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Thursday, June 29, 2023

Van der Waals equation

From Wikipedia, the free encyclopedia

In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules.

The ideal gas law treats gas molecules as point particles that interact with their containers but not each other, meaning they neither take up space nor change kinetic energy during collisions (i.e. all collisions are perfectly elastic). The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant:

To account for the volume occupied by real gas molecules, the Van der Waals equation replaces in the ideal gas law with , where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole:

Van der Waals equation on a wall in Leiden

The second modification made to the ideal gas law accounts for interaction between molecules of the gas. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form , where a is a constant whose value depends on the gas.

The complete Van der Waals equation is therefore:

For n moles of gas, it can also be written as:

When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT.

This equation approximates the behavior of real fluids above their critical temperatures and is qualitatively reasonable for their liquid and low-pressure gaseous states at low temperatures. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. In particular, p is a constant function of V at given temperatures in these regions. As such, the Van der Waals model is not useful for calculations intended to predict real behavior in regions near critical points. Corrections to address these predictive deficiencies include the equal area rule and the principle of corresponding states.

The equation was named for its developer, the Dutch physicist Johannes Diderik van der Waals.

Overview and history

The Van der Waals equation is a thermodynamic equation of state based on the theory that fluids are composed of particles with non-zero volumes, and subject to a (not necessarily pairwise) inter-particle attractive force. It was based on work in theoretical physical chemistry performed in the late 19th century by Johannes Diderik van der Waals, who did related work on the attractive force that also bears his name. The equation is known to be based on a traditional set of derivations deriving from Van der Waals' and related efforts, as well as a set of derivation based in statistical thermodynamics, see below.

Van der Waals' early interests were primarily in the field of thermodynamics, where a first influence was Rudolf Clausius's published work on heat in 1857; other significant influences were the writings by James Clerk Maxwell, Ludwig Boltzmann, and Willard Gibbs. After initial pursuit of teaching credentials, Van der Waals' undergraduate coursework in mathematics and physics at the University of Leiden in the Netherlands led (with significant hurdles) to his acceptance for doctoral studies at Leiden under Pieter Rijke. While his dissertation helps to explain the experimental observation in 1869 by Irish professor of chemistry Thomas Andrews (Queen's University Belfast) of the existence of a critical point in fluids, science historian Martin J. Klein states that it is not clear whether Van der Waals was aware of Andrews' results when he began his doctorate work.

Van der Waals' doctoral research culminated in an 1873 dissertation that provided a semi-quantitative theory describing the gas-liquid change of state and the origin of a critical temperature, Over de Continuïteit van den Gas- en Vloeistoftoestand (Dutch; in English, On the Continuity of the Gas and Liquid State). It was in this dissertation that the first derivations of what we now refer to as the Van der Waals equation appeared. James Clerk Maxwell reviewed and lauded its published content in the British science journal Nature, and Van der Waals began independent work that would result in his receipt of the Nobel Prize in 1910, which emphasized the contribution of his formulation of this "equation of state for gases and liquids".

Equation

Van der Waals isotherms. The model correctly predicts a mostly incompressible liquid phase, but the oscillations in the phase transition zone do not fit experimental data.
 
Van der Waals isotherms and oscillations, temperatures, T, below and above TC. Shown in this PV diagram are a series of 5 isotherms for temperatures ranging from below (blue) to above (red) TC (the critical temperature), where the upward arise and downward dip in the isotherm predicted by the Van der Waals equation (the "oscillation", especially evident for the two isotherms for values of T < TC) is evident in the region of the gas-liquid phase transition.

The equation relates four state variables: the pressure of the fluid p, the total volume of the fluid's container V, the number of particles N, and the absolute temperature of the system T.

The intensive, microscopic form of the equation is:

where

is the volume of the container occupied by each particle (not the velocity of a particle), and kB is the Boltzmann constant. It introduces two new parameters: a′, a measure of the average attraction between particles, and b′, the volume excluded from v by one particle.

The equation can be also written in extensive, molar form:

or also:

where

is a measure of the average attraction between particles,

is the volume excluded by a mole of particles,

is the number of moles,

is the universal gas constant, kB is the Boltzmann constant, and NA is the Avogadro constant,

is the specific molar volume.

Also the constant a, b can be expressed in terms of the critical constants:

And the critical constants can be expressed in terms of a, b:

A careful distinction must be drawn between the volume available to a particle and the volume of a particle. In the intensive equation, v equals the total space available to each particle, while the parameter b′ is proportional to the proper volume of a single particle – the volume bounded by the atomic radius. This is subtracted from v because of the space taken up by one particle. In Van der Waals' original derivation, given below, b' is four times the proper volume of the particle. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb.

Gas mixture

If a mixture of gases is being considered, and each gas has its own (attraction between molecules) and (volume occupied by molecules) values, then and for the mixture can be calculated as

= total number of moles of gas present,
for each , = number of moles of gas present, and

and the rule of adding partial pressures becomes invalid if the numerical result of the equation is significantly different from the ideal gas equation .

Reduced form

The Van der Waals equation can also be expressed in terms of reduced properties:

The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states.

This yields a critical compressibility factor of 3/8. Reasons for modification of ideal gas equation: The equation state for ideal gas is PV=RT. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made.

Compressibility factor

The compressibility factor for the Van der Waals equation is:

Or in reduced form by substitution of :

At the critical point:

Validity

The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour. It also adequately predicts and explains the Joule–Thomson effect (temperature change during adiabatic expansion), which is not possible in ideal gas.

Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. This apparent discrepancy is resolved in the context of vapour–liquid equilibrium: at a particular temperature, there exist two points on the Van der Waals isotherm that have the same chemical potential, and thus a system in thermodynamic equilibrium will appear to traverse a straight line on the pV diagram as the ratio of vapour to liquid changes. However, in such a system, there are really only two points present (the liquid and the vapour) rather than a series of states connected by a line, so connecting the locus of points is incorrect: it is not an equation of multiple states, but an equation of (a single) state. It is indeed possible to compress a gas beyond the point at which it would typically condense, given the right conditions, and it is also possible to expand a liquid beyond the point at which it would usually boil. Such states are called "metastable" states. Such behaviour is qualitatively (though perhaps not quantitatively) predicted by the Van der Waals equation of state.

However, the values of physical quantities as predicted with the Van der Waals equation of state "are in very poor agreement with experiment", so the model's utility is limited to qualitative rather than quantitative purposes. Empirically-based corrections can easily be inserted into the Van der Waals model (see Maxwell's correction, below), but in so doing, the modified expression is no longer as simple an analytical model; in this regard, other models, such as those based on the principle of corresponding states, achieve a better fit with roughly the same work. Even with its acknowledged shortcomings, the pervasive use of the Van der Waals equation in standard university physical chemistry textbooks makes clear its importance as a pedagogic tool to aid understanding fundamental physical chemistry ideas involved in developing theories of vapour–liquid behavior and equations of state. In addition, other (more accurate) equations of state such as the Redlich–Kwong and Peng–Robinson equation of state are essentially modifications of the Van der Waals equation of state.

Derivation

Textbooks in physical chemistry generally give two derivations of the title equation. One is the conventional derivation that goes back to Van der Waals, a mechanical equation of state that cannot be used to specify all thermodynamic functions; the other is a statistical mechanics derivation that makes explicit the intermolecular potential neglected in the first derivation. A particular advantage of the statistical mechanical derivation is that it yields the partition function for the system, and allows all thermodynamic functions to be specified (including the mechanical equation of state).

Conventional derivation

Consider one mole of gas composed of non-interacting point particles that satisfy the ideal gas law:(see any standard Physical Chemistry text, op. cit.)

Next, assume that all particles are hard spheres of the same finite radius r (the Van der Waals radius). The effect of the finite volume of the particles is to decrease the available void space in which the particles are free to move. We must replace V by V − b, where b is called the excluded volume (per mole) or "co-volume". The corrected equation becomes

The excluded volume is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. To see this, we must realize that a particle is surrounded by a sphere of radius 2r (two times the original radius) that is forbidden for the centers of the other particles. If the distance between two particle centers were to be smaller than 2r, it would mean that the two particles penetrate each other, which, by definition, hard spheres are unable to do.

The excluded volume for the two particles (of average diameter d or radius r) is

,

which, divided by two (the number of colliding particles), gives the excluded volume per particle:

,

So b′ is four times the proper volume of the particle. It was a point of concern to Van der Waals that the factor four yields an upper bound; empirical values for b′ are usually lower. Of course, molecules are not infinitely hard, as Van der Waals thought, and are often fairly soft. To obtain the excluded volume per mole we just need to multiply by the number of molecules in a mole, i.e. by the avogadro number:

.

Next, we introduce a (not necessarily pairwise) attractive force between the particles. Van der Waals assumed that, notwithstanding the existence of this force, the density of the fluid is homogeneous; furthermore, he assumed that the range of the attractive force is so small that the great majority of the particles do not feel that the container is of finite size. Given the homogeneity of the fluid, the bulk of the particles do not experience a net force pulling them to the right or to the left. This is different for the particles in surface layers directly adjacent to the walls. They feel a net force from the bulk particles pulling them into the container, because this force is not compensated by particles on the side where the wall is (another assumption here is that there is no interaction between walls and particles, which is not true, as can be seen from the phenomenon of droplet formation; most types of liquid show adhesion). This net force decreases the force exerted onto the wall by the particles in the surface layer. The net force on a surface particle, pulling it into the container, is proportional to the number density. On considering one mole of gas, the number of particles will be NA

.

The number of particles in the surface layers is, again by assuming homogeneity, also proportional to the density. In total, the force on the walls is decreased by a factor proportional to the square of the density, and the pressure (force per unit surface) is decreased by

,

so that

Upon writing n for the number of moles and nVm = V, the equation obtains the second form given above,

It is of some historical interest to point out that Van der Waals, in his Nobel prize lecture, gave credit to Laplace for the argument that pressure is reduced proportional to the square of the density.

Statistical thermodynamics derivation

The canonical partition function Z of an ideal gas consisting of N = nNA identical (non-interacting) particles, is:

where is the thermal de Broglie wavelength,

with the usual definitions: h is the Planck constant, m the mass of a particle, k the Boltzmann constant and T the absolute temperature. In an ideal gas z is the partition function of a single particle in a container of volume V. In order to derive the Van der Waals equation we assume now that each particle moves independently in an average potential field offered by the other particles. The averaging over the particles is easy because we will assume that the particle density of the Van der Waals fluid is homogeneous. The interaction between a pair of particles, which are hard spheres, is taken to be

r is the distance between the centers of the spheres and d is the distance where the hard spheres touch each other (twice the Van der Waals radius). The depth of the Van der Waals well is .

Because the particles are not coupled under the mean field Hamiltonian, the mean field approximation of the total partition function still factorizes,

,

but the intermolecular potential necessitates two modifications to z. First, because of the finite size of the particles, not all of V is available, but only V − Nb', where (just as in the conventional derivation above)

.

Second, we insert a Boltzmann factor exp[ - ϕ/2kT] to take care of the average intermolecular potential. We divide here the potential by two because this interaction energy is shared between two particles. Thus

The total attraction felt by a single particle is

where we assumed that in a shell of thickness dr there are N/V 4π r2dr particles. This is a mean field approximation; the position of the particles is averaged. In reality the density close to the particle is different than far away as can be described by a pair correlation function. Furthermore, it is neglected that the fluid is enclosed between walls. Performing the integral we get

Hence, we obtain,

From statistical thermodynamics we know that

,

so that we only have to differentiate the terms containing . We get

Maxwell equal area rule

Van der Waals isotherm oscillation and Maxwell's equal areas. Maxwell's rule eliminates the oscillating behavior of the isotherm in the phase transition zone by defining it as a certain isobar in that zone. The above isotherm is for a reduced temperature of TR=0.9. The Maxwell correction is at a vapor pressure of pV≈0.64700 between the reduced volume of the pure liquid VL≈0.60340 and the pure gas VG≈2.3488 at the vapor pressure.

Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. Unlike for ideal gases, the p-V isotherms oscillate with a relative minimum (d) and a relative maximum (e). Any pressure between pd and pe appears to have 3 values for the volume, contradicting the experimental observation that two state variables completely determine a one-component system's state. Moreover, the isothermal compressibility is negative between d and e (equivalently ), which cannot describe a system at equilibrium.

To address these problems, James Clerk Maxwell replaced the isotherm between points a and c with a horizontal line positioned so that the areas of the two shaded regions would be equal (replacing the a-d-b-e-c) curve with a straight line from a to c); this portion of the isotherm corresponds to the liquid-vapor equilibrium. The regions of the isotherm from ad and from ce are interpreted as metastable states of super-heated liquid and super-cooled vapor, respectively. The equal area rule can be expressed as:

where pV is the vapor pressure (flat portion of the curve), VL is the volume of the pure liquid phase at point a on the diagram, and VG is the volume of the pure gas phase at point c on the diagram. A two-phase mixture at pV will occupy a total volume between VL and VG, as determined by Maxwell's lever rule.

Maxwell justified the rule based on the fact that the area on a pV diagram corresponds to mechanical work, saying that work done on the system in going from c to b should equal work released on going from a to b. This is because the change in free energy A(T,V) equals the work done during a reversible process, and, as a state variable, the free energy must be path-independent. In particular, the value of A at point b should be the same regardless of whether the path taken is from left or right across the horizontal isobar, or follows the original Van der Waals isotherm.

This derivation is not entirely rigorous, since it requires a reversible path through a region of thermodynamic instability, while b is unstable. Nevertheless, modern derivations from chemical potential reach the same conclusion, and it remains a necessary modification to the Van der Waals and to any other analytic equation of state.

From chemical potential

The Maxwell equal area rule can also be derived from an assumption of equal chemical potential μ of coexisting liquid and vapour phases. On the isotherm shown in the above plot, points a and c are the only pair of points which fulfill the equilibrium condition of having equal pressure, temperature and chemical potential. It follows that systems with volumes intermediate between these two points will consist of a mixture of the pure liquid and gas with specific volumes equal to the pure liquid and gas phases at points a and c.

The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. Since:

where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as:

is

Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes.

Locus of coexistence for two phases of Van der Waals fluid

A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. One sees that the two locii meet at the critical point (1,1,1) smoothly. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal.

Other parameters, forms and applications

Other thermodynamic parameters

We reiterate that the extensive volume V  is related to the volume per particle v=V/N  where N = nNA  is the number of particles in the system. The equation of state does not give us all the thermodynamic parameters of the system. We can take the equation for the Helmholtz energy A 

From the equation derived above for lnQ, we find

Where Φ is an undetermined constant, which may be taken from the Sackur–Tetrode equation for an ideal gas to be:

This equation expresses A  in terms of its natural variables V  and T , and therefore gives us all thermodynamic information about the system. The mechanical equation of state was already derived above

The entropy equation of state yields the entropy (S )

from which we can calculate the internal energy

Similar equations can be written for the other thermodynamic potential and the chemical potential, but expressing any potential as a function of pressure p  will require the solution of a third-order polynomial, which yields a complicated expression. Therefore, expressing the enthalpy and the Gibbs energy as functions of their natural variables will be complicated.

Reduced form

Although the material constant a and b in the usual form of the Van der Waals equation differs for every single fluid considered, the equation can be recast into an invariant form applicable to all fluids.

Defining the following reduced variables (fR, fC are the reduced and critical variable versions of f, respectively),

,

where

as shown by Salzman.

The first form of the Van der Waals equation of state given above can be recast in the following reduced form:

)

This equation is invariant for all fluids; that is, the same reduced form equation of state applies, no matter what a and b may be for the particular fluid.

This invariance may also be understood in terms of the principle of corresponding states. If two fluids have the same reduced pressure, reduced volume, and reduced temperature, we say that their states are corresponding. The states of two fluids may be corresponding even if their measured pressure, volume, and temperature are very different. If the two fluids' states are corresponding, they exist in the same regime of the reduced form equation of state. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly.

Cubic equation

The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is:

At the critical temperature, where we get as expected

For TR < 1, there are 3 values for vR. For TR > 1, there is 1 real value for vR.

The solution of this equation for the case where there are three separate roots may be found at Maxwell construction.

Application to compressible fluids

The equation is also usable as a PVT equation for compressible fluids (e.g. polymers). In this case specific volume changes are small and it can be written in a simplified form:

where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters.

Activity coefficient

From Wikipedia, the free encyclopedia

In thermodynamics, an activity coefficient is a factor used to account for deviation of a mixture of chemical substances from ideal behaviour. 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, , of a substance B in an ideal mixture of liquids or an ideal solution is given by

,

where μo
B
is the chemical potential of a pure substance , and is the mole fraction of the substance in the mixture.

This is generalised to include non-ideal behavior by writing

when is the activity of the substance in the mixture,

,

where is the activity coefficient, which may itself depend on . As approaches 1, the substance behaves as if it were ideal. For instance, if  ≈ 1, then Raoult's law is accurate. For  > 1 and  < 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 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.

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 in the liquid phase,

with the convention In other words: Pure liquids represent the ideal case.

At infinite dilution, the activity coefficient approaches its limiting value, . Comparison with Henry's law,

immediately gives

In other words: The compound shows nonideal behavior in the dilute case.

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

with and

The symbol has been used here to distinguish between the two kinds of activity coefficients. Usually it is omitted, as it is clear from the context which kind is meant. But there are cases where both kinds of activity coefficients are needed and may even appear in the same equation, e.g., for solutions of salts in (water + alcohol) mixtures. This is sometimes a source of errors.

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 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.

Ionic solutions

For solution of substances which ionize in solution the activity coefficients of the cation and anion cannot be experimentally determined independently of each other because solution properties depend on both ions. Single ion activity coefficients must be linked to the activity coefficient of the dissolved electrolyte as if undissociated. In this case a mean stoichiometric activity coefficient of the dissolved electrolyte, γ±, is used. It is called stoichiometric because it expresses both the deviation from the ideality of the solution and the incomplete ionic dissociation of the ionic compound which occurs especially with the increase of its concentration.

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

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

More generally, the mean activity coefficient of a compound of formula is given by

Single-ion activity coefficients can be calculated theoretically, for example by using the Debye–Hückel equation. The theoretical equation can be tested by combining the calculated single-ion activity coefficients to give mean values which can be compared to experimental values.

The prevailing view that single ion activity coefficients are unmeasurable independently, or perhaps even physically meaningless, has its roots in the work of Guggenheim in the late 1920s. However, chemists have never been able to give up the idea of single ion activities, and by implication single ion activity coefficients. For example, pH is defined as the negative logarithm of the hydrogen ion activity. If the prevailing view on the physical meaning and measurability of single ion activities is correct then defining pH as the negative logarithm of the hydrogen ion activity places the quantity squarely in the unmeasurable category. Recognizing this logical difficulty, International Union of Pure and Applied Chemistry (IUPAC) states that the activity-based definition of pH is a notional definition only. Despite the prevailing negative view on the measurability 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 solutions

For concentrated ionic solutions the hydration of ions must be taken into consideration, as done by Stokes and Robinson in their hydration model from 1948. The activity coefficient of the electrolyte is split into electric and statistical components by E. Glueckauf who modifies the Robinson–Stokes model.

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

Concentrated solution statistical part of the activity coefficient is:

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

Experimental determination of activity coefficients

Activity coefficients may be determined experimentally by making measurements on non-ideal mixtures. Use may be made of Raoult's law or Henry's law to provide a value for an ideal mixture against which the experimental value may be compared to obtain the activity coefficient. Other colligative properties, such as osmotic pressure may also be used.

Radiochemical methods

Activity coefficients can be determined by radiochemical methods.

At infinite dilution

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

Binary solutions with water
X γx (K) γW (K)
Ethanol 4.3800 (283.15) 3.2800 (298.15)
Acetone
6.0200 (307.85)

Theoretical calculation of activity coefficients

UNIQUAC Regression of activity coefficients (chloroform/methanol mixture)

Activity coefficients of electrolyte solutions may be calculated theoretically, using the Debye–Hückel equation or extensions such as the Davies equation, Pitzer equations or TCPC model. Specific ion interaction theory (SIT) may also be used.

For non-electrolyte solutions correlative methods such as UNIQUAC, NRTL, MOSCED or UNIFAC may be employed, provided fitted component-specific or model parameters are available. COSMO-RS is a theoretical method which is less dependent on model parameters as required information is obtained 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:

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 as solvent, the activity aw can be calculated using:

where ν is the number of ions produced from the dissociation of one molecule of the dissolved salt, b 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.

Link to ionic diameter

The ionic activity coefficient is connected to the ionic diameter by the formula obtained from Debye–Hückel theory of electrolytes:

where A and B are constants, zi is the valence number of the ion, and I is ionic strength.

Dependence on state parameters

The derivative of an activity coefficient with respect to temperature is related to excess molar enthalpy by

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

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, ΔrG, is equal to the difference between these sums and therefore, at equilibrium, is equal to zero. Thus, for an equilibrium such as

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

Upon rearrangement this expression becomes

The sum σμo
S
+ τμo
T
αμo
A
βμo
B
is the standard free energy change for the reaction, .

Therefore,

where 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

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

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

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