Colligative Properties
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In
chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
, colligative properties are those properties of
solution Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Soluti ...
s that depend on the ratio of the number of solute particles to the number of
solvent A solvent (s) (from the Latin '' solvō'', "loosen, untie, solve") is a substance that dissolves a solute, resulting in a solution. A solvent is usually a liquid but can also be a solid, a gas, or a supercritical fluid. Water is a solvent for ...
particles In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
in a solution, and not on the nature of the chemical species present. The number ratio can be related to the various units for
concentration In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', ''number concentration'', an ...
of a solution such as
molarity Molar concentration (also called molarity, amount concentration or substance concentration) is a measure of the concentration of a chemical species, in particular of a solute in a solution, in terms of amount of substance per unit volume of solut ...
,
molality Molality is a measure of the number of moles of solute in a solution corresponding to 1 kg or 1000 g of solvent. This contrasts with the definition of molarity which is based on a specified volume of solution. A commonly used unit for molali ...
,
normality (chemistry) In chemistry, the equivalent concentration or normality (N) of a solution is defined as the molar concentration ''ci'' divided by an equivalence factor ''f''eq: :Normality = Definition Normality is defined as the number of gramme or mole equ ...
, etc. The assumption that solution properties are independent of nature of solute particles is exact only for ideal solutions, which are solutions that exhibit
thermodynamic Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of ther ...
properties analogous to those of an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
, and is approximate for dilute real solutions. In other words, colligative properties are a set of solution properties that can be reasonably approximated by the assumption that the solution is ideal. Only properties which result from the dissolution of a nonvolatile solute in a volatile liquid solvent are considered.KL Kapoor ''Applications of Thermodynamics'' Volume 3 They are essentially solvent properties which are changed by the presence of the solute. The solute particles displace some solvent molecules in the liquid phase and thereby reduce the concentration of solvent and increase its entropy, so that the colligative properties are independent of the nature of the solute. The word colligative is derived from the Latin ''colligatus'' meaning ''bound together''. This indicates that all colligative properties have a common feature, namely that they are related only to the number of solute molecules relative to the number of solvent molecules and not to the nature of the solute. Colligative properties include: * Relative lowering of vapor pressure ( Raoult's law) * Elevation of boiling point * Depression of freezing point *
Osmotic pressure Osmotic pressure is the minimum pressure which needs to be applied to a solution to prevent the inward flow of its pure solvent across a semipermeable membrane. It is also defined as the measure of the tendency of a solution to take in a pure ...
For a given solute-solvent mass ratio, all colligative properties are inversely proportional to solute molar mass. Measurement of colligative properties for a dilute solution of a non-ionized solute such as
urea Urea, also known as carbamide, is an organic compound with chemical formula . This amide has two amino groups (–) joined by a carbonyl functional group (–C(=O)–). It is thus the simplest amide of carbamic acid. Urea serves an important r ...
or
glucose Glucose is a simple sugar with the molecular formula . Glucose is overall the most abundant monosaccharide, a subcategory of carbohydrates. Glucose is mainly made by plants and most algae during photosynthesis from water and carbon dioxide, using ...
in water or another solvent can lead to determinations of relative
molar mass In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, p ...
es, both for small molecules and for
polymers A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic an ...
which cannot be studied by other means. Alternatively, measurements for ionized solutes can lead to an estimation of the percentage of
dissociation Dissociation, in the wide sense of the word, is an act of disuniting or separating a complex object into parts. Dissociation may also refer to: * Dissociation (chemistry), general process in which molecules or ionic compounds (complexes, or salts ...
taking place. Colligative properties are studied mostly for dilute solutions, whose behavior may be approximated as that of an ideal solution. In fact, all of the properties listed above are colligative only in the dilute limit: at higher concentrations, the freezing point depression, boiling point elevation, vapor pressure elevation or depression, and osmotic pressure are all dependent on the chemical nature of the solvent and the solute.


Relative lowering of vapor pressure

A
vapor In physics, a vapor (American English) or vapour (British English and Canadian English; American and British English spelling differences#-our, -or, see spelling differences) is a substance in the gas phase at a temperature lower than its critic ...
is a substance in a gaseous state at a temperature lower than its critical point.
Vapor Pressure Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases ...
is the pressure exerted by a vapor in thermodynamic equilibrium with its solid or liquid state. The vapor pressure of a solvent is lowered when a non-volatile solute is dissolved in it to form a solution. For an ideal solution, the equilibrium vapor pressure is given by Raoult's law as p = p^_ x_ + p^_ x_ + \cdots, where p^_ is the vapor pressure of the ''pure'' component (i= A, B, ...) and x_ is the
mole fraction In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of a constituent (expressed in moles), ''ni'', divided by the total amount of all constituents in a mixture (also expressed in moles), ''n''tot. This ex ...
of the component in the solution For a solution with a solvent (A) and one non-volatile solute (B), p^_ = 0 and p = p^_ x_ The vapor pressure ''lowering'' relative to pure solvent is \Delta p = p^_ - p = p^_ (1 - x_) = p^_ x_, which is proportional to the mole fraction of solute. If the solute dissociates in solution, then the number of moles of solute is increased by the
van 't Hoff factor The van 't Hoff factor (named after Dutch chemist Jacobus Henricus van 't Hoff) is a measure of the effect of a solute on colligative properties such as osmotic pressure, relative lowering in vapor pressure, boiling-point elevation and freez ...
i, which represents the true number of solute particles for each formula unit. For example, the strong electrolyte MgCl2 dissociates into one Mg2+ ion and two Cl ions, so that if ionization is complete, i = 3 and \Delta p = p^_ x_, where x_ is calculated with moles of solute i times initial moles and moles of solvent same as initial moles of solvent before dissociation. The measured colligative properties show that ''i'' is somewhat less than 3 due to
ion association In chemistry, ion association is a chemical reaction whereby ions of opposite electric charge come together in solution to form a distinct chemical entity. Ion associates are classified, according to the number of ions that associate with each ...
.


Boiling point and freezing point

Addition of solute to form a solution stabilizes the solvent in the liquid phase, and lowers the solvent's
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
so that solvent molecules have less tendency to move to the gas or solid phases. As a result, liquid solutions slightly above the solvent boiling point at a given pressure become stable, which means that the boiling point increases. Similarly, liquid solutions slightly below the solvent freezing point become stable meaning that the freezing point decreases. Both the
boiling point elevation Boiling-point elevation describes the phenomenon that the boiling point of a liquid (a solvent) will be higher when another compound is added, meaning that a solution has a higher boiling point than a pure solvent. This happens whenever a non-volat ...
and the
freezing point depression Freezing-point depression is a drop in the minimum temperature at which a substance freezing, freezes, caused when a smaller amount of another, non-Volatility (chemistry), volatile substance is added. Examples include adding salt into water (u ...
are proportional to the lowering of vapor pressure in a dilute solution. These properties are colligative in systems where the solute is essentially confined to the liquid phase. Boiling point elevation (like vapor pressure lowering) is colligative for non-volatile solutes where the solute presence in the gas phase is negligible. Freezing point depression is colligative for most solutes since very few solutes dissolve appreciably in solid solvents.


Boiling point elevation (ebullioscopy)

The
boiling point The boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor. The boiling point of a liquid varies depending upon the surrounding envir ...
of a liquid at a given external pressure is the temperature (T_) at which the vapor pressure of the liquid equals the external pressure. The ''normal boiling point'' is the boiling point at a pressure equal to 1 atm. The boiling point of a pure solvent is increased by the addition of a non-volatile solute, and the elevation can be measured by
ebullioscopy An ebullioscope (from the Latin ''ēbullīre'' (to boil) + -scope) is an instrument for measuring the boiling point of a liquid. This can be used for determining the alcoholic strength of a mixture, or for determining the molecular weight of a non- ...
. It is found that :\Delta T_ = T_ - T_ = i\cdot K_b \cdot m Here ''i'' is the
van 't Hoff factor The van 't Hoff factor (named after Dutch chemist Jacobus Henricus van 't Hoff) is a measure of the effect of a solute on colligative properties such as osmotic pressure, relative lowering in vapor pressure, boiling-point elevation and freez ...
as above, ''K''b is the
ebullioscopic constant In thermodynamics, the ebullioscopic constant relates molality to boiling point elevation. It is the ratio of the latter to the former: :\Delta T = iK_\text b * is the van 't Hoff factor, the number of particles the solute splits into or forms ...
of the solvent (equal to 0.512 °C kg/mol for water), and ''m'' is the
molality Molality is a measure of the number of moles of solute in a solution corresponding to 1 kg or 1000 g of solvent. This contrasts with the definition of molarity which is based on a specified volume of solution. A commonly used unit for molali ...
of the solution. The boiling point is the temperature at which there is equilibrium between liquid and gas phases. At the boiling point, the number of gas molecules condensing to liquid equals the number of liquid molecules evaporating to gas. Adding a solute dilutes the concentration of the liquid molecules and reduces the rate of evaporation. To compensate for this and re-attain equilibrium, the boiling point occurs at a higher temperature. If the solution is assumed to be an ideal solution, ''K''b can be evaluated from the
thermodynamic Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of ther ...
condition for liquid-vapor equilibrium. At the boiling point the
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
μA of the solvent in the solution phase equals the chemical potential in the pure vapor phase above the solution. :\mu _A(T_b) = \mu_A^(T_b) + RT\ln x_A\ = \mu_A^(g, 1 \,\mathrm), where the asterisks indicate pure phases. This leads to the result K_b = RMT_b^2/\Delta H_, where R is the
molar gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
, M is the solvent
molar mass In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, p ...
and Δ''H''vap is the solvent molar
enthalpy of vaporization The enthalpy of vaporization (symbol ), also known as the (latent) heat of vaporization or heat of evaporation, is the amount of energy (enthalpy) that must be added to a liquid substance to transform a quantity of that substance into a gas. T ...
.T. Engel and P. Reid, Physical Chemistry (Pearson Benjamin Cummings 2006) p.204-5


Freezing point depression (cryoscopy)

The freezing point (T_) of a pure solvent is lowered by the addition of a solute which is insoluble in the solid solvent, and the measurement of this difference is called ''cryoscopy''. It is found that :\Delta T_ = T_ - T_ = - i\cdot K_f \cdot m (which can also be written as \Delta T_ = T_ - T_ = i\cdot K_f \cdot m ) Here ''Kf'' is the
cryoscopic constant In thermodynamics, the cryoscopic constant, , relates molality to freezing point depression (which is a colligative property). It is the ratio of the latter to the former: :\Delta T_f = -i \cdot K_f \cdot b * is the van ‘t Hoff factor, the ...
(equal to 1.86 °C kg/mol for the freezing point of water), ''i'' is the van 't Hoff factor, and ''m'' the molality (in mol/kg). This predicts the melting of ice by
road salt Sodium chloride , commonly known as salt (although sea salt also contains other chemical salts), is an ionic compound with the chemical formula NaCl, representing a 1:1 ratio of sodium and chloride ions. With molar masses of 22.99 and 35.45 g/ ...
. In the liquid solution, the solvent is diluted by the addition of a solute, so that fewer molecules are available to freeze. Re-establishment of equilibrium is achieved at a lower temperature at which the rate of freezing becomes equal to the rate of liquefying. At the lower freezing point, the vapor pressure of the liquid is equal to the vapor pressure of the corresponding solid, and the chemical potentials of the two phases are equal as well. The equality of chemical potentials permits the evaluation of the cryoscopic constant as K_f = RMT_f^2/\Delta_H, where Δfus''H'' is the solvent molar
enthalpy of fusion In thermodynamics, the enthalpy of fusion of a substance, also known as (latent) heat of fusion, is the change in its enthalpy resulting from providing energy, typically heat, to a specific quantity of the substance to change its state from a s ...
.


Osmotic pressure

The osmotic pressure of a solution is the difference in pressure between the solution and the pure liquid solvent when the two are in equilibrium across a
semipermeable membrane Semipermeable membrane is a type of biological or synthetic, polymeric membrane that will allow certain molecules or ions to pass through it by osmosis. The rate of passage depends on the pressure, concentration, and temperature of the molecul ...
, which allows the passage of solvent molecules but not of solute particles. If the two phases are at the same initial pressure, there is a net transfer of solvent across the membrane into the solution known as
osmosis Osmosis (, ) is the spontaneous net movement or diffusion of solvent molecules through a selectively-permeable membrane from a region of high water potential (region of lower solute concentration) to a region of low water potential (region of ...
. The process stops and equilibrium is attained when the pressure difference equals the osmotic pressure. Two laws governing the osmotic pressure of a dilute solution were discovered by the German botanist W. F. P. Pfeffer and the Dutch chemist J. H. van’t Hoff: # The
osmotic pressure Osmotic pressure is the minimum pressure which needs to be applied to a solution to prevent the inward flow of its pure solvent across a semipermeable membrane. It is also defined as the measure of the tendency of a solution to take in a pure ...
of a dilute solution at constant temperature is directly proportional to its concentration. # The osmotic pressure of a solution is directly proportional to its absolute temperature. These are analogous to
Boyle's law Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as: The ...
and
Charles's law Charles's law (also known as the law of volumes) is an experimental gas law that describes how gases tend to expand when heated. A modern statement of Charles's law is: When the pressure on a sample of a dry gas is held constant, the Kelvin t ...
for gases. Similarly, the combined
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
, PV = nRT, has as an analogue for ideal solutions \Pi V = n R T i, where \Pi is osmotic pressure; ''V'' is the volume; ''n'' is the number of moles of solute; ''R'' is the molar
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
8.314 J K−1 mol−1; ''T'' is absolute temperature; and ''i'' is the
Van 't Hoff factor The van 't Hoff factor (named after Dutch chemist Jacobus Henricus van 't Hoff) is a measure of the effect of a solute on colligative properties such as osmotic pressure, relative lowering in vapor pressure, boiling-point elevation and freez ...
. The osmotic pressure is then proportional to the
molar concentration Molar concentration (also called molarity, amount concentration or substance concentration) is a measure of the concentration of a chemical species, in particular of a solute in a solution, in terms of amount of substance per unit volume of solut ...
c = n/V, since :\Pi = \frac {V} = c R T i The osmotic pressure is proportional to the concentration of solute particles ''ci'' and is therefore a colligative property. As with the other colligative properties, this equation is a consequence of the equality of solvent chemical potentials of the two phases in equilibrium. In this case the phases are the pure solvent at pressure ''P'' and the solution at total pressure (''P'' + \Pi).


History

The word colligative (Latin: co, ligare) was introduced in 1891 by
Wilhelm Ostwald Friedrich Wilhelm Ostwald (; 4 April 1932) was a Baltic German chemist and German philosophy, philosopher. Ostwald is credited with being one of the founders of the field of physical chemistry, with Jacobus Henricus van 't Hoff, Walther Nernst, ...
. Ostwald classified solute properties in three categories: H.W. Smith, Circulation 21, 808 (1960) ''Theory of Solutions: A Knowledge of the Laws of Solutions ...'' # ''colligative'' properties, which depend only on solute concentration and temperature and are independent of the nature of the solute particles # ''additive'' properties such as mass, which are the sums of properties of the constituent particles and therefore depend also on the composition (or molecular formula) of the solute, and # ''constitutional'' properties, which depend further on the molecular structure of the given solute.


References

Solutions Physical chemistry Amount of substance