Raoults law

The relation between Raoult s law and the definition of an ideal solution given by Equation (8.57) is obtained by a study of Equation (10.35) or (10.40). If a solution is ideal, then A/i must be zero and the right-hand side of both equations must be zero. If we write Pyt in both equations as Pt, the partial pressure of the component, and Pyj in Equation (10.40) as P[, then the logarithmic term becomes lnfP P ), which is zero when Raoult s law, given in the form Pl = P[xl, is obeyed. We then see that to define an ideal solution in terms of Raoult s law and still be consistent with Equation (10.57) requires that the experimental measurements be made at the same total pressure and that the vapor behaves as an ideal gas. 

Equation (15.13) can describe either an ideal solution [see Equation (14.7)] or a solution sufficiently dilute that Henry s law is followed [see Equation (15.5)]. In either case, it follows that 

The value of k is constant because the standard chemical potentials in the two solvents are constants at a fixed temperature. Nemst s distribution law also can be stated in terms of molality. 

We can show that if the solute obeys Henry s law in very dilute solutions, the solvent follows Raoult s law in the same solutions. Let us start from the Gibbs-Duhem Equation (9.34), which relates changes in the chemical potential of the solute to changes in the chemical potential of the solvent that is, for a two-component system 

Let us apply Equation (15.20), which is a general relationship for any two-component system, to a solution for which Henry s law describes the behavior of the solute. Erom Equation (15.5), 

These equations are the same as Equation (14.6) and Equation (14.7), statements of Raoult s law thus, the solvent obeys Raoult s law when the solute obeys Henry s law. As Henry s law is a limiting law for the solute in dilute solution, Raoult s law 

For the purpose of this monograph, it is desirable to recognize two aspects of Raoult s experimental work, for the expression Raoult s law is glibly used altogether too loosely in the original literature and in the textbooks. His experimental work on the depression of the freezing point of a pure liquid by the addition of another substance is described in the papers cited in my list of references which have dates from 

A brief report on Raoult s work was published by Cahours, Berthelot, and Debray in which they drew attention to certain historical items such as the observations of Blagden. It should be noticed that when Raoult refers to a solution containing 1 g-molecule in 2 liters (aqueous) the Xa value is about 0.01. 

There were, of course, discussions and counter suggestions. Auwers and Meyerfor example, suggested that glacial acetic acid should always be used as the solvent instead of benzene or water. They believed the Beckmann determinations of the molecular weight of oximes were probably wrong because he had used benzene. It was pointed out that when it was absolutely necessary to use benzene or water, blank experiments with a compound of known molecular weight should be carried out. 

RaouWs Experiments on the Depression of the Vapor Pressure of a Liquid 

Raoult s Data for Five Liquids S and Diethyl Ether E 

The above characterization of ideal solutions does not require the fluid to behave as an ideal gas. In fact, certain liquid mixtures behave as ideal solutions, but they obviously do not obey the ideal gas law. In a mixture forming a vapor phase and a liquid phase at equilibrium with each other, either one of the phases, or both phases, may approach ideal solution behavior. Ideal solution behavior is approached at low pressures and usually with mixtures of chemically similar components. Referring to Equation 1.24, if the vapor phase is assumed to behave as an ideal gas, 0/ = 1, and the left-hand side of the equation reduces to T , where P is the total pressure. Moreover, for ideal liquid solutions, 

The vapor-liquid distribution coefficient in an ideal gas, ideal liquid solution system is therefore 

Since p is a function of temperature only, the vapor-liquid distribution coefficient, or A -value, is also a function of temperature only at constant total pressure. It bears a simple relationship to the total pressure and is independent of the composition. 

A fundamental property of a substance is the tendency for its atoms or molecules to spread into the surrounding space. A consequence of this property is the observed vapor pressure of liquids and solids. In order to understand the effects of the formation of a solution on this property, reference may be drawn to a solution consisting of two substances, A and B, with A being the solvent and B the solute. If the vapor pressure, PA, of the solvent over the solution is considered, it is clear that it must be proportional to the amount of A present in the solution. Thus, 

The ratio NjJ (NA + NB) represents the mole fraction of the solvent in the solution, where Nb and Na are the numbers of moles of the solute B and the solvent A respectively present in the solution, and K is a constant. In the case of the pure solvent NB = 0 and the fraction Na/(Nb + Na) is equal to 1. This leads to the equality K - PA (where PA is the vapor pressure of the pure solvent A). The expression for the vapor pressure of the solvent now takes the following form 

By similar arguments, the expression for the vapour pressure of the solute can be written as 

All these are mathematical expressions of Raoult s law. According to this law, the vapor pressure of a component of the solution is directly proportional to the mole fraction of that component in the solution. The constant of proportionality is the vapor pressure of the component in its pure state. Usually, Raoult s law is expressed as 

The partial pressure is defined as the pressure each gas would exert if it alone were to occupy the entire volume occupied by the mixture at the same temperature. Thus, the total pressure exerted by A and B is equal to the sum of the partial pressure of A, PA, and partial pressure of B, PB 

The liquid-vapor equilibrium situation with two or more components was not fully investigated until much later (Raoult, 1887). After a great deal of experimental work, which extended well into the 1900s, the relationship between vapor partial pressures and liquid compositions was well established, and forms what is now the best avenue of introduction to the understanding of activities. 

This work was done by mixing two or more liquid components in known proportions, then equilibrating the liquid with its own vapor and measuring the composition of the vapor. Because the total vapor pressure (also measured) was relatively low (generally well below one atmosphere) the vapor behaved as an ideal gas solution, and the partial pressures of the components could be calculated from their compositions using Dalton s Law. Of the many systems investigated, a very few were found to exhibit a particularly simple relationship between the vapor partial pressures and the liquid composition. In these systems, for all compositions, the partial pressures of the gas constituents were found to be a linear function of their mole fractions in the liquid. That is, in the binary system A-B, 

As mentioned earlier, the only way these simple relationships can hold is for the intermolecular forces between A-A, B-B and A-B to be identical, so that a molecule A behaves in the same way whether it is surrounded mostly by A or mostly by B. Solutions in which this happens are called ideal solutions, and the relation 

To facilitate discussions to follow, we should emphasize that the normal Raoult s law diagram (11.4b) shows vapor partial pressures plotted against liquid compositions. Obviously we cannot substitute for Xgin this diagram as they are 

There are not many systems that even approximately follow Raoult s Law. Even those systems that do approximately follow Raoult s Law such as benzene-toluene and ethylene bromide-propylene bromide will be found to have small deviations if very accurate measurements are made. Raoult s Law is an ideal concept that real systems are compared to. 

In 1888, the French physical chemist Francois Raoult published his finding that when a dilute liquid solution of a volatile solvent and a nonelectrolyte solute is equilibrated with a gas phase, the partial pressure pA of the solvent in the gas phase is proportional to the mole fraction xa of the solvent in the solution  

Here p is the saturation vapor pressure of the pure solvent (the pressure at which the pure liquid and pure gas phases are in equilibrium). 

In order to place Raoult s law in a rigorous thermodynamic framework, consider the two systems depicted in Fig. 9.5 on the next page. The liquid phase of system 1 is a binary 

Thermodynamics and Chemistry, second edition, version 3 2011 by Howard DeVoe. Latest version www.chem.umd.edu/thermobook 

Suppose that we vary the composition of the solution in system 1 at constant temperature, while adjusting the partial pressure of C so as to keep p constant. If we find that the partial pressure of the solvent over a range of composition is given by pa = XapX where p is the partial pressure of A in system 2 at the same T and p, we will say that the solvent obeys Raoult s law for partial pressure in this range. This is the same as the original Raoult s law, except that p is now the vapor pressure of pure liquid A at the pressure p of the liquid mixture. Section 12.8.1 will show that unless p is much greater than p is practically the same as the saturation vapor pressure of pure liquid A, in which case Raoult s law for partial pressure becomes identical to the original law. 

We saw in Section 11.13.2 that in an ideal solution all activity coefficients are equal to 1.0. Under these conditions Eq. 13.5.1 gives  

This rq resents the well-known Raoult s law, and indicates that the partial pressure, of component i is simply proportional to its mole fraction in the liquid phase and the proportionality constant is the vapor pressure of pure i at the system temperature, independently of what the other components of the mixture are. 

In other words, the tendency of component i to leave the liquid state, its liquid phase fugacity, is not effected by the other components of the mixture just by its own vapor pressure in the pure state and its mole fraction in the liquid mixture. In terms of molecular characteristics, this would require (Section 11.12.2) that  

An ideal solution represents, therefore, a hypothetical case. Because size and shape differences, however, are important only in extreme cases -such as polymer-solvent systems - we would expect that Raoult s law will be a reasonable approximation for mixtures of similar compounds, such as methanol-ethanol,hexane-heptane, etc. Indeed, the activity coefficients for the methanol-ethanol system at atmospheric pressure, for example, are to within 10% from one. 

In the next six Examples we use Raoult s law in some typical VLE problems including applications to distillation column design. 

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Henry s law Raoults law Osmotic pressure Boiling point... 

By assuming EDC and C2H4 form an ideal solution, the mol fraction of ethylene dissolved in the liquid can be estimated, from Raoults Law (see Chapter 8). 

The chemical potential jU, of the components of an ideal mixture of liquids (the components of an ideal mixture of liquids obey the Raoult law over the whole range of mole fractions and are completely miscible) is... 

The standard term p is the chemical potential of the pure component i (i.e. when Xj = 1) at the temperature of the system and the corresponding saturated vapour pressure. According to the Raoult law, in an ideal mixture the partial pressure of each component above the liquid is proportional to its mole fraction in the liquid,... 

The activities of the polymer and monomer of the hypothetical solutions given in Figure 9.6(a) are shown in Figure 9.6(b). While r = 1 corresponds to Raoult law behaviour, strong negative deviations are observed for r = 10, 100 and 1000. 

The Raoult law, the decrease of vapour pressure ps of a solution proportional to the solute concentration is a consequence of the model too. Solute molecules of which the own vapour pressure can be neglected have to be in the holes of the model (Fig. 1 right). Therefore, ps of the solvent decreases corresponding to Raoult s law. Now this effect is reduced by the increase of the sum of pair potentials because the coordina-... 

In more fundamental terms, the solubility of a chemical in water is determined by the activity coefficient in water yw which can be viewed as a "correction factor" to Raoults Law, i.e.,... 

Besides, let us note the automatic observance (certainly with correctly set initial data) and, hence, needlessness of the formalized descriptions in equilibrium modeling of such important regularities of macroscopic system behavior as the Gibbs phase rule, the Le Chatelier-Brown principle, mass action laws, the Henry law, the Raoult law, etc. 

The solvent and the key component that show most similar liquid-ph ase behavior tend to exhibit little molecular interactions. These components form an ideal or nearly ideal liquid solution. The activity coefficient of this key approaches unity, or may even show negative deviations from Raoults law if solvating or complexing interactions occur. On the other hand, the dissimilar key and the solvent demonstrate unfavorable molecular interactions, and the activity coefficient of this key increases. The positive deviations from Raoults law are further enhanced by the diluting effect of the high-solvent concentration, and the value of the activity coefficient of this key may approach the infinite dilution value, often a very large number. 

The relation Rpx = P"/x[ is simply a reformulation of Raoult Law when applied to the solvent. Any of the other relations are equivalent to Henry s Law when applied to the solute. Rxx = x [ x[ is also known as the Nernst Distribution Law. 

A comparison between Gy and Gy for the methanol—water system showed that in that case they were close to each other. However, as was already pointed out, the use of a reference state is particularly important for the systems with small deviations from the Raoult law. In such cases, the reference state based on Gy provides results very different from those based on Gjf. 

Glycol mononitrate has aroused a certain amount of interest. Twist and Baughan (109) examined the vapour pressure of the solution of this substance, and of a number of other nitrate esters, and found the deviations from Raoult laws to be of the same order as those observed by Chedin and Vandoni for nitrocellulose [110]. Prior to this work Marans and Zelinski (III] prepared a number of mixed esters of the type XI where R are unsaturated acyls apt to polymerize and thus to give combustible polymers. 

Rgure 8-9. Acetone (1)-chloroform (2) system at 760 mm Hg. Maximum boiling azeotrope formed by negative deviations from Raoult Law (dashed lines). Used by permission. Smith, B.D., Design of Equilibrium Stage Processes, McGraw-Hiil, New York, (1963), all rights reserved. 

In order to better demonstrate whether a system follows Raoult s law, a diagram of the phase equilibrium called T-x-y should be plotted. This plot (Figure 2) shows the equilibrium temperatures at which either a liquid solution will start bubbling (bubble curve) or a vapor mixture starts condensing (dew curve). The two systems with their experimental data and the calculation curve of the ideal solution is shown in Figure 2. In Figure 2, the system of hexane-benzene at the pressure of 101.33 kPa [10] and the system of ethylacetate-benzene [11] show negative deviations from RaoulTs law. 

Example 3 Detv and Bubble Point Calculations As indicated by Example 2a, a binary system in vapor/liquid equilibrium has 2 degrees of freedom. Thus of the four phase rule variables T, P, x, and t/i, two must be fixed to allow calculation of the other two, regardless of the formulation of the equilibrium equations. Modified Raoults law [Eq. (4-307)] may therefore be applied to the calculation of any pair of phase rule variables, given the other two. 

By equation (31.5), the activity of the solvent is equivalent to fi/fi where fi is the fugacity in a given solution and / is numerically equal to that in the standard state, i.e., pure liquid at 1 atm. pressure at the given temperature. Hence, it is seen from equation (34.1) that for an ideal solution the activity of the solvent should always be equal to its mole fraction, provided the total pressure is 1 e m. In other words, in these circumstances the activity coefficient ui/ni should be inity at all concentrations. For a nonideal solution, therefore, the deviation of ai/Ni from unity at 1 atm. pressure may be taken as a measure of the departure from ideal (Raoult law) behavior. Since the activities of liquids are not greatly affected by pressure, this conclusion may be accepted as generally applicable, provided the pressure is not too high. 

Estimate the vapor pressure of the two components in a regular solution for which cAh/(RT) = 1 and = 0.4 given that the vapor pressure of pure component A is 15.0 kPa and that of pure B, 20.0 kPa. Also calculate the Raoult law activity coefficients. Repeat the calculation for the case that cAh/(RT) = -1. 

Fig. 1.9 Vapor pressure for a hypothetical regular solution for which cAh(RT = 1 plotted against the mole fraction of component B. The vapor pressure of pure component B is 26.7 kPa, and that of component A, 20.0 kPa. The broken lines show Raoult law behavior.

Using Wilson s parameters for the carbon tetrachloride-acetonitrile system, estimate the Raoult law activity coefficients for each component in a equimolar solution. Then estimate the molar Gibbs energy of mixing. 

From the theory for regular solutions, the Raoult law activity coefficient for component B in a solution of A and B is... 

Table 1.4 Raoult Law and Henry Law Activity Coefficients for Dilute Solutions of Methanol in...

This is the form of the Gibbs-Duhem equation needed to relate the activity of component B in solution to that of component A. Choosing the Raoult law activity for the solvent A, and the Henry law activity for the solute B, equation (1.13.4) may be rewritten as... 

It should be noted that the values of quickly become non-unity and are greater than one. This is indicative of strong attractive solute-solvent interactions and negative deviations from Raoult law behavior. In the case of the methanol water system for which positive deviations from Raoult s law is observed (table 1.4), the Hemy law activity coefficients are less than one. 

Use an interpolation method to obtain the values of the mole fraetion of CCI4 in the vapor and the total vapor pressure for values of the mole fraetion in the liquid phase equal to 0.2, 0.4, 0.6, and 0.8. Then calculate the vapor pressure and the activity coefficient of each component for the same values. Finally estimate the molar Gibbs energy of mixing on the Raoult law scale at these four points. 

Calculate the Raoult law activity coefficients of both components and plot them as a function of the mole fraction of acetone. Determine the range of composition with respect to acetone that the solution can be regarded as regular. Calculate the enthalpy parameter for acetone-chloroform interactions on the basis of a one-parameter least-squares fit of the data in this range using an appropriate plot. 

Calculate the Raoult law and Henry law activity coeiScients for methanol on the mole fraction scale. 

The symmetrical ideal behavior is equivalent to the well-known Raoult law. Suppose that a mixture of A and B is in equilibrium with an ideal-gas phase let PA be the partial pressure of A. The chemical potential of A in the gas phase is... 

To integrate this we must know of some relation between p and n This is given by the approximate Raoult Law, namely, that—... 

Non-ideal Solution A solution formed by mixing two liquids is said to be non-ideal if it does not obey Raoults law or the interactions of A and B molecules in the solution are not similar to those of pure A and pure B or D V 1 0 and... 

Taking 80°C as the normal boiling point of the mixture, applying Raoults law,... 

The next step for the calculation of the number of theoretical trays is to make a material balance around the tower bottom (it can also be started from the tower top). The concentrations of the components in the ascending vapor stream are calculated from equilibrium equation (Dalton Raoult laws) for the existing mixture composition in the tower bottom (5.3). 

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