View source for Molality

{{Distinguish|Molarity}}
'''Molality''', also called '''molal concentration''', is a measure of the [[concentration]] of a [[Wiktionary:solute|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 [[Mole (unit)|mol]]/[[kilogram|kg]]. A solution of concentration 1&nbsp;mol/kg is also sometimes denoted as ''1 molal''.

==Definition==
The molality (''b''), of a [[solution]] is defined as the [[amount of substance]] (in [[mole (unit)|mole]]s) of solute, ''n''<sub>solute</sub>, divided by the [[mass]] (in [[kg]]) of the [[solvent]], ''m''<sub>solvent</sub>:<ref>{{GoldBookRef | file = M03970 | title = molality}}</ref>

:<math>b = \frac{n_\mathrm{solute}}{m_\mathrm{solvent}}</math>

In the cases 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.<ref>Journal of Solution Chemistry 5 (1976), 575</ref>

== Origin ==
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 [[Merle Randall|M. Randall]] in the 1923 publication of ''Thermodynamics and the Free Energies of Chemical Substances.''<ref name=OED>{{cite book|title=www.OED.com|year=2011|publisher=Oxford University Press}}</ref> Though the two terms are subject to being confused with one another, the molality and molarity of a weak [[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 ==
The [[SI]] unit for molality is moles per kilogram of solvent.

A solution with a molality of 3&nbsp;mol/kg is often described as "3 molal" or "3&nbsp;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.<ref>{{cite web|url=http://physics.nist.gov/Pubs/SP811/sec11.html |title=NIST Guide to SI Units |accessdate=2007-12-17|at=sec. 8.6.8 }}</ref>  This recommendation has not been universally implemented in academia yet.

== 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]] or [[mass concentration (chemistry)|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 ==
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''<sub>0</sub>, is found to be nothing more than the reciprocal of its molar mass, ''M''<sub>0</sub> (expressed as kg/mol):
:<math>b_0=\frac{n_0}{n_0 M_0}=\frac{1}{M_0}.</math>

=== Mass fraction===
The conversions to and from the [[mass fraction (chemistry)|mass fraction]], ''w'', of the solute in a single-solute solution are
:<math>w=\frac{1}{1+\dfrac{1}{b M}},\quad b=\frac{w}{(1-w)M} ,</math>

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<sub>i</sub>'' and ''w<sub>i</sub>'' be, respectively, the molality and mass fraction of the ''i''-th solute,
:<math>w_i=w_0 b_i M_i,\quad b_i=\frac{w_i}{w_0 M_i},</math>

where ''M<sub>i</sub>'' is the molar mass of the ''i''th solute, and ''w''<sub>0</sub> 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,
:<math>w_0=\frac{1}{1+\displaystyle\sum_{j=1}^{n}{b_j M_j}}=1-\sum_{j=1}^{n}{w_j}.</math>

=== Mole fraction ===
The conversions to and from the [[mole fraction]], ''x'', of the solute in a single-solute solution are
:<math>x=\frac{1}{1+\dfrac{1}{M_0 b}},\quad b=\frac{x}{M_0(1-x)},</math>
where ''M''<sub>0</sub> is the molar mass of the solvent.

More generally, for an ''n''-solute/one-solvent solution, letting ''x<sub>i</sub>'' be the mole fraction of the ''i''th solute,
:<math>x_i=x_0 M_0 b_i,\quad b_i=\frac{b_0 x_i}{x_0},</math>

where ''x''<sub>0</sub> 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:
:<math>x_0=\frac{1}{1+M_0\displaystyle\sum_{j=1}^{n}{b_j}}=1-\sum_{j=1}^{n}{x_j}.</math>

=== Molar concentration (molarity)===
The conversions to and from the [[molar concentration]], ''c'', for one-solute solutions are
:<math>c = \frac{\rho b}{1+ b M},\quad b=\frac{c}{\rho-cM},</math>

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
:<math>c_i =c_0 M_0 b_i,\quad b_i=\frac{b_0 c_i}{c_0},</math>

where the molar concentration of the solvent ''c''<sub>0</sub> is expressible both as a function of the molalities as well as a function of the molarities:
:<math>c_0=\frac{\rho b_0}{1+\displaystyle\sum_{j=1}^{n}{b_j M_j}}=\frac{\rho-\displaystyle\sum_{j=1}^{n}{c_i M_i}}{M_0}.</math>

=== Mass concentration ===
The conversions to and from the [[mass concentration (chemistry)|mass concentration]], ''ρ''<sub>solute</sub>, of a single-solute solution are
:<math>\rho_\mathrm{solute} = \frac{\rho b M}{1+b M},\quad b=\frac{\rho_\mathrm{solute}}{M \left(\rho-\rho_\mathrm{solute}\right)},</math>

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<sub>i</sub>'', is related to its molality, ''b<sub>i</sub>'', as follows:
:<math>\rho_i = \rho_0 b_i M_i,\quad b_i=\frac{\rho_i}{\rho_0 M_i},</math>

where the mass concentration of the solvent, ''ρ''<sub>0</sub>, is expressible both as a function of the molalities as well as a function of the mass concentrations:
:<math>\rho_0=\frac{\rho}{1+\displaystyle\sum_{j=1}^n b_j M_j}=\rho-\sum_{j=1}^{n}{\rho_i}.</math>

=== Equal ratios ===
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:
:<math>\frac{b_i}{b_j}=\frac{x_i}{x_j}=\frac{c_i}{c_j}=\frac{\rho_i M_j}{\rho_j M_i}=\frac{w_i M_j}{w_j M_i},</math>

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

=== Example of conversion ===
An acid mixture consists of 0.76/0.04/0.20 mass fractions of 70% HNO<sub>3</sub> / 49% HF / H<sub>2</sub>O, where the percentages refer to mass fractions of the bottled acids carrying a balance of H<sub>2</sub>O. The first step is determining the mass fractions of the constituents:
:<math>\begin{align}
w_\mathrm{HNO_3}&=0.70\times 0.76=0.532\\
w_\mathrm{HF}&=0.49\times 0.04=0.0196\\
w_\mathrm{H_2O}&=1-w_\mathrm{HNO_3}-w_\mathrm{HF}=0.448\\
\end{align}</math>

The approximate molar masses in kg/mol are
:<math>M_\mathrm{HNO_3}=0.063\ \mathrm{kg/mol},\quad M_\mathrm{HF}=0.020\ \mathrm{kg/mol},\ M_\mathrm{H_2O}=0.018\ \mathrm{kg/mol}.</math>
First derive the molality of the solvent, in mol/kg,
:<math>b_\mathrm{H_2O}=\frac{1}{M_\mathrm{H_2O}}=\frac{1}{0.018}\ \mathrm{mol/kg},</math>
and use that to derive all the others by use of the equal ratios:
:<math>\frac{b_\mathrm{HNO_3}}{b_\mathrm{H_2O}}=\frac{w_\mathrm{HNO_3}M_\mathrm{H_2O}}{w_\mathrm{H_2O}M_\mathrm{HNO_3}}
\quad \therefore b_\mathrm{HNO_3}=18.83\ \mathrm{mol/kg}.</math>
Actually, ''b''<sub>H<sub>2</sub>O</sub> 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:
:<math>b_\mathrm{HF}=\frac{w_\mathrm{HF}}{w_\mathrm{H_2O}M_\mathrm{HF}}=2.19\ \mathrm{mol/kg}.</math>
The mole fractions may be derived from this result:
:<math>x_\mathrm{H_2O}=\frac{1}{1+M_\mathrm{H_2O}\left(b_\mathrm{HNO_3}+b_\mathrm{HF}\right)}=0.726,</math>
:<math>\frac{x_\mathrm{HNO_3}}{x_\mathrm{H_2O}}=\frac{b_\mathrm{HNO_3}}{b_\mathrm{H_2O}}\quad \therefore x_\mathrm{HNO_3}=0.246,</math>
:<math>x_\mathrm{HF}=1-x_\mathrm{HNO_3}-x_\mathrm{H_2O}=0.029.</math>

=== Osmolality ===
Osmolality is a variation of molality that takes into account only solutes that contribute to a solution's [[osmotic pressure]]. It is measured in [[osmole]]s 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==
Molality appears in the expression of the [[apparent molar property|apparent (molar) volume]] of a solute as a function of the molality ''b'' of that solute (and density of the solution and solvent):

:<math> {}^\phi\tilde{V}_1 = \frac{1}{b}\left( \frac{1}{\rho} - \frac{1}{\rho_0^0}\right) + \frac{M_1}{\rho}</math>{{clarification|date=August 2018}}

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

:<math> {}^\phi\tilde{V}_i = \frac{1}{\sum b_j}\left( \frac{1}{\rho} - \frac{1}{\rho_0^0}\right) + \frac{\sum b_j M_j}{\sum b_j \rho}</math>

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:

:<math> {}^\phi\tilde{V}_{12..} = \frac{1}{b_T}\left( \frac{1}{\rho} - \frac{1}{\rho_0^0}\right) + \frac{M}{\rho}</math>, <math> M = \sum y_i M_i</math>

:<math> y_i = \frac {b_i}{b_T}</math>, y<sub>i,j</sub> being ratios involving molalities of solutes i,j and the total molality b<sub>T</sub>.

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.<ref>Harned Owen, Physical Chemistry of Electrolytic Solutions, third edition 1958, p. 398-399</ref>.

:<math> {}^\phi\tilde{V}_{123..} (b_1 + b_2 + b_3 + ...) = b_1 {}^\phi\tilde{V}_{1} + b_2 {}^\phi\tilde{V}_{2} + b_3 {}^\phi\tilde{V}_{3}+...</math>,

=== 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, [[solvation|hydration index number h]], the number of ions from the dissociation and the ratio r<sub>a</sub> between the [[apparent molar property|apparent molar volume]] of the electrolyte and the molar volume of water.

Concentrated solution statistical part of the activity coefficient is:
:<math>\ln \gamma_s = \frac{h- \nu}{\nu} \ln \left(1 + \frac{br_a}{55.5}\right) - \frac{h}{\nu} \ln \left(1 - \frac{br_a}{55.5}\right) + \frac{br_a\left(r_a + h -\nu\right)}{55.5 \left(1 + \frac{br_a}{55.5}\right)}</math><ref>{{Cite journal |doi = 10.1039/TF9555101235|title = The influence of ionic hydration on activity coefficients in concentrated electrolyte solutions|journal = Transactions of the Faraday Society|volume = 51|pages = 1235|year = 1955|last1 = Glueckauf|first1 = E.}}</ref>,<ref>{{Cite journal |doi = 10.1039/TF9575300305|title = The influence of ionic hydration on activity coefficients in concentrated electrolyte solutions|journal = Transactions of the Faraday Society|volume = 53|pages = 305|year = 1957|last1 = Glueckauf|first1 = E.}}</ref><ref name="Kortüm1960">{{cite journal|last1=Kortüm|first1=G.|title=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|journal=[[Angewandte Chemie]]|volume=72|issue=24|year=1960|page=97|issn=0044-8249|doi=10.1002/ange.19600722427}}</ref>

== References ==
{{Reflist}}

{{Wiktionary}}
{{Mole concepts}}
{{Chemical solutions}}

[[Category:Chemical properties]]

[[es:Concentración#Molalidad]]