Conjugate gradient method

An activity coefficient is a factor used in thermodynamics to account for deviations from ideal behaviour 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 solution and volume variation in mixing is zero) and, as a result, properties of the mixtures can be expressed directly in terms of simple concentrations or partial pressures of the substances present e.g. Raoult's law. Deviations from ideality are accommodated by modifying the concentration by an activity coefficient. Analogously, expressions involving gases can be adjusted for non-ideality by scaling partial pressures by a fugacity coefficient.

The concept of activity coefficient is closely linked to that of activity in chemistry.

Thermodynamic definition

The chemical potential, $\mu _{B}$, of a substance B in an ideal mixture is given by

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

where $\mu _{B}^{\ominus }$ is the chemical potential in the standard state and x_B is the mole fraction of the substance in the mixture.

This is generalised to include non-ideal behavior by writing

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

when $a_{B}$ is the activity of the substance in the mixture with

$a_{B}=x_{B}\gamma _{B}$

where $\gamma _{B}$ is the activity coefficient. As $\gamma _{B}$ approaches 1, the substance behaves as if it were ideal. For instance, if $\gamma _{B}\approx 1$, then Raoult's Law is accurate. For $\gamma _{B}>1$ and $\gamma _{B}<1$, substance B shows positive and negative deviation from Raoult's law, respectively. A positive deviation implies that substance B is more volatile.

In many cases, as $x_{B}$ goes to zero, the activity coefficient of substance B approaches a constant; this relationship is Henry's Law for the solvent. These relationships are related to each other through the Gibbs-Duhem equation. Note that in general activity coefficients are dimensionless.

Modifying mole fractions or concentrations by activity coefficients gives the effective activities of the components, and hence allows expressions such as Raoult's law and equilibrium constants constants to be applied to both ideal and non-ideal mixtures.

Knowledge of activity coefficients is particularly important in the context of electrochemistry since the behaviour of electrolyte solutions is often far from ideal, due to the effects of the ionic atmosphere. Additionally, they are particularly important in the context of soil chemistry due to the low volumes of solvent and, consequently, the high concentration of electrolytes.

Dependence on state parameters

The derivative of the activity coefficient to temperature and respectively pressure are connected to the excess molar properties.

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

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

Application to chemical equilibrium

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 the reactions, $\Delta _{r}G$, is equal to the difference between these sums and therefore, at equilibrium, is equal to zero. Thus, for an equilibrium such as

$\alpha A+\beta B\rightleftharpoons \sigma S+\tau T$

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

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

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

Upon rearrangement this expression becomes

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

The sum $\left(\sigma \mu _{S}^{\ominus }+\tau \mu _{T}^{\ominus }-\alpha \mu _{A}^{\ominus }-\beta \mu _{B}^{\ominus }\right)$ is the standard free energy change for the reaction, $\Delta _{r}G^{\ominus }$. Therefore

$\Delta _{r}G^{\ominus }=-RT\ln K$

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

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

$K={\frac {[S]^{\sigma }[T]^{\tau }}{[A]^{\alpha }[B]^{\beta }}}\times {\frac {\gamma _{S}^{\sigma }\gamma _{T}^{\tau }}{\gamma _{A}^{\alpha }\gamma _{B}^{\beta }}}$

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

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

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

Measurement and prediction of activity coefficients

UNIQUAC Regression of Activity Coefficients (Chloroform/Methanol Mixture)

Activity coefficients may be measured experimentally or calculated theoretically, using the Debye-Hückel equation or extensions such as Davies equation, Pitzer equations or TCPC model. Specific ion interaction theory (SIT) may also be used. Alternatively correlative methods such as UNIQUAC, NRTL, MOSCED or UNIFAC may be employed, provided fitted component-specific or model parameters are available.

A new alternative for activity coefficients prediction, which is less dependent on model parameters, is the COSMO-RS method. In this methods the required information comes from quantum mechanics calculations specific to each molecule (sigma profiles) combined with a statistical thermodynamics treatment of surface segments.

For uncharged species, the activity coefficient γ₀ mostly follows a salting-out model:

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

This simple model predicts activities of many species (dissolved undissociated gases such as CO₂, H₂S, NH₃, undissociated acids and bases) to high ionic strengths (up to 5 mol/kg). The value of the constant b for CO₂ is 0.11 at 10 °C and 0.20 at 330 °C.

For water (solvent), the activity a_w can be calculated using:

$\ln(a_{w})={\frac {-\nu m}{55.51}}$φ

where ν is the number of ions produced from the dissociation of one molecule of the dissolved salt, m is the molality of the salt dissolved in water, φ 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 particles of salt versus that of the solvent.

References

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