Compared to the \(Px_{\text{B}}\) diagram of Figure \(\PageIndex{3}\), the phases are now in reversed order, with the liquid at the bottom (low temperature), and the vapor on top (high Temperature). Under these conditions therefore, solid nitrogen also floats in its liquid. Attention has been directed to mesophases because they enable display devices and have become commercially important through the so-called liquid-crystal technology. We write, dy2 dy1 = dy2 dt dy1 dt = g l siny1 y2, (the phase-plane equation) which can readily be solved by the method of separation of variables . We can reduce the pressure on top of a liquid solution with concentration \(x^i_{\text{B}}\) (see Figure 13.3) until the solution hits the liquidus line. The figure below shows the experimentally determined phase diagrams for the nearly ideal solution of hexane and heptane. Overview[edit] A phase diagram is often considered as something which can only be measured directly. The corresponding diagram is reported in Figure 13.1. liquid. \tag{13.3} The second type is the negative azeotrope (right plot in Figure 13.8). The partial vapor pressure of a component in a mixture is equal to the vapor pressure of the pure component at that temperature multiplied by its mole fraction in the mixture. y_{\text{A}}=\frac{0.02}{0.05}=0.40 & \qquad y_{\text{B}}=\frac{0.03}{0.05}=0.60 A similar concept applies to liquidgas phase changes. The chemical potential of a component in the mixture is then calculated using: \[\begin{equation} A simple example diagram with hypothetical components 1 and 2 in a non-azeotropic mixture is shown at right. For most substances Vfus is positive so that the slope is positive. Raoults law applied to a system containing only one volatile component describes a line in the \(Px_{\text{B}}\) plot, as in Figure 13.1. That means that there are only half as many of each sort of molecule on the surface as in the pure liquids. Some organic materials pass through intermediate states between solid and liquid; these states are called mesophases. Low temperature, sodic plagioclase (Albite) is on the left; high temperature calcic plagioclase (anorthite) is on the right. This fact can be exploited to separate the two components of the solution. This coefficient is either larger than one (for positive deviations), or smaller than one (for negative deviations). Metastable phases are not shown in phase diagrams as, despite their common occurrence, they are not equilibrium phases. [4], For most substances, the solidliquid phase boundary (or fusion curve) in the phase diagram has a positive slope so that the melting point increases with pressure. This happens because the liquidus and Dew point lines coincide at this point. Figure 13.1: The PressureComposition Phase Diagram of an Ideal Solution Containing a Single Volatile Component at Constant Temperature. \end{equation}\], where \(i\) is the van t Hoff factor introduced above, \(m\) is the molality of the solution, \(R\) is the ideal gas constant, and \(T\) the temperature of the solution. which relates the chemical potential of a component in an ideal solution to the chemical potential of the pure liquid and its mole fraction in the solution. Since the vapors in the gas phase behave ideally, the total pressure can be simply calculated using Dalton's law as the sum of the partial pressures of the two components P TOT = P A + P B. Based on the ideal solution model, we have defined the excess Gibbs energy ex G m, which . \mu_i^{\text{vapor}} = \mu_i^{{-\kern-6pt{\ominus}\kern-6pt-}} + RT \ln \frac{P_i}{P^{{-\kern-6pt{\ominus}\kern-6pt-}}}. (solid, liquid, gas, solution of two miscible liquids, etc.). Starting from a solvent at atmospheric pressure in the apparatus depicted in Figure 13.11, we can add solute particles to the left side of the apparatus. For example, the strong electrolyte \(\mathrm{Ca}\mathrm{Cl}_2\) completely dissociates into three particles in solution, one \(\mathrm{Ca}^{2+}\) and two \(\mathrm{Cl}^-\), and \(i=3\). When both concentrations are reported in one diagramas in Figure \(\PageIndex{3}\)the line where \(x_{\text{B}}\) is obtained is called the liquidus line, while the line where the \(y_{\text{B}}\) is reported is called the Dew point line. \end{equation}\]. - Ideal Henrian solutions: - Derivation and origin of Henry's Law in terms of "lattice stabilities." - Limited mutual solubility in terminal solid solutions described by ideal Henrian behaviour. concrete matrix holds aggregates and fillers more than 75-80% of its volume and it doesn't contain a hydrated cement phase. The obtained phase equilibria are important experimental data for the optimization of thermodynamic parameters, which in turn . If you have a second liquid, the same thing is true. Raoults law acts as an additional constraint for the points sitting on the line. B) for various temperatures, and examine how these correlate to the phase diagram. I want to start by looking again at material from the last part of that page. A line on the surface called a triple line is where solid, liquid and vapor can all coexist in equilibrium. This fact, however, should not surprise us, since the equilibrium constant is also related to \(\Delta_{\text{rxn}} G^{{-\kern-6pt{\ominus}\kern-6pt-}}\) using Gibbs relation. As can be tested from the diagram the phase separation region widens as the . At any particular temperature a certain proportion of the molecules will have enough energy to leave the surface. As such, a liquid solution of initial composition \(x_{\text{B}}^i\) can be heated until it hits the liquidus line. (b) For a solution containing 1 mol each of hexane and heptane molecules, estimate the vapour pressure at 70 C when vaporization on reduction of the external pressure Show transcribed image text Expert Answer 100% (4 ratings) Transcribed image text: Thus, the substance requires a higher temperature for its molecules to have enough energy to break out of the fixed pattern of the solid phase and enter the liquid phase. If the forces were any different, the tendency to escape would change. The phase diagram for carbon dioxide shows the phase behavior with changes in temperature and pressure. The axes correspond to the pressure and temperature. In other words, it measures equilibrium relative to a standard state. You may have come cross a slightly simplified version of Raoult's Law if you have studied the effect of a non-volatile solute like salt on the vapor pressure of solvents like water. Once again, there is only one degree of freedom inside the lens. At a molecular level, ice is less dense because it has a more extensive network of hydrogen bonding which requires a greater separation of water molecules. The book systematically discusses phase diagrams of all types, the thermodynamics behind them, their calculations from thermodynamic . mixing as a function of concentration in an ideal bi-nary solution where the atoms are distributed at ran-dom. \begin{aligned} In other words, the partial vapor pressure of A at a particular temperature is proportional to its mole fraction. For non-ideal solutions, the formulas that we will derive below are valid only in an approximate manner. If we assume ideal solution behavior,the ebullioscopic constant can be obtained from the thermodynamic condition for liquid-vapor equilibrium. If you follow the logic of this through, the intermolecular attractions between two red molecules, two blue molecules or a red and a blue molecule must all be exactly the same if the mixture is to be ideal. This definition is equivalent to setting the activity of a pure component, \(i\), at \(a_i=1\). \end{equation}\]. Raoults law states that the partial pressure of each component, \(i\), of an ideal mixture of liquids, \(P_i\), is equal to the vapor pressure of the pure component \(P_i^*\) multiplied by its mole fraction in the mixture \(x_i\): \[\begin{equation} The minimum (left plot) and maximum (right plot) points in Figure 13.8 represent the so-called azeotrope. For example, the water phase diagram has a triple point corresponding to the single temperature and pressure at which solid, liquid, and gaseous water can coexist in a stable equilibrium (273.16K and a partial vapor pressure of 611.657Pa). The activity of component \(i\) can be calculated as an effective mole fraction, using: \[\begin{equation} However, some liquid mixtures get fairly close to being ideal. (13.9) as: \[\begin{equation} Calculate the mole fraction in the vapor phase of a liquid solution composed of 67% of toluene (\(\mathrm{A}\)) and 33% of benzene (\(\mathrm{B}\)), given the vapor pressures of the pure substances: \(P_{\text{A}}^*=0.03\;\text{bar}\), and \(P_{\text{B}}^*=0.10\;\text{bar}\). Phase diagrams can use other variables in addition to or in place of temperature, pressure and composition, for example the strength of an applied electrical or magnetic field, and they can also involve substances that take on more than just three states of matter. Have seen that if d2F/dc2 everywhere 0 have a homogeneous solution. The mole fraction of B falls as A increases so the line will slope down rather than up. \end{equation}\]. \mu_i^{\text{solution}} = \mu_i^* + RT \ln \left(\gamma_i x_i\right), If you triple the mole fraction, its partial vapor pressure will triple - and so on. We can also report the mole fraction in the vapor phase as an additional line in the \(Px_{\text{B}}\) diagram of Figure \(\PageIndex{2}\). However, doing it like this would be incredibly tedious, and unless you could arrange to produce and condense huge amounts of vapor over the top of the boiling liquid, the amount of B which you would get at the end would be very small. The total vapor pressure, calculated using Daltons law, is reported in red. The increase in concentration on the left causes a net transfer of solvent across the membrane. 2. (a) Label the regions of the diagrams as to which phases are present. The main advantage of ideal solutions is that the interactions between particles in the liquid phase have similar mean strength throughout the entire phase. There is also the peritectoid, a point where two solid phases combine into one solid phase during cooling. B) with g. liq (X. By Debbie McClinton Dr. Miriam Douglass Dr. Martin McClinton. \end{aligned} \end{equation}\label{13.1.2} \] The total pressure of the vapors can be calculated combining Daltons and Roults laws: \[\begin{equation} \begin{aligned} P_{\text{TOT}} &= P_{\text{A}}+P_{\text{B}}=x_{\text{A}} P_{\text{A}}^* + x_{\text{B}} P_{\text{B}}^* \\ &= 0.67\cdot 0.03+0.33\cdot 0.10 \\ &= 0.02 + 0.03 = 0.05 \;\text{bar} \end{aligned} \end{equation}\label{13.1.3} \] We can then calculate the mole fraction of the components in the vapor phase as: \[\begin{equation} \begin{aligned} y_{\text{A}}=\dfrac{P_{\text{A}}}{P_{\text{TOT}}} & \qquad y_{\text{B}}=\dfrac{P_{\text{B}}}{P_{\text{TOT}}} \\ y_{\text{A}}=\dfrac{0.02}{0.05}=0.40 & \qquad y_{\text{B}}=\dfrac{0.03}{0.05}=0.60 \end{aligned} \end{equation}\label{13.1.4} \] Notice how the mole fraction of toluene is much higher in the liquid phase, \(x_{\text{A}}=0.67\), than in the vapor phase, \(y_{\text{A}}=0.40\). If a liquid has a high vapor pressure at a particular temperature, it means that its molecules are escaping easily from the surface. Solutions are possible for all three states of matter: The number of degrees of freedom for binary solutions (solutions containing two components) is calculated from the Gibbs phase rules at \(f=2-p+2=4-p\). For example, if the solubility limit of a phase needs to be known, some physical method such as microscopy would be used to observe the formation of the second phase. This result also proves that for an ideal solution, \(\gamma=1\). We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. That is exactly what it says it is - the fraction of the total number of moles present which is A or B. You can see that we now have a vapor which is getting quite close to being pure B. This behavior is observed at \(x_{\text{B}} \rightarrow 0\) in Figure 13.6, since the volatile component in this diagram is \(\mathrm{A}\). { Fractional_Distillation_of_Ideal_Mixtures : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Fractional_Distillation_of_Non-ideal_Mixtures_(Azeotropes)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Immiscible_Liquids_and_Steam_Distillation : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Liquid-Solid_Phase_Diagrams:_Salt_Solutions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Liquid-Solid_Phase_Diagrams:_Tin_and_Lead" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Non-Ideal_Mixtures_of_Liquids" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Phases_and_Their_Transitions : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Phase_Diagrams_for_Pure_Substances : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Raoults_Law_and_Ideal_Mixtures_of_Liquids : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "Acid-Base_Equilibria" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Chemical_Equilibria : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Dynamic_Equilibria : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Heterogeneous_Equilibria : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Le_Chateliers_Principle : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Physical_Equilibria : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Solubilty : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, Raoult's Law and Ideal Mixtures of Liquids, [ "article:topic", "fractional distillation", "Raoult\'s Law", "authorname:clarkj", "showtoc:no", "license:ccbync", "licenseversion:40" ], https://chem.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FPhysical_and_Theoretical_Chemistry_Textbook_Maps%2FSupplemental_Modules_(Physical_and_Theoretical_Chemistry)%2FEquilibria%2FPhysical_Equilibria%2FRaoults_Law_and_Ideal_Mixtures_of_Liquids, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), Ideal Mixtures and the Enthalpy of Mixing, Constructing a boiling point / composition diagram, The beginnings of fractional distillation, status page at https://status.libretexts.org. Let's focus on one of these liquids - A, for example. If you boil a liquid mixture, you can find out the temperature it boils at, and the composition of the vapor over the boiling liquid. A notorious example of this behavior at atmospheric pressure is the ethanol/water mixture, with composition 95.63% ethanol by mass. \mu_{\text{solution}} < \mu_{\text{solvent}}^*. If the proportion of each escaping stays the same, obviously only half as many will escape in any given time. The \(T_{\text{B}}\) diagram for two volatile components is reported in Figure \(\PageIndex{4}\). (11.29), it is clear that the activity is equal to the fugacity for a non-ideal gas (which, in turn, is equal to the pressure for an ideal gas). You might think that the diagram shows only half as many of each molecule escaping - but the proportion of each escaping is still the same. Every point in this diagram represents a possible combination of temperature and pressure for the system. Two types of azeotropes exist, representative of the two types of non-ideal behavior of solutions. As is clear from Figure \(\PageIndex{4}\), the mole fraction of the \(\text{B}\) component in the gas phase is lower than the mole fraction in the liquid phase. 1. 1. \Delta T_{\text{m}}=T_{\text{m}}^{\text{solution}}-T_{\text{m}}^{\text{solvent}}=-iK_{\text{m}}m, In addition to temperature and pressure, other thermodynamic properties may be graphed in phase diagrams. In practice, this is all a lot easier than it looks when you first meet the definition of Raoult's Law and the equations! The page will flow better if I do it this way around. Since the vapors in the gas phase behave ideally, the total pressure can be simply calculated using Daltons law as the sum of the partial pressures of the two components \(P_{\text{TOT}}=P_{\text{A}}+P_{\text{B}}\). \end{equation}\]. Phase diagrams are used to describe the occurrence of mesophases.[16]. As is clear from the results of Exercise \(\PageIndex{1}\), the concentration of the components in the gas and vapor phases are different. The open spaces, where the free energy is analytic, correspond to single phase regions. Notice that the vapor over the top of the boiling liquid has a composition which is much richer in B - the more volatile component. To remind you - we've just ended up with this vapor pressure / composition diagram: We're going to convert this into a boiling point / composition diagram. For non-ideal gases, we introduced in chapter 11 the concept of fugacity as an effective pressure that accounts for non-ideal behavior. Now we'll do the same thing for B - except that we will plot it on the same set of axes. At this temperature the solution boils, producing a vapor with concentration \(y_{\text{B}}^f\). Using the phase diagram. That would give you a point on the diagram. Raoults law acts as an additional constraint for the points sitting on the line. One type of phase diagram plots temperature against the relative concentrations of two substances in a binary mixture called a binary phase diagram, as shown at right. Instead, it terminates at a point on the phase diagram called the critical point. A triple point identifies the condition at which three phases of matter can coexist. These two types of mixtures result in very different graphs. Raoult's Law only works for ideal mixtures. The definition below is the one to use if you are talking about mixtures of two volatile liquids. A tie line from the liquid to the gas at constant pressure would indicate the two compositions of the liquid and gas respectively.[13]. How these work will be explored on another page. \mu_{\text{non-ideal}} = \mu^{{-\kern-6pt{\ominus}\kern-6pt-}} + RT \ln a, The data available for the systems are summarized as follows: \[\begin{equation} \begin{aligned} x_{\text{A}}=0.67 \qquad & \qquad x_{\text{B}}=0.33 \\ P_{\text{A}}^* = 0.03\;\text{bar} \qquad & \qquad P_{\text{B}}^* = 0.10\;\text{bar} \\ & P_{\text{TOT}} = ? (9.9): \[\begin{equation} In equation form, for a mixture of liquids A and B, this reads: In this equation, PA and PB are the partial vapor pressures of the components A and B. &= \mu_{\text{solvent}}^* + RT \ln x_{\text{solution}}, The solidliquid phase boundary can only end in a critical point if the solid and liquid phases have the same symmetry group. A two component diagram with components A and B in an "ideal" solution is shown. For two particular volatile components at a certain pressure such as atmospheric pressure, a boiling-point diagram shows what vapor (gas) compositions are in equilibrium with given liquid compositions depending on temperature. An ideal mixture is one which obeys Raoult's Law, but I want to look at the characteristics of an ideal mixture before actually stating Raoult's Law. You would now be boiling a new liquid which had a composition C2. Positive deviations on Raoults ideal behavior are not the only possible deviation from ideality, and negative deviation also exits, albeit slightly less common. The following two colligative properties are explained by reporting the changes due to the solute molecules in the plot of the chemical potential as a function of temperature (Figure 12.1). Employing this method, one can provide phase relationships of alloys under different conditions. The diagram is divided into three areas, which represent the solid, liquid . \\ y_{\text{A}}=? \end{equation}\]. \tag{13.8} [11][12] For example, for a single component, a 3D Cartesian coordinate type graph can show temperature (T) on one axis, pressure (p) on a second axis, and specific volume (v) on a third. This is also proven by the fact that the enthalpy of vaporization is larger than the enthalpy of fusion. 3) vertical sections.[14]. The concept of an ideal solution is fundamental to chemical thermodynamics and its applications, such as the explanation of colligative properties . \tag{13.12} These are mixtures of two very closely similar substances. Therefore, the liquid and the vapor phases have the same composition, and distillation cannot occur. There may be a gap between the solidus and liquidus; within the gap, the substance consists of a mixture of crystals and liquid (like a "slurry").[1]. The chilled water leaves at the same temperature and warms to 11C as it absorbs the load. \[ P_{methanol} = \dfrac{2}{3} \times 81\; kPa\], \[ P_{ethanol} = \dfrac{1}{3} \times 45\; kPa\]. Legal. \tag{13.14} The corresponding diagram is reported in Figure \(\PageIndex{2}\). For a pure component, this can be empirically calculated using Richard's Rule: Gfusion = - 9.5 ( Tm - T) Tm = melting temperature T = current temperature \end{equation}\]. The global features of the phase diagram are well represented by the calculation, supporting the assumption of ideal solutions. With diagram .In a steam jet refrigeration system, the evaporator is maintained at 6C. The temperature scale is plotted on the axis perpendicular to the composition triangle. Another type of binary phase diagram is a boiling-point diagram for a mixture of two components, i. e. chemical compounds. They must also be the same otherwise the blue ones would have a different tendency to escape than before. Suppose that you collected and condensed the vapor over the top of the boiling liquid and reboiled it. As emerges from Figure \(\PageIndex{1}\), Raoults law divides the diagram into two distinct areas, each with three degrees of freedom.\(^1\) Each area contains a phase, with the vapor at the bottom (low pressure), and the liquid at the top (high pressure). When the forces applied across all molecules are the exact same, irrespective of the species, a solution is said to be ideal. where \(P_i^{\text{R}}\) is the partial pressure calculated using Raoults law. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. various degrees of deviation from ideal solution behaviour on the phase diagram.) \qquad & \qquad y_{\text{B}}=? Comparing eq. This page looks at the phase diagrams for non-ideal mixtures of liquids, and introduces the idea of an azeotropic mixture (also known as an azeotrope or constant boiling mixture). The x-axis of such a diagram represents the concentration variable of the mixture. Figure 13.2: The PressureComposition Phase Diagram of an Ideal Solution Containing Two Volatile Components at Constant Temperature. If you plot a graph of the partial vapor pressure of A against its mole fraction, you will get a straight line. \tag{13.20} P_{\text{solvent}}^* &- P_{\text{solution}} = P_{\text{solvent}}^* - x_{\text{solvent}} P_{\text{solvent}}^* \\ As the mole fraction of B falls, its vapor pressure will fall at the same rate. Because of the changes to the phase diagram, you can see that: the boiling point of the solvent in a solution is higher than that of the pure solvent; 1 INTRODUCTION. K_{\text{b}}=\frac{RMT_{\text{b}}^{2}}{\Delta_{\mathrm{vap}} H}, The liquidus is the temperature above which the substance is stable in a liquid state. Its difference with respect to the vapor pressure of the pure solvent can be calculated as: \[\begin{equation} \begin{aligned} An azeotrope is a constant boiling point solution whose composition cannot be altered or changed by simple distillation. The critical point remains a point on the surface even on a 3D phase diagram. where \(R\) is the ideal gas constant, \(M\) is the molar mass of the solvent, and \(\Delta_{\mathrm{vap}} H\) is its molar enthalpy of vaporization. There is actually no such thing as an ideal mixture! As the mixtures are typically far from dilute and their density as a function of temperature is usually unknown, the preferred concentration measure is mole fraction. &= \underbrace{\mu_{\text{solvent}}^{{-\kern-6pt{\ominus}\kern-6pt-}} + RT \ln P_{\text{solvent}}^*}_{\mu_{\text{solvent}}^*} + RT \ln x_{\text{solution}} \\
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