaning through the interpretive process.
The process of Communication may be summarized as follows:

Communication is thus a network of interactions and naturally the sender and the receiver keep on changing their roles.
Another aspect of communication is the deployment of a code consisting of arbitrarily evolved symbols and the determination of the appropriateness of their use in given situations, leading to the emergence of diverse communication patterns. A number of factors come into play in shaping these patterns. In fact, communication is often but not always momentary. At times communication is a cumulative process that starts before the actual communicative event takes place and continues after it has occurred. Thus communication therefore must acquire a true perspective of not only the present requirements of the situation but also its relationship with the past and its impact on the future.

2 Components of Communication


Communication is a process where one sets out to convey a message to another person through the medium of words, gestures and / or pictures. The process of conveying the message is fulfilled only when the person receiving it has understood the message entirely.

The cycle gives the process of communication. It would be observed that the entire event takes place within a common frame of reference, also called as communication environment. The source refers to the point of origin of a message which is encoded by the sender and transmitted through the channel to the receiver. The receipt of the message exercises an impact in communication environment leading to some result. The observance of the result by the sender is called ‘feedback’.
During feedback the direction of the communication process is reversed. When providing feedback, the original receiver goes through the same process as did the original sender with the same factors influencing the receiver. The receiver may use the same channel / a different channel for feedback.
The message sent is not the same as the message received. It is also to be noted that all the messages do not produce the intended result. Thus, the success of communication is measured in terms of not only the effective transmission of the message but also the achievement of the intended result.

3 Barriers to Communication

Communication is not always successful. Several things can prevent the message from reaching the intended recipient or from having the desired effect on the recipient. There may be some fault in the communication system which may prevent the message from reaching. Some of these defects are in the mechanical devices used for transmitting – medium, some are in tools we use for communication – language, and some are in nature of persons who are engaged in communication – the sender and recipient / receiver. It can be divided into three broad groups: Listening, Speaking and Environmental.

Listening barriers:

• Interrupting the speaker
• Not maintaining eye contact with the speaker
• Rushing the speaker to complete what he/she has to say
• Making the speaker feel as though he/she is wasting the listener's time
• Being distracted by something that is not part of the ongoing communication
• Getting ahead of the speaker and completing his/her thoughts
• Ignoring the speaker's requests
• Topping the speaker's story with one's own set of examples
• Forgetting what is being discussed
• Asking too many questions, for the sake of probing

Barriers while speaking:

• Unclear messages
• Lack of consistency in the communication process
• Incomplete sentences
• Not understanding the receiver
• Not seeking clarifications while communicating

Environmental barriers include:

• An individual's subjective viewpoint towards issues/people, which leads to assumptions
• An emotional block, which can lead to an attitude of indifference, suspicion or hostility towards the subject
• An emotional block or bias that is based on a third party's view point, or on what you have read/heard
• Words can have different meanings to different people, thus blocking communication
• Use of negative words

4 Patterns of Communication



Communication can be one way and two way process. Both the patterns are followed in various circumstances effectively and efficiently by the people in the society. There is also horizontal and vertical movement of information from one source to another. The horizontal flow keeps individuals of the same status and peer groups informed of what others are doing and what is expected of them. The vertical communication is both downward and upward. It is essential to have both upward and downward movement since mere downward flow is like talking to a person continuously without giving him a chance to respond. In such situation the pattern will be as given in One Way communication and when mutual chance is given to the listener, the pattern is similar to two way communication.

-One way Communication

• A person always instructing another
• Always speaking without giving chance for the other to respond
• Always directing the other to do something
• Conveying some information to other and so on.

-Two way Communication

 

• A person named X instructs Y
• Y reports to X
• X speaks to Y
• Y responds to X and so on

This two way communication could be classified into two: One to many and many to one.


(a)One to One Communication

 In one to one communication there is only one sender and one receiver wherein the sender passes on some information to the receiver and the receiver passes on to the sender in return. Such kind of communication is one to one communication in two way communication pattern.

 (b)One to Many Communication


In one to many communication there is only one sender and one or more receiver wherein the sender passes on some information to all the receivers and each of the receiver passes on / replies to the sender in return. Such kind of communication is One to Many Communication in two way communication pattern.


(c)Many to One Communication


In Many to One Communication there are several senders and one receiver wherein all the senders pass on some information to the receiver and the receiver replies for them. Such kind of communication is Many to One Communication in two way communication pattern.
It is obvious that conveying all information to everybody would be a meaningless exercise. For proper functioning of a Communication system the following questions should be asked and the answers constantly reviewed:

• What information is to be conveyed?
• Who requires it?
• What should be its form?
• What techniques of dissemination should be used?
• What technological aids should be used? 

5 Types of Communication

 Communication takes place by exchanging symbols to describe our ideas and experience. Language is a common symbol system which is used for sharing our experiences with others. We can also use other symbols like pictures, colours, signs and sounds to communicate. We do communicate a number of things by our facial expressions, movements, clothing and so on, though we do not speak. Thus communications through words are called Verbal Communication; communications through symbols are called non-verbal communication.