# Molality

**Molality**, also called **molal concentration**, is a measure of the concentration of a solute in a solution in terms of amount of substance in a specified amount of mass of the solvent. This contrasts with the definition of molarity which is based on a specified volume of solution.

A commonly used unit for molality in chemistry is mol/kg. A solution of concentration 1 mol/kg is also sometimes denoted as *1 molal*.

## Definition[edit]

The molality (*b*), of a solution is defined as the amount of substance (in moles) of solute, *n*_{solute}, divided by the mass (in kg) of the solvent, *m*_{solvent}:^{[1]}

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.^{[2]}

## Origin[edit]

The term *molality* is formed in analogy to *molarity* which is the molar concentration of a solution. The earliest known use of the intensive property 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.*^{[3]} Though the two terms are subject to being confused with one another, the molality and molarity of a dilute aqueous solution 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[edit]

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" or "3 m". However, following the SI system of units, the National Institute of Standards and Technology, the United States authority on measurement, considers the term "molal" and the unit symbol "m" to be obsolete, and suggests mol/kg or a related unit of the SI.^{[4]} This recommendation has not been universally implemented in academia yet.

## Usage considerations[edit]

- 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 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 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, 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 properties[edit]

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):

### Mass fraction[edit]

The conversions to and from the mass fraction, *w*, of the solute in a single-solute solution are

where *b* is the molality and *M* is the molar mass of the solute.

More generally, for an *n*-solute/one-solvent solution, letting *b _{i}* and

*w*be, respectively, the molality and mass fraction of the

_{i}*i*-th solute,

where *M _{i}* 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,

### Mole fraction[edit]

The conversions to and from the mole fraction, *x*, of the solute in a single-solute solution are

where *M*_{0} is the molar mass of the solvent.

More generally, for an *n*-solute/one-solvent solution, letting *x _{i}* be the mole fraction of the

*i*th solute,

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:

### Molar concentration (molarity)[edit]

The conversions to and from the molar concentration, *c*, for one-solute solutions are

where *ρ* is the mass density of the solution, *b* is the molality, and *M* is the molar mass (in kg/mol) of the solute.

For solutions with *n* solutes, the conversions are

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 molarities:

### Mass concentration[edit]

The conversions to and from the mass concentration, *ρ*_{solute}, of a single-solute solution are

where *ρ* is the mass density of the solution, *b* is the molality, and *M* is the molar mass of the solute.

For the general *n*-solute solution, the mass concentration of the *i*th solute, *ρ _{i}*, is related to its molality,

*b*, as follows:

_{i}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 mass concentrations:

### Equal ratios[edit]

Alternatively, we 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:

where *i* and *j* are subscripts representing *all* the constituents, the *n* solutes plus the solvent.

### Example of conversion[edit]

An acid mixture consists of 0.76/0.04/0.20 mass fractions of 70% HNO_{3} / 49% HF / H_{2}O, where the percentages refer to mass fractions of the bottled acids carrying a balance of H_{2}O. The first step is determining the mass fractions of the constituents:

The approximate molar masses in kg/mol are

First derive the molality of the solvent, in mol/kg,

and use that to derive all the others by use of the equal ratios:

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:

The mole fractions may be derived from this result:

### Osmolality[edit]

Osmolality is a variation of molality that takes into account only solutes that contribute to a solution's osmotic pressure. It is measured in osmoles of the solute per kilogram of water. This unit is frequently used in medical laboratory results in place of osmolarity, because it can be measured simply by depression of the freezing point of a solution, or cryoscopy (see also: osmostat and colligative properties).

## Relation to apparent (molar) properties[edit]

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):

^{[clarification needed]}

For multicomponent systems the relation is slightly modified by the sum of molalities of solutes.

For multicomponent systems with more than one solute a total molality an mean apparent molar volume can be defined for the solutes together and also a mean molar mass 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:

- ,

- , y
_{i,j}being ratios involving molalities of solutes i,j and the total molality b_{T}.

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 of multicomponent solution.^{[5]}.

- ,

### Relation to apparent molar properties and activity coefficients[edit]

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 r_{a} between the apparent molar volume of the electrolyte and the molar volume of water.

Concentrated solution statistical part of the activity coefficient is:

^{[6]},^{[7]}^{[8]}

## References[edit]

**^**IUPAC,*Compendium of Chemical Terminology*, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "molality". doi:10.1351/goldbook.M03970**^**Sangster, James; Teng, Tjoon-Tow; Lenzi, Fabio (1976). "Molal volumes of sucrose in aqueous solutions of NaCl, KCl, or urea at 25°C".*Journal of Solution Chemistry*.**5**(8): 575–585. doi:10.1007/BF00647379.**^***www.OED.com*. Oxford University Press. 2011.**^**"NIST Guide to SI Units". sec. 8.6.8. Retrieved 2007-12-17.**^**Harned Owen, Physical Chemistry of Electrolytic Solutions, third edition 1958, p. 398-399**^**Glueckauf, E. (1955). "The influence of ionic hydration on activity coefficients in concentrated electrolyte solutions".*Transactions of the Faraday Society*.**51**: 1235. doi:10.1039/TF9555101235.**^**Glueckauf, E. (1957). "The influence of ionic hydration on activity coefficients in concentrated electrolyte solutions".*Transactions of the Faraday Society*.**53**: 305. doi:10.1039/TF9575300305.**^**Kortüm, G. (1960). "The Structure of Electrolytic Solutions, herausgeg. von W. J. Hamer. John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London 1959. 1. Aufl., XII, 441 S., geb. $ 18.50".*Angewandte Chemie*.**72**(24): 97. doi:10.1002/ange.19600722427. ISSN 0044-8249.

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