Ideal solution partial molar properties

Foi an ideal solution, G, = 0, and tlieiefoie 7 = 1- Compatison shows that equation 203 relates to exactiy as equation 163 relates ( ) to GG Moreover, just as ( ) is a partial property with respect to G /E.T, so In y is a partial property with respect to G /RT. Equation 116, the defining equation for a partial molar property, in this case becomes equation 204 ... 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

All three quantities are for the same T, P, and physical state. Eq. (4-126) defines a partial molar property change of mixing, and Eq. (4-125) is the summability relation for these properties. Each of Eqs. (4-93) through (4-96) is an expression for an ideal solution property, and each may be combined with the defining equation for an excess property (Eq. [4-99]), yielding ... 

The partial molar properties of a component i of an ideal solution are readily obtained ... 

The standard state is here also a hypothetical one it is equivalent to a 1 molal solution in which the solute has some of the partial molar properties, e.g., heat content and heat capacity, of the infinitely dilute solution. It has been referred to as the hypothetical ideal 1 molal solution At high dilutions the molality of a solution is directly proportional to its mole fraction ( 32f), and hence dilute solutions in which the activity of the solute is equal to its molality also satisfy Henry s law. Under such conditions, the departure from unity of the activity coefiicient 7 , equal to a2/m, like that of 7n, is a measure of the deviation from Henry s law. 

Most solutions do not exhibit ideal behavior, and the actual curve corresponding to the variation of the molar volume or enthalpy of the mixture deviates from a straight line (e.g., the solid line in Fig. 5.1). When the curve for the molar volume lies above the ideal mixture line, the system expands upon mixing when the curve lies below the line, the system contracts. In the case of the molar enthalpy, a curve that lies above the ideal mixture line corresponds to the system that absorbs heat (e.g., mixing lead bromide and water) a curve that lies below the line corresponds to the system releasing heat (e.g., mixing sulfuric acid and water). This non-ideal mixing in the case of the molar enthalpy is the principle used in cold packs and heat packs. We will develop mathematical models to describe non-ideal mixtures. We use partial molar properties in more detail later. 

Note that in this case the. pure component and partial molar enthalpies differ considerably. Con-sequendy. we say that this solution is quite nonideal, where, as we shall see in Chapter 9. an ideal solution is one in wh ch some partial molar properties (in particular the enthalpy, internal energy, and volume) are equal to the pure component values. Further, here the solution is so nonideal that at the temperature chosen the purecomponenf and partial molar. enthalpies are even of different signs for both water and sulfuric acid. For later reference we note that, at a h,so4 = 0.5, we have ... 

In practice, pure-component molar enthalpies are employed to approximate A/7rx. This approximation is exact for ideal solutions only, when partial molar properties reduce to pure-component molar properties. In general, one accounts for more than the making and breaking of chemical bonds in (3-35). Nonidealities such as heats of solution and ionic interactions are also accounted for when partial molar enthalpies are employed. Now, the first law of thermodynamics for open systems, which contains the total differential of specific enthalpy, is written in a form that allows one to calculate temperature profiles in a tubular reactor ... 

Partial Molar Properties of Lewis-Randall Ideal Solutions... 

To obtain expressions for the partial molar properties of ideal solutions, we first determine the chemical potential. Using the ideal-solution fugacity (5.1.6) in the integrated definition of fugacity (4.3.12) we find... 

With the partial molar properties now known, expressions for the total properties of ideal solutions can be formed from the generic relation between a mixture property and its corresponding component partial molar properties ... 

We have now considered both ideal solution behavior and deviations from this, but in a rather generalized way, using activity coefficients. We now have to start to consider how to measure these things, and doing this means we have to consider partial molar properties in much more detail. 

Excess properties, the difference between the property in a real solution and in an ideal solution, are generally expressed as a relative or relative partial molar properties, such as the relative enthalpy, L, or relative partial molar enthalpy, L. The Gibbs energy is treated differently. The fact that Gj-p is a thermodynanoic potential leads naturally to the definition of a relative partial molar Gibbs energy (q. - /a°) which is not the difference from an ideal solution (/A — pL° is not zero even for an ideal solution) but the difference from a standard state, which in this chapter is a pure phase, but may also be some hypothetical state. The form of the equation relating q, - to composition then... 

A note on partial molar properties In case you are beginning to wonder why there are so many questions and problems about concentrations I will answer by telling you that you need concentrations in about four out of every five problems in physical chemistry. The matter of fact is that a lot of chemistry and all of biochemistry takes place in solutions. Then there are problems inherent to solutions. Solutions are considered simple physical mixtures of two or more different kinds of molecules, with no chemical bonds made or broken. For a really well-behaved solution physical chemists have a name, by analogy with the gas laws an ideal solution. Yet solutions are actually complicated systems whose molecular nature we are only now beginning to understand [1, 2, 3, 4]. Two solvents, when mixed, often release heat (or absorb heat) and undergo change in volume. Think of a water sulfuric acid (caution]) mixture or a water DMSO (dimethyl sulfoxide) mixture. After the solvent mixture equilibrates you will find that its volume is not equal to the sum of the volumes of the pure solvents (it is usually smaller). In physical chemistry we treat these problems by using the concept of molar volume, V. Molar volumes are empirical numbers - they are determined by experimental measurements for different solvent compositions. Read the next problem. 

Gibbs functions for a real salt solution and the corresponding ideal salt solution containing m2 moles of salt in a kilogram of solvent. GE can be calculated for many aqueous salt solutions from published values of 0 and y . In the same way, the corresponding excess enthalpy HE can be defined and this equals the apparent partial molar enthalpy. Thus the properties of salt solutions can be examined in plots of GE, HE, and T SE against m2, where SE is the... 

Those properties that depend only on the concentration of solute molecules and not on the nature of the solute are called colligative. A colligative property is also a measure of the chemical potential (partial molar Gibbs free energy) of the solvent in the solution. We consider ideal solutions first and then show how allowances are made for real solutions. 

An analysis of the cosolvent concentration dependence of the osmotic second virial coefficient (OSVC) in water—protein—cosolvent mixtures is developed. The Kirkwood—Buff fluctuation theory for ternary mixtures is used as the main theoretical tool. On its basis, the OSVC is expressed in terms of the thermodynamic properties of infinitely dilute (with respect to the protein) water—protein—cosolvent mixtures. These properties can be divided into two groups (1) those of infinitely dilute protein solutions (such as the partial molar volume of a protein at infinite dilution and the derivatives of the protein activity coefficient with respect to the protein and water molar fractions) and (2) those of the protein-free water—cosolvent mixture (such as its concentrations, the isothermal compressibility, the partial molar volumes, and the derivative of the water activity coefficient with respect to the water molar fraction). Expressions are derived for the OSVC of ideal mixtures and for a mixture in which only the binary mixed solvent is ideal. The latter expression contains three contributions (1) one due to the protein—solvent interactions which is connected to the preferential binding parameter, (2) another one due to protein/protein interactions (B p ), and (3) a third one representing an ideal mixture contribution The cosolvent composition dependencies of these three contributions... 

The standard state is here a purely hypothetical one, just as is the case with gases ( 30b) it might be regarded as the state in which the mole fraction of the solute is unity, but certain thermodynamic properties, e.g., partial molar heat content and heat capacity, are those of the solute in the reference state, he., infinite dilution (cf. 37d). If the solution behaved ideally over the whole range of compodtion, the activity would always be equal to the mole fraction, even when n = 1, i.e., for the pure solute (cf. Fig. 24,1). In this event, the proposed standard state would represent the pure liquid solute at 1 atm. pressure. For nonideal solutions, however, the standard state has no reality, and so it is preferable to define it in terms of a reference state. 

In order to indicate the fact that the value of G as given by equation (42.1) applies to the constituent 2, i.e., the solute, a subscript 2 is sometimes included. However, this is usually omitted, for in the great majority of cases it is understood that the apparent molar property refers to the solute. It i.s seen from equation (42.1) that o is the apparent contribution of 1 mole of the component 2 to the property G of the mixture. If the particular property were strictly additive for the two components, e.g., volume and heat content for ideal gas and liquid solutions, the value of 4>q would be equal to the actual molar contribution, and hence also to the partial molar value. For nonideal systems, however, the quantities are all different. 

A substance in solution has a chemical potential, which is the partial molar free energy of the substance, which determines its reactivity. At constant pressure and temperature, reactivity is given by the thermodynamic activity of the substance for a so-called ideal system, this equals the mole fraction. Most food systems are nonideal, and then activity equals mole fraction times an activity coefficient, which may markedly deviate from unity. In many dilute solutions, the solute behaves as if the system were ideal. For such ideally dilute systems, simple relations exist for the solubility of substances, partitioning over phases, and the so-called colligative properties (lowering of vapor pressure, boiling point elevation, freezing point depression, osmotic pressure). 

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