Osmolality
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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 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 ...
which is based on a specified
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of solution. A commonly used unit for molality 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 ...
is mol/ kg. A solution of concentration 1 mol/kg is also sometimes denoted as ''1 molal''. The unit ''mol/kg'' requires that molar mass be expressed in ''kg/mol'', instead of the usual ''g/mol'' or ''kg/kmol''.


Definition

The molality (''b''), of a
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 ...
is defined as the
amount of substance In chemistry, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, ions, ...
(in
mole Mole (or Molé) may refer to: Animals * Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America * Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpida ...
s) of solute, ''n''solute, divided by the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
(in kg) of the
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 ...
, ''m''solvent: :b = \frac In the case of solutions with more than one solvent, molality can be defined for the mixed solvent considered as a pure pseudo-solvent. Instead of mole solute per kilogram solvent as in the binary case, units are defined as mole solute per kilogram mixed solvent.


Origin

The term ''molality'' is formed in analogy to ''molarity'' which is 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 ...
of a solution. The earliest known use of the
intensive property Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is on ...
molality and of its adjectival unit, the now-deprecated ''molal'', appears to have been published by G. N. Lewis and M. Randall in the 1923 publication of ''Thermodynamics and the Free Energies of Chemical Substances.'' Though the two terms are subject to being confused with one another, the molality and molarity of a dilute
aqueous solution An aqueous solution is a solution in which the solvent is water. It is mostly shown in chemical equations by appending (aq) to the relevant chemical formula. For example, a solution of table salt, or sodium chloride (NaCl), in water would be re ...
are nearly the same, as one kilogram of water (solvent) occupies the volume of 1 liter at room temperature and a small amount of solute has little effect on the volume.


Unit

The SI unit for molality is moles per kilogram of solvent. A solution with a molality of 3 mol/kg is often described as "3 molal", "3 m" or "3 ''m''". However, following the SI system of units, the
National Institute of Standards and Technology The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
, the
United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
authority on
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
, considers the term "molal" and the unit symbol "m" to be obsolete, and suggests mol/kg or a related unit of the SI.


Usage considerations


Advantages

The primary advantage of using molality as a measure of concentration is that molality only depends on the masses of solute and solvent, which are unaffected by variations in temperature and pressure. In contrast, solutions prepared volumetrically (e.g.
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 ...
or mass concentration) are likely to change as temperature and pressure change. In many applications, this is a significant advantage because the mass, or the amount, of a substance is often more important than its volume (e.g. in a
limiting reagent The limiting reagent (or limiting reactant or limiting agent) in a chemical reaction is a reactant that is totally consumed when the chemical reaction is completed. The amount of product formed is limited by this reagent, since the reaction canno ...
problem). Another advantage of molality is the fact that the molality of one solute in a solution is independent of the presence or absence of other solutes.


Problem areas

Unlike all the other compositional properties listed in "Relation" section (below), molality ''depends'' on the choice of the substance to be called “solvent” in an arbitrary mixture. If there is only one pure liquid substance in a mixture, the choice is clear, but not all solutions are this clear-cut: in an alcohol–water solution, either one could be called the solvent; in an alloy, or
solid solution A solid solution, a term popularly used for metals, is a homogenous mixture of two different kinds of atoms in solid state and have a single crystal structure. Many examples can be found in metallurgy, geology, and solid-state chemistry. The word ...
, there is no clear choice and all constituents may be treated alike. In such situations, mass or mole fraction is the preferred compositional specification.


Relation to other compositional quantities

In what follows, the solvent may be given the same treatment as the other constituents of the solution, such that the molality of the solvent of an ''n''-solute solution, say ''b''0, is found to be nothing more than the reciprocal of its molar mass, ''M''0 (expressed as kg/mol): :b_0=\frac=\frac. For the solutes the expression of molalities is similar: :b_i=\frac=\frac = \frac The expressions linking molalities to mass fractions and mass concentrations contain the molar masses of the solutes ''M''i: :b_i = \frac = \frac = \frac Similarly the equalities below are obtained from the definitions of the molalities and of the other compositional quantities. The mole fraction of solvent can be obtained from the definition by dividing the numerator and denominator to the amount of solvent n0: :x_0 = \frac = \frac Then the sum of ratios of the other mole amounts to the amount of solvent is substituted with expressions from below containing molalities: :\frac = b_i M_0 :\sum_i^n \frac = M_0 \sum_i^n b_i giving the result :x_0 = \frac = \frac


Mass fraction

The conversions to and from the mass fraction, ''w1'', of the solute in a single-solute solution are :w_1=\frac,\quad b_1=\frac , where ''b1'' is the molality and ''M1'' is the
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 ...
of the solute. More generally, for an ''n''-solute/one-solvent solution, letting ''bi'' and ''wi'' be, respectively, the molality and mass fraction of the ''i''-th solute, :w_i=w_0 b_i M_i,\quad b_i=\frac, where ''Mi'' is the molar mass of the ''i''th solute, and ''w''0 is the mass fraction of the solvent, which is expressible both as a function of the molalities as well as a function of the other mass fractions, :w_0=\frac=1-\sum_^. Substitution gives: :w_i = \frac, \quad b_i = \frac


Mole fraction

The conversions to and from 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 ...
, ''x1'' mole fraction of the solute in a single-solute solution are :x_1=\frac,\quad b_1=\frac, where ''M''0 is the molar mass of the solvent. More generally, for an ''n''-solute/one-solvent solution, letting ''xi'' be the mole fraction of the ''i''th solute, :x_i=x_0 M_0 b_i,\quad b_i=\frac = \frac, where ''x''0 is the mole fraction of the solvent, expressible both as a function of the molalities as well as a function of the other mole fractions: :x_0=\frac=1-\sum_^. Substitution gives: :x_i = \frac, \quad b_i = \frac


Molar concentration (molarity)

The conversions to and from 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 ...
, ''c1'', for one-solute solutions are :c_1 = \frac,\quad b_1=\frac, where ''ρ'' is the
mass density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
of the solution, ''b1'' is the molality, and ''M1'' is the molar mass (in kg/mol) of the solute. For solutions with ''n'' solutes, the conversions are :c_i =c_0 M_0 b_i,\quad b_i=\frac, where the molar concentration of the solvent ''c''0 is expressible both as a function of the molalities as well as a function of the other molarities: :c_0=\frac=\frac. Substitution gives: :c_i=\frac,\quad b_i=\frac,


Mass concentration

The conversions to and from the mass concentration, ''ρ''solute, of a single-solute solution are :\rho_\mathrm = \frac,\quad b=\frac, or :\rho_1 = \frac,\quad b_1=\frac, where ''ρ'' is the mass density of the solution, ''b1'' is the molality, and ''M1'' is the molar mass of the
solute 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 ...
. For the general ''n''-solute solution, the mass concentration of the ''i''th solute, ''ρi'', is related to its molality, ''bi'', as follows: :\rho_i = \rho_0 b_i M_i,\quad b_i=\frac, where the mass concentration of the solvent, ''ρ''0, is expressible both as a function of the molalities as well as a function of the other mass concentrations: :\rho_0=\frac=\rho-\sum_^. Substitution gives: :\rho_i=\frac,\quad b_i=\frac,


Equal ratios

Alternatively, one may use just the last two equations given for the compositional property of the solvent in each of the preceding sections, together with the relationships given below, to derive the remainder of properties in that set: :\frac=\frac=\frac=\frac=\frac, where ''i'' and ''j'' are subscripts representing ''all'' the constituents, the ''n'' solutes plus the solvent.


Example of conversion

An acid mixture consists of 0.76, 0.04, and 0.20 mass fractions of 70% HNO3, 49% HF, and H2O, where the percentages refer to mass fractions of the bottled acids carrying a balance of H2O. The first step is determining the mass fractions of the constituents: :\begin w_\mathrm &= 0.70 \times 0.76 = 0.532\\ w_\mathrm &= 0.49 \times 0.04 = 0.0196\\ w_\mathrm &= 1 - w_\mathrm - w_\mathrm = 0.448\\ \end The approximate molar masses in kg/mol are :M_\mathrm=0.063\ \mathrm,\quad M_\mathrm=0.020\ \mathrm,\ M_\mathrm=0.018\ \mathrm. First derive the molality of the solvent, in mol/kg, :b_\mathrm=\frac=\frac\ \mathrm, and use that to derive all the others by use of the equal ratios: :\frac=\frac\quad \therefore b_\mathrm=18.83\ \mathrm. Actually, ''b''H2O cancels out, because it is not needed. In this case, there is a more direct equation: we use it to derive the molality of HF: :b_\mathrm=\frac=2.19\ \mathrm. The mole fractions may be derived from this result: :x_\mathrm=\frac=0.726, :\frac=\frac\quad \therefore x_\mathrm=0.246, :x_\mathrm=1-x_\mathrm-x_\mathrm=0.029.


Osmolality

Osmolality is a variation of molality that takes into account only solutes that contribute to a solution's
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 ...
. It is measured in
osmole Osmotic concentration, formerly known as osmolarity, is the measure of solute concentration, defined as the number of osmoles (Osm) of solute per litre (L) of solution (osmol/L or Osm/L). The osmolarity of a solution is usually expressed as Osm/L ...
s of the solute per
kilogram The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially ...
of water. This unit is frequently used in
medical laboratory A medical laboratory or clinical laboratory is a laboratory where tests are conducted out on clinical specimens to obtain information about the health of a patient to aid in diagnosis, treatment, and prevention of disease. Clinical Medical labor ...
results in place of
osmolarity Osmotic concentration, formerly known as osmolarity, is the measure of solute concentration, defined as the number of osmoles (Osm) of solute per litre (L) of solution (osmol/L or Osm/L). The osmolarity of a solution is usually expressed as Osm/L ...
, because it can be measured simply by depression of the freezing point of a solution, or
cryoscopy Freezing-point depression is a drop in the minimum temperature at which a substance freezes, caused when a smaller amount of another, non- volatile substance is added. Examples include adding salt into water (used in ice cream makers and for ...
(see also:
osmostat The osmostat is the regulatory center in the hypothalamus that controls the osmolality of the extracellular fluid. The area in the anterior region of the hypothalamus contains the osmoreceptors, cells that control osmolality via the secretion of a ...
and
colligative properties In chemistry, colligative properties are those properties of solutions that depend on the ratio of the number of solute particles to the number of solvent particles in a solution, and not on the nature of the chemical species present. The number ...
).


Relation to apparent (molar) properties

Molality appears in the expression of the apparent (molar) volume of a solute as a function of the molality ''b'' of that solute (and density of the solution and solvent): :^\phi\tilde_1 = \frac = \left(\frac - \frac\right) \frac = \left(\frac - \frac\right) \frac = \left(\frac - \frac\right)\frac + \frac :^\phi\tilde_1 = \frac\left( \frac - \frac\right) + \frac For multicomponent systems the relation is slightly modified by the sum of molalities of solutes. Also a total molality and a mean apparent molar volume can be defined for the solutes together and also a mean
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 ...
of the solutes as if they were a single solute. In this case the first equality from above is modified with the mean molar mass M of the pseudosolute instead of the molar mass of the single solute: :^\phi\tilde_ = \frac\left( \frac - \frac\right) + \frac, M = \sum y_i M_i :y_i = \frac , yi,j being ratios involving molalities of solutes i,j and the total molality bT. The sum of products molalities - apparent molar volumes of solutes in their binary solutions equals the product between the sum of molalities of solutes and apparent molar volume in ternary or multicomponent solution. :^\phi\tilde_ (b_1 + b_2 + b_3 + \ldots) = b_ ^\phi\tilde_1 + b_ ^\phi\tilde_ + b_ ^\phi\tilde_ + \ldots,


Relation to apparent molar properties and activity coefficients

For concentrated ionic solutions the activity coefficient of the electrolyte is split into electric and statistical components. The statistical part includes molality b, hydration index number h, the number of ions from the dissociation and the ratio ra between the apparent molar volume of the electrolyte and the molar volume of water. Concentrated solution statistical part of the activity coefficient is: :\ln \gamma_s = \frac \ln \left(1 + \frac\right) - \frac \ln \left(1 - \frac\right) + \frac


Molalities of a ternary or multicomponent solution

The molalities of solutes b1, b2 in a ternary solution obtained by mixing two binary aqueous solutions with different solutes (say a sugar and a salt or two different salts) are different than the initial molalities of the solutes bii in their binary solutions. b_=\frac = \frac b_=\frac = \frac b_=\frac = \frac b_=\frac = \frac The content of solvent in mass fractions w01 and w02 from each solution of masses ms1 and ms2 to be mixed as a function of initial molalities is calculated. Then the amount (mol) of solute from each binary solution is divided by the sum of masses of water after mixing: b_1 = \frac\frac = \frac\frac = \frac \frac b_2 = \frac\frac = \frac\frac = \frac \frac Mass fractions of each solute in the initial solutions w11 and w22 are expressed as a function of the initial molalities b11, b22 : w_=\frac w_=\frac These expressions of mass fractions are substituted in the final molalitaties. b_1 = \frac \frac b_2 = \frac \frac The results for a ternary solution can be extended to a multicomponent solution (with more than two solutes).


From the molalities of the binary solutions

The molalities of the solutes in a ternary solution can be expressed also from molalities in the binary solutions and their masses: b_=\frac = \frac b_=\frac = \frac The binary solution molalities are: b_=\frac = \frac b_=\frac = \frac The masses of the solutes determined from the molalities of the solutes and the masses of water can be substituted in the expressions of the masses of solutions: m_ = m_ + m_ = m_ (1 + b_ M_1) Similarly for the mass of the second solution: m_ = m_ + m_ = m_ (1 + b_ M_2) From here one can obtain the masses of water to be summed in the denominator of the molalitities of the solutes in the ternary solutions. m_ = \frac m_ = \frac Thus the ternary molalities are: b_ = \frac = \frac = \frac b_ = \frac = \frac = \frac


See also

*
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 ...


References

{{Chemical solutions Chemical properties es:Concentración#Molalidad