# Activity Coefficient

An activity coefficient is a factor used in thermodynamics to account for deviations from ideal behavior in a mixture of chemical substances. In an ideal mixture, the microscopic interactions between each pair of chemical species are the same (or macroscopically equivalent, the enthalpy change of the solution and the volume variation in the mixture are zero) and, as a result, the properties of the mixture can be Expressed directly as simple concentrations or partial pressures of substances present , as in Raoult’s law . an activity coefficient. Deviation from the norm is adjusted by modifying the concentration by . Similarly , expressions involving gases can be adjusted for non- ideality by scaling the partial pressures by a fugacity coefficient.

The concept of the activity coefficient is closely associated with activity in chemistry .

thermodynamic definition

The chemical potential of substance B in an ideal mixture of liquids or an ideal solution, μB, is given by

{\displaystyle \mu _{\mathrm {B} }=\mu _{\mathrm {B} }^{\ominus }+RT\ln x_{\mathrm {B} }\,}

where μob A is the chemical potential of a pure substance and X is the mole fraction of the substance in the mixture. B

This is generalized to include non-ideal behavior by writing

{\displaystyle \mu _{\mathrm {B} }=\mu _{\mathrm {B} }^{\ominus }+RT\ln a_{\mathrm {B} }\,}

When B is the activity of the substance in the mixture

{\displaystyle a_{\mathrm {B} }=x_{\mathrm {B} }\gamma _{\mathrm {B} }}

where b activity coefficient, which by itself can depend on x b . As B approaches 1, matter behaves as if it were ideal. For example, if b 1, then Raoult ‘s law is accurate. For B > 1 and B < 1, positive and negative deviations from Raoult ‘s law from the performance of substance B, respectively . A positive deviation implies that substance B is more volatile.

In many cases, the activity coefficient of substance B becomes constant as soon as X becomes zero; This relation is Henry’s law for the solvent . These relations are related to each other through the Gibbs–Duheim equation. Note that in normal activity the coefficients are dimensionless.

In detail: Raoult’s law states that the partial pressure of component B is related to its vapor pressure (saturation pressure) and its mole fraction x B in the liquid phase ,

{\displaystyle p_{\mathrm {B} }=x_{\mathrm {B} }\gamma _{\mathrm {B} }p_{\mathrm {B} }^{\sigma }\;,}

In other words with convention : Pure liquids represent ideal conditions.

{\displaystyle \lim _{x_{\mathrm {B} }\to 1}\gamma _{\mathrm {B} }=1\;.}

At infinite dilution, the activity coefficient approaches its limiting value , B . Compare with Henry’s law ,

{\displaystyle p_{\mathrm {B} }=K_{\mathrm {H,B} }x_{\mathrm {B} }\quad {\text{for}}\quad x_{\mathrm {B} }\to 0\;,}

gives immediately

{\displaystyle K_{\mathrm {H,B} }=p_{\mathrm {B} }^{\sigma }\gamma _{\mathrm {B} }^{\infty }\;.}

In other words: the compound shows non-ideal behavior in the dilute case.

The above definition of activity coefficient is impractical if the compound does not exist as a pure liquid. This is often the case with electrolytes or biochemical compounds. In such cases, a different definition is used that considers infinite dilution to be the ideal condition:

{\displaystyle \gamma _{\mathrm {B} }^{\dagger }\equiv \gamma _{\mathrm {B} }/\gamma _{\mathrm {B} }^{\infty }}

together and

{\displaystyle \lim _{x_{\mathrm {B} }\to 0}\gamma _{\mathrm {B} }^{\dagger }=1\;,}
{\displaystyle \mu _{\mathrm {B} }=\underbrace {\mu _{\mathrm {B} }^{\ominus }+RT\ln \gamma _{\mathrm {B} }^{\infty }} _{\mu _{\mathrm {B} }^{\ominus \dagger }}+RT\ln \left(x_{\mathrm {B} }\gamma _{\mathrm {B} }^{\dagger }\right)}

The symbol is used here to differentiate between the two types of activity coefficients. It is usually omitted, as it is clear from the context what type is meant. But there are cases where both types of activity coefficients are needed and can also appear in the same equation, for example, for solutions of salts in (water + alcohol) mixtures. This is sometimes the source of errors.

Modifying the mole fractions or concentrations by activity coefficients gives the effective activities of the components , and so expressions such as Raoult’s law and equilibrium constant apply to both ideal and non-ideal mixtures.

Knowledge of the activity coefficient is particularly important in the context of electrochemistry because the behavior of electrolyte solutions is often far from ideal due to the effects of the ionic environment . Additionally, they are particularly important in the context of soil chemistry due to their low solvent content and, consequently, high concentration of electrolytes .

### ionic solution

For solutions of substances that are ionized in solution, the activity coefficients of the cation and anion cannot be experimentally determined independently of each other because the solution properties depend on both ions. The single ion activity coefficient should be linked to the activity coefficient of the dissolved electrolyte as undissolved. In this case a mean stoichiometric activity coefficient of the dissolved electrolyte, ± gamma , is used. It is called stoichiometric because it expresses both the deviation from the ideality of the solution and the incomplete ionic dissociation of an ionic compound that typically occurs with an increase in its concentration.

For a 1:1 electrolyte, such as NaCl it is given by:

\gamma _{\pm }={\sqrt {\gamma _{+}\gamma _{-}}}

where γ+ and γ are the activity coefficients of the cation and anion respectively.

More generally, the mean activity coefficient of a compound of the formula A p B q is given by.

{\displaystyle \gamma _{\pm }={\sqrt[{p+q}]{\gamma _{\mathrm {A} }^{p}\gamma _{\mathrm {B} }^{q}}}}

The single-ion activity coefficient can be calculated theoretically, for example by using the Debye–Huckel equation . The theoretical equation can be tested by combining the calculated single-ion activity coefficients to give a mean value that can be compared with the experimental values.

The prevailing view that single ion activity coefficients are independently scalable, or perhaps even physically meaningless, has its roots in the work of Guggenheim in the late 1920s.  However, chemists have never been able to abandon the idea of ​​single ion activities and the implication single ion activity coefficient. For example, pH is defined as the negative logarithm of hydrogen ion activity. If the prevailing view on the physical meaning and scalability of single ion activities is correct, then defining pH as the negative logarithm of hydrogen ion activity places the quantity in the scalable range. Recognizing this logistical difficulty, the International Union of Pure and Applied Chemistry (IUPAC) states that the activity-based definition of pH is only a hypothetical definition. Despite the prevailing negative view on the scalability of single ion coefficients, the concept of single ion activities continues to be discussed in the literature, and at least one author presents a definition of single ion activity in terms of purely thermodynamic quantities and proposes a method of measuring single ion activity coefficients based on purely thermodynamic processes.

### concentrated ionic solution

For concentrated ionic solutions the hydration of ions must be taken into account, as did Stokes and Robinson in their hydration model from 1948.

The activity coefficient of the electrolyte is divided into electrical and statistical components by E. Glueckoff who modifies the Robinson–Stokes model.

The statistical part also includes the hydration index number h , the number of ions by dissociation and the ratio r between the apparent molar volume of the electrolyte and the molar volume of water and the molality b.

The concentrated solution statistical part of the activity coefficient is:

{\displaystyle \ln \gamma _{s}={\frac {h-\nu }{\nu }}\ln \left(1+{\frac {br}{55.5}}\right)-{\frac {h}{\nu }}\ln \left(1-{\frac {br}{55.5}}\right)+{\frac {br(r+h-\nu )}{55.5\left(1+{\frac {br}{55.5}}\right)}}}

The Stokes–Robinson model has also been analyzed and improved by other investigators.

## Experimental determination of activity coefficient

The activity coefficients can be determined experimentally by measuring them on non-ideal mixtures. Use can be made of Raoult’s law or Henry’s law to provide a value for an ideal mixture against which the experimental value can be compared to obtain the activity coefficient. Other collinear properties such as osmotic pressure can also be used.

The activity coefficient can be determined by radiochemical methods. 

### at infinite dilution

Activity coefficients for binary mixtures are often reported at infinite dilutions of each component. Because activity coefficient models are simplified at infinite dilution, such empirical values ​​can be used to estimate the interaction energy. Examples are given for water:

Theoretical Calculation of Activity Coefficient

The activity coefficient of an electrolyte solution can be calculated theoretically, using the Debye–Huckel equation or extensions such as the Davis equation, the Pitzer equation or the TCPC model. Specific ion interaction theory (SIT) can also be used.

For non-electrolyte solutions, correlating methods such as UNIQUAC, NRTL, MOSCED or UNIFAC can be employed, provided that fitted component-specific or model parameters are available. COSMO-RS is a theoretical method that is less dependent on model parameters because the required information is obtained from specific quantum mechanics calculations for each molecule (sigma profile), combined with statistical thermodynamics treatment of surface segments.

For unplanted species, the activity coefficient 0 is mostly a variant of the lay-out model:

\log _{{10}}(\gamma _{{0}})=bI

This simple model predicts the activities of many species (such as CO 2 , H 2 S, NH 3 , dissociated acids and bases) for high ionic strength (up to 5 mol/kg) . The value of constant b for CO 2 is 0.11 at 10 °C and 0.20 at 330 °C.

For water as the solvent, the activity w can be calculated using:

{\displaystyle \ln(a_{\mathrm {w} })={\frac {-\nu b}{55.51}}\varphi }

where is the number of ions produced by dissociation of a molecule of dissolved salt, b is the molality of the salt dissolved in water, is the osmotic coefficient of water, and the constant 55.51 represents the molality of water. In the above equation, the activity of a solvent (here water) is represented as inversely proportional to the number of salt particles versus the number of solvent.

## link to ionic diameter

The ionic activity coefficient is linked to the ionic diameter formula obtained by the Debye-Hückel theory of electrolytes:

{\displaystyle \log(\gamma _{i})=-{\frac {Az_{i}^{2}{\sqrt {I}}}{1+Ba{\sqrt {I}}}}}

where a and b are constants, z is the valence number of the i ion, and i is the ionic strength.

## dependency on state

The derivative of an activity coefficient with respect to temperature is related to the additional molar enthalpy

{\displaystyle {\bar {H}}_{i}^{\mathsf {E}}=-RT^{2}{\frac {\partial }{\partial T}}\ln(\gamma _{i})}

Similarly, the derivative of an activity coefficient with respect to pressure can be related to the additional molar volume.

{\displaystyle {\bar {V}}_{i}^{\mathsf {E}}=RT{\frac {\partial }{\partial P}}\ln(\gamma _{i})}

## Application for chemical equilibration

At equilibrium, the sum of the chemical potentials of the reactants is equal to the sum of the chemical potentials of the products. The Gibbs free energy change for reactions, g , is equal to the difference between these sums and therefore, at equilibrium, is equal to zero. Thus, for an equilibrium such as

α A + β B ⇌ σ S + τ T

{\displaystyle \Delta _{\mathrm {r} }G=\sigma \mu _{\mathrm {S} }+\tau \mu _{\mathrm {T} }-(\alpha \mu _{\mathrm {A} }+\beta \mu _{\mathrm {B} })=0\,}

Substitute in the expressions for the chemical potential of each reactant:

{\displaystyle \Delta _{\mathrm {r} }G=\sigma \mu _{S}^{\ominus }+\sigma RT\ln a_{\mathrm {S} }+\tau \mu _{\mathrm {T} }^{\ominus }+\tau RT\ln a_{\mathrm {T} }-(\alpha \mu _{\mathrm {A} }^{\ominus }+\alpha RT\ln a_{\mathrm {A} }+\beta \mu _{\mathrm {B} }^{\ominus }+\beta RT\ln a_{\mathrm {B} })=0}

On rearrangement this expression becomes

{\displaystyle \Delta _{\mathrm {r} }G=\left(\sigma \mu _{\mathrm {S} }^{\ominus }+\tau \mu _{\mathrm {T} }^{\ominus }-\alpha \mu _{\mathrm {A} }^{\ominus }-\beta \mu _{\mathrm {B} }^{\ominus }\right)+RT\ln {\frac {a_{\mathrm {S} }^{\sigma }a_{\mathrm {T} }^{\tau }}{a_{\mathrm {A} }^{\alpha }a_{\mathrm {B} }^{\beta }}}=0}
The ~~sum~~\sigma  \mu^ \vartheta _s~+~T \mu^ \vartheta _T~-~ \alpha   \mu^ \vartheta _A~-~ \beta  \mu^ \vartheta _ \beta

The standard free energy change for the reaction is,

ΔrG^o~~~Therefore,

K is the equilibrium constant. Note that the activities and equilibrium constants are dimensionless numbers.

This derivation serves two purposes. This standard shows the relationship between the free energy change and the equilibrium constant. It also shows that an equilibrium constant is defined as the quotient of activities. In practice it is inconvenient. When each activity is replaced by the product of a concentration and an activity coefficient, the equilibrium constant is defined as

{\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 }}}}

where [S] denotes the concentration of S, etc. In practice the equilibrium constants are determined in a medium such that the quotient of the activity coefficient is constant and can be ignored, giving the general expression

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

which applies under the condition that the activity quotient has a particular (constant) value.