Ideal solution

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https://en.wikipedia.org/wiki/Ideal_solution

In chemistry, an ideal solution or ideal mixture is a solution that exhibits thermodynamic properties analogous to those of a mixture of ideal gases. The enthalpy of mixing is zero as is the volume change on mixing by definition; the closer to zero the enthalpy of mixing is, the more "ideal" the behavior of the solution becomes. The vapor pressures of the solvent and solute obey Raoult's law and Henry's law, respectively, and the activity coefficient (which measures deviation from ideality) is equal to one for each component.

The concept of an ideal solution is fundamental to chemical thermodynamics and its applications, such as the explanation of colligative properties.

Physical origin

Ideality of solutions is analogous to ideality for gases, with the important difference that intermolecular interactions in liquids are strong and cannot simply be neglected as they can for ideal gases. Instead we assume that the mean strength of the interactions are the same between all the molecules of the solution.

More formally, for a mix of molecules of A and B, then the interactions between unlike neighbors (U_AB) and like neighbors U_AA and U_BB must be of the same average strength, i.e., 2 U_AB = U_AA + U_BB and the longer-range interactions must be nil (or at least indistinguishable). If the molecular forces are the same between AA, AB and BB, i.e., U_AB = U_AA = U_BB, then the solution is automatically ideal.

If the molecules are almost identical chemically, e.g., 1-butanol and 2-butanol, then the solution will be almost ideal. Since the interaction energies between A and B are almost equal, it follows that there is only a very small overall energy (enthalpy) change when the substances are mixed. The more dissimilar the nature of A and B, the more strongly the solution is expected to deviate from ideality.

Formal definition

Different related definitions of an ideal solution have been proposed. The simplest definition is that an ideal solution is a solution for which each component obeys Raoult's law ${\displaystyle p_i=x_i{p}_i^{\ast }}$ for all compositions. Here ${\displaystyle p_i}$ is the vapor pressure of component ${\displaystyle i}$ above the solution, ${\displaystyle x_i}$ is its mole fraction and ${\displaystyle p_i^{\ast }}$ is the vapor pressure of the pure substance ${\displaystyle i}$ at the same temperature.

This definition depends on vapor pressure, which is a directly measurable property, at least for volatile components. The thermodynamic properties may then be obtained from the chemical potential μ (which is the partial molar Gibbs energy g) of each component. If the vapor is an ideal gas,

${\displaystyle \mu (T,p_i)=g(T,p_i)=g^{\mathrm{u}}(T,p^u)+RT\ln \frac{p_i}{p^u}.}$

The reference pressure ${\displaystyle p^u}$ may be taken as ${\displaystyle P^o}$ = 1 bar, or as the pressure of the mix, whichever is simpler.

On substituting the value of ${\displaystyle p_i}$ from Raoult's law,

${\displaystyle \mu (T,p_i)=g^{\mathrm{u}}(T,p^u)+RT\ln \frac{p_i^{\ast }}{p^u}+RT\ln x_i={\mu }_i^{\ast }+RT\ln x_i.}$

This equation for the chemical potential can be used as an alternate definition for an ideal solution.

However, the vapor above the solution may not actually behave as a mixture of ideal gases. Some authors therefore define an ideal solution as one for which each component obeys the fugacity analogue of Raoult's law ${\displaystyle f_i=x_i{f}_i^{\ast }}$. Here ${\displaystyle f_i}$ is the fugacity of component ${\displaystyle i}$ in solution and ${\displaystyle f_i^{\ast }}$ is the fugacity of ${\displaystyle i}$ as a pure substance. Since the fugacity is defined by the equation

${\displaystyle \mu (T,P)=g(T,P)=g^{\mathrm{u}}(T,p^u)+RT\ln \frac{f_i}{p^u}}$

this definition leads to ideal values of the chemical potential and other thermodynamic properties even when the component vapors above the solution are not ideal gases. An equivalent statement uses thermodynamic activity instead of fugacity.

Thermodynamic properties

Volume

If we differentiate this last equation with respect to ${\displaystyle p}$ at ${\displaystyle T}$ constant we get:

${\displaystyle {\left(\frac{\mathrm{\partial }g(T,P)}{\mathrm{\partial }P}\right)}_T=RT{\left(\frac{\mathrm{\partial }\ln f}{\mathrm{\partial }P}\right)}_T.}$

Since we know from the Gibbs potential equation that:

${\displaystyle {\left(\frac{\mathrm{\partial }g(T,P)}{\mathrm{\partial }P}\right)}_T=v}$

with the molar volume ${\displaystyle v}$, these last two equations put together give:

${\displaystyle {\left(\frac{\mathrm{\partial }\ln f}{\mathrm{\partial }P}\right)}_T=\frac{v}{RT}.}$

Since all this, done as a pure substance, is valid in an ideal mix just adding the subscript ${\displaystyle i}$ to all the intensive variables and changing ${\displaystyle v}$ to ${\displaystyle \overline{v_i}}$, with optional overbar, standing for partial molar volume:

${\displaystyle {\left(\frac{\mathrm{\partial }\ln f_i}{\mathrm{\partial }P}\right)}_{T,x_i}=\frac{\overline{v_i}}{RT}.}$

Applying the first equation of this section to this last equation we find:

${\displaystyle v_i^{\ast }={\overline{v}}_i}$

which means that the partial molar volumes in an ideal mix are independent of composition. Consequently, the total volume is the sum of the volumes of the components in their pure forms:

${\displaystyle V=\sum _i{V}_i^{\ast }.}$

Enthalpy and heat capacity

Proceeding in a similar way but taking the derivative with respect to ${\displaystyle T}$ we get a similar result for molar enthalpies:

${\displaystyle \frac{g(T,P)-g^{\mathrm{g}\mathrm{a}\mathrm{s}}(T,p^u)}{RT}=\ln \frac{f}{p^u}.}$

Remembering that ${\displaystyle {\left(\frac{\mathrm{\partial }\frac{g}{T}}{\mathrm{\partial }T}\right)}_P=-\frac{h}{T^2}}$ we get:

${\displaystyle -\frac{\overline{h_i}-h_i^{\mathrm{g}\mathrm{a}\mathrm{s}}}{R}=-\frac{h_i^{\ast }-h_i^{\mathrm{g}\mathrm{a}\mathrm{s}}}{R}}$

which in turn means that ${\displaystyle \overline{h_i}=h_i^{\ast }}$ and that the enthalpy of the mix is equal to the sum of its component enthalpies.

Since ${\displaystyle \overline{u_i}=\overline{h_i}-p\overline{v_i}}$ and ${\displaystyle u_i^{\ast }=h_i^{\ast }-p{v}_i^{\ast }}$, similarly

${\displaystyle u_i^{\ast }=\overline{u_i}.}$

It is also easily verifiable that

${\displaystyle C_{pi}^{\ast }=\overline{C_{pi}}.}$

Entropy of mixing

Finally since

${\displaystyle \overline{g_i}={\mu }_i=g_i^{\mathrm{g}\mathrm{a}\mathrm{s}}+RT\ln \frac{f_i}{p^u}=g_i^{\mathrm{g}\mathrm{a}\mathrm{s}}+RT\ln \frac{f_i^{\ast }}{p^u}+RT\ln x_i={\mu }_i^{\ast }+RT\ln x_i}$

we find that

${\displaystyle \mathrm{\Delta }{g}_{i,\mathrm{m}\mathrm{i}\mathrm{x}}=RT\ln x_i.}$

Since the Gibbs free energy per mole of the mixture ${\displaystyle G_m}$ is

$${\displaystyle G_m=\sum _i{x}_i{g}_i}$$

then

${\displaystyle \mathrm{\Delta }{G}_{\mathrm{m},\mathrm{m}\mathrm{i}\mathrm{x}}=RT\sum _i{x}_i\ln x_i.}$

At last we can calculate the molar entropy of mixing since ${\displaystyle g_i^{\ast }=h_i^{\ast }-T{s}_i^{\ast }}$ and ${\displaystyle \overline{g_i}=\overline{h_i}-T\overline{s_i}}$

${\displaystyle \mathrm{\Delta }{s}_{i,\mathrm{m}\mathrm{i}\mathrm{x}}=-R\sum _i\ln x_i}$

${\displaystyle \mathrm{\Delta }{S}_{\mathrm{m},\mathrm{m}\mathrm{i}\mathrm{x}}=-R\sum _i{x}_i\ln x_i.}$

Consequences

Solvent–solute interactions are the same as solute–solute and solvent–solvent interactions, on average. Consequently, the enthalpy of mixing (solution) is zero and the change in Gibbs free energy on mixing is determined solely by the entropy of mixing. Hence the molar Gibbs free energy of mixing is

${\displaystyle \mathrm{\Delta }{G}_{\mathrm{m},\mathrm{m}\mathrm{i}\mathrm{x}}=RT\sum _i{x}_i\ln x_i}$

or for a two-component ideal solution

${\displaystyle \mathrm{\Delta }{G}_{\mathrm{m},\mathrm{m}\mathrm{i}\mathrm{x}}=RT(x_A\ln x_A+x_B\ln x_B)}$

where m denotes molar, i.e., change in Gibbs free energy per mole of solution, and ${\displaystyle x_i}$ is the mole fraction of component ${\displaystyle i}$. Note that this free energy of mixing is always negative (since each ${\displaystyle x_i\in [0,1]}$, each ${\displaystyle \ln x_i}$ or its limit for ${\displaystyle x_i\to 0}$ must be negative (infinite)), i.e., ideal solutions are miscible at any composition and no phase separation will occur.

The equation above can be expressed in terms of chemical potentials of the individual components

${\displaystyle \mathrm{\Delta }{G}_{\mathrm{m},\mathrm{m}\mathrm{i}\mathrm{x}}=\sum _i{x}_i\mathrm{\Delta }{\mu }_{i,\mathrm{m}\mathrm{i}\mathrm{x}}}$

where ${\displaystyle \mathrm{\Delta }{\mu }_{i,\mathrm{m}\mathrm{i}\mathrm{x}}=RT\ln x_i}$ is the change in chemical potential of ${\displaystyle i}$ on mixing. If the chemical potential of pure liquid ${\displaystyle i}$ is denoted ${\displaystyle {\mu }_i^{\ast }}$, then the chemical potential of ${\displaystyle i}$ in an ideal solution is

${\displaystyle {\mu }_i={\mu }_i^{\ast }+RT\ln x_i.}$

Any component ${\displaystyle i}$ of an ideal solution obeys Raoult's Law over the entire composition range:

${\displaystyle p_i=(p_i{)}_{\text{pure}}{x}_i}$

where ${\displaystyle (p_i{)}_{\text{pure}}}$ is the equilibrium vapor pressure of pure component ${\displaystyle i}$ and ${\displaystyle x_i}$is the mole fraction of component ${\displaystyle i}$ in solution.

Non-ideality

Deviations from ideality can be described by the use of Margules functions or activity coefficients. A single Margules parameter may be sufficient to describe the properties of the solution if the deviations from ideality are modest; such solutions are termed regular.

In contrast to ideal solutions, where volumes are strictly additive and mixing is always complete, the volume of a non-ideal solution is not, in general, the simple sum of the volumes of the component pure liquids and solubility is not guaranteed over the whole composition range. By measurement of densities, thermodynamic activity of components can be determined.

Solution (chemistry)

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https://en.wikipedia.org/wiki/Solution_(chemistry)

Making a saline water solution by dissolving table salt (NaCl) in water. The salt is the solute and the water the solvent.

In chemistry, a solution is a special type of homogeneous mixture composed of two or more substances. In such a mixture, a solute is a substance dissolved in another substance, known as a solvent. If the attractive forces between the solvent and solute particles are greater than the attractive forces holding the solute particles together, the solvent particles pull the solute particles apart and surround them. These surrounded solute particles then move away from the solid solute and out into the solution. The mixing process of a solution happens at a scale where the effects of chemical polarity are involved, resulting in interactions that are specific to solvation. The solution usually has the state of the solvent when the solvent is the larger fraction of the mixture, as is commonly the case. One important parameter of a solution is the concentration, which is a measure of the amount of solute in a given amount of solution or solvent. The term "aqueous solution" is used when one of the solvents is water.

Characteristics

• A solution is a homogeneous mixture of two or more substances.
• The particles of solute in a solution cannot be seen by the naked eye. By contrast, particles may be visible in a suspension.
• A solution does not cause beams of light to scatter. By contrast, the particles in a suspension or colloid can cause Tyndall scattering or Rayleigh scattering.
• A solution is stable, and solutes will not precipitate unless added in excess of the mixture's solubility, at which point the excess would remain in its solid phase. A solution containing more dissolved solutes than at equilibrium is referred to as supersaturated.
• The solutes and solvents in a solution cannot be separated by filtration (or mechanically).
• It is composed of only one phase.

Types

Homogeneous means that the components of the mixture form a single phase. Heterogeneous means that the components of the mixture are of different phase. The properties of the mixture (such as concentration, temperature, and density) can be uniformly distributed through the volume but only in absence of diffusion phenomena or after their completion. Usually, the substance present in the greatest amount is considered the solvent. Solvents can be gases, liquids, or solids. One or more components present in the solution other than the solvent are called solutes. The solution has the same physical state as the solvent.

Gaseous mixtures

If the solvent is a gas, only gases (non-condensable) or vapors (condensable) are dissolved under a given set of conditions. An example of a gaseous solution is air (oxygen and other gases dissolved in nitrogen). Since interactions between gaseous molecules play almost no role, non-condensable gases form rather trivial solutions. In the literature, they are not even classified as solutions, but simply addressed as homogeneous mixtures of gases. The Brownian motion and the permanent molecular agitation of gas molecules guarantee the homogeneity of the gaseous systems. Non-condensable gases mixtures (e.g., air/CO₂, or air/xenon) do not spontaneously demix, nor sediment, as distinctly stratified and separate gas layers as a function of their relative density. Diffusion forces efficiently counteract gravitation forces under normal conditions prevailing on Earth. The case of condensable vapors is different: once the saturation vapor pressure at a given temperature is reached, vapor excess condenses into the liquid state.

Liquid solutions

If the solvent is a liquid, then almost all gases, liquids, and solids can be dissolved. Here are some examples:

• Gas in liquid:
  • Oxygen in water
  • Carbon dioxide in water – a less simple example, because the solution is accompanied by a chemical reaction (formation of ions). The visible bubbles in carbonated water are not the dissolved gas, but only an effervescence of carbon dioxide that has come out of solution; the dissolved gas itself is not visible since it is dissolved on a molecular level.
• Liquid in liquid:
  • The mixing of two or more substances of the same chemistry but different concentrations to form a constant. (Homogenization of solutions)
  • Alcoholic beverages are basically solutions of ethanol in water.
• Solid in liquid:
  • Sucrose (table sugar) in water
  • Sodium chloride (NaCl) (table salt) or any other salt in water, which forms an electrolyte: When dissolving, salt dissociates into ions.
• Solutions in water are especially common, and are called aqueous solutions.
• Non-aqueous solutions are when the liquid solvent involved is not water.

Counterexamples are provided by liquid mixtures that are not homogeneous: colloids, suspensions, emulsions are not considered solutions.

Body fluids are examples of complex liquid solutions, containing many solutes. Many of these are electrolytes since they contain solute ions, such as potassium. Furthermore, they contain solute molecules like sugar and urea. Oxygen and carbon dioxide are also essential components of blood chemistry, where significant changes in their concentrations may be a sign of severe illness or injury.

Solid solutions

If the solvent is a solid, then gases, liquids, and solids can be dissolved.

• Gas in solids:
  • Hydrogen dissolves rather well in metals, especially in palladium; this is studied as a means of hydrogen storage.
• Liquid in solid:
  • Mercury in gold, forming an amalgam
  • Water in solid salt or sugar, forming moist solids
  • Hexane in paraffin wax
  • Polymers containing plasticizers such as phthalate (liquid) in PVC (solid)
• Solid in solid:
  • Steel, basically a solution of carbon atoms in a crystalline matrix of iron atoms
  • Alloys like bronze and many others
  • Radium sulfate dissolved in barium sulfate: a true solid solution of Ra in BaSO₄

Solubility

Main articles: Solubility and Solvation

The ability of one compound to dissolve in another compound is called solubility.^{[clarification needed]} When a liquid can completely dissolve in another liquid the two liquids are miscible. Two substances that can never mix to form a solution are said to be immiscible.

All solutions have a positive entropy of mixing. The interactions between different molecules or ions may be energetically favored or not. If interactions are unfavorable, then the free energy decreases with increasing solute concentration. At some point, the energy loss outweighs the entropy gain, and no more solute particles can be dissolved; the solution is said to be saturated. However, the point at which a solution can become saturated can change significantly with different environmental factors, such as temperature, pressure, and contamination. For some solute-solvent combinations, a supersaturated solution can be prepared by raising the solubility (for example by increasing the temperature) to dissolve more solute and then lowering it (for example by cooling).

Usually, the greater the temperature of the solvent, the more of a given solid solute it can dissolve. However, most gases and some compounds exhibit solubilities that decrease with increased temperature. Such behavior is a result of an exothermic enthalpy of solution. Some surfactants exhibit this behaviour. The solubility of liquids in liquids is generally less temperature-sensitive than that of solids or gases.

Properties

The physical properties of compounds such as melting point and boiling point change when other compounds are added. Together they are called colligative properties. There are several ways to quantify the amount of one compound dissolved in the other compounds collectively called concentration. Examples include molarity, volume fraction, and mole fraction.

The properties of ideal solutions can be calculated by the linear combination of the properties of its components. If both solute and solvent exist in equal quantities (such as in a 50% ethanol, 50% water solution), the concepts of "solute" and "solvent" become less relevant, but the substance that is more often used as a solvent is normally designated as the solvent (in this example, water).

Liquid solution characteristics

See also: Solvent § Solvent classifications

In principle, all types of liquids can behave as solvents: liquid noble gases, molten metals, molten salts, molten covalent networks, and molecular liquids. In the practice of chemistry and biochemistry, most solvents are molecular liquids. They can be classified into polar and non-polar, according to whether their molecules possess a permanent electric dipole moment. Another distinction is whether their molecules can form hydrogen bonds (protic and aprotic solvents). Water, the most commonly used solvent, is both polar and sustains hydrogen bonds.

Water is a good solvent because the molecules are polar and capable of forming hydrogen bonds (1).

Salts dissolve in polar solvents, forming positive and negative ions that are attracted to the negative and positive ends of the solvent molecule, respectively. If the solvent is water, hydration occurs when the charged solute ions become surrounded by water molecules. A standard example is aqueous saltwater. Such solutions are called electrolytes. Whenever salt dissolves in water ion association has to be taken into account.

Polar solutes dissolve in polar solvents, forming polar bonds or hydrogen bonds. As an example, all alcoholic beverages are aqueous solutions of ethanol. On the other hand, non-polar solutes dissolve better in non-polar solvents. Examples are hydrocarbons such as oil and grease that easily mix, while being incompatible with water.

An example of the immiscibility of oil and water is a leak of petroleum from a damaged tanker, that does not dissolve in the ocean water but rather floats on the surface.

Preparation from constituent ingredients

It is common practice in laboratories to make a solution directly from its constituent ingredients. There are three cases in practical calculation:

• Case 1: amount of solvent volume is given.
• Case 2: amount of solute mass is given.
• Case 3: amount of final solution volume is given.

In the following equations, A is solvent, B is solute, and C is concentration. Solute volume contribution is considered through the ideal solution model.

• Case 1: amount (mL) of solvent volume V_A is given. Solute mass m_B = C V_A d_A /(100-C/d_B)
• Case 2: amount of solute mass m_B is given. Solvent volume V_A = m_B (100/C-1/ d_B )
• Case 3: amount (mL) of final solution volume Vt is given. Solute mass m_B = C Vt /100; Solvent volume V_A=(100/C-1/ d_B) m_B
• Case 2: solute mass is known, V_A = m_B 100/C
• Case 3: total solution volume is known, same equation as case 1. V_A=Vt; m_B = C V_A /100

Example: Make 2 g/100mL of NaCl solution with 1 L water. The density of the resulting solution is considered to be equal to that of water, statement holding especially for dilute solutions, so the density information is not required.

m_B = C V_A = ( 2 / 100 ) g/mL × 1000 mL = 20 g

Chemists often make concentrated stock solutions that may then be diluted as needed for laboratory applications. Standard solutions are those where concentrations of solutes are accurately and precisely known.