# Mean molar mass

The mean molar mass ( symbol :) , also referred to as mean molar mass , mean mass related to the amount of substance or molar mass mean , is a physico-chemical quantity which transfers the term molar mass of pure substances to substance mixtures / mixed phases (e.g. solutions ). Here, the total mass is related to the total amount of substances in the substance mixture , so it is a molar quantity . ${\ displaystyle {\ overline {M}}}$

## Definition and characteristics

The molar mass (molar mass, substance- related mass) M i of a certain considered pure substance i is defined as the quotient of its mass m i and its substance amount n i :

${\ displaystyle M_ {i} = {\ frac {m_ {i}} {n_ {i}}}}$

Similarly, the mean molar mass of a mixture of substances results as the quotient of its total mass m and its total amount of substance n : ${\ displaystyle {\ overline {M}}}$

${\ displaystyle {\ overline {M}} = {\ frac {m} {n}}}$

The total mass m and the total amount n of the substance mixture are each the sums of the individual masses or individual substance amounts of all mixture components , formulated below for a general mixture of a total of Z components (index z as a general index for the totalization , includes component i ):

${\ displaystyle m = \ sum _ {z = 1} ^ {Z} m_ {z} \ qquad n = \ sum _ {z = 1} ^ {Z} n_ {z}}$

The individual components of the mixture must be specified with regard to their " particles " - on which the concept of the amount of substance is based - material elementary objects such as atoms , molecules , ions or formula units come into question. The value of the total amount of substance (and thus also the value of the target value mean molar mass) depends on this definition.

The derived SI unit of the mean molar mass is kg / mol , in practice the unit g / mol is also common.

The mean molar mass corresponds to the mean value of the molar masses of the individual mixture components weighted with the molar proportions x or the equally large particle number proportions X ( N stands for the number of particles ):

${\ displaystyle {\ overline {M}} = {\ frac {m} {n}} = {\ frac {\ sum _ {z = 1} ^ {Z} m_ {z}} {\ sum _ {z = 1} ^ {Z} n_ {z}}} = {\ frac {\ sum _ {z = 1} ^ {Z} (n_ {z} \ cdot M_ {z})} {\ sum _ {z = 1 } ^ {Z} n_ {z}}} = \ sum _ {z = 1} ^ {Z} (x_ {z} \ cdot M_ {z}) = \ sum _ {z = 1} ^ {Z} ( X_ {z} \ cdot M_ {z}) = {\ frac {\ sum _ {z = 1} ^ {Z} (N_ {z} \ cdot M_ {z})} {\ sum _ {z = 1} ^ {Z} N_ {z}}}}$

In the case of polymers , the mean molar mass is a parameter of the molar mass distribution , the molar masses of all chain lengths occurring being averaged. The mean molar mass corresponding to the above definition is also called number-weighted mean molar mass or number- average molar mass in order to distinguish it from differently weighted mean values.

## Relationships with salary sizes

The following table shows the relationships between the mean molar mass and the content quantities defined in DIN 1310 for mixtures of substances / mixed phases in the form of size equations . The formula symbols M and ρ provided with an index stand for the molar mass or density (at the same pressure and temperature as in the substance mixture) of the respective pure substance identified by the index . The symbol ρ without an index represents the density of the mixed phase. As above, the index z serves as a general index for the sums (consideration of a general mixture of substances from a total of Z components) and includes i . N A is Avogadro's constant ( N A ≈ 6.022 · 10 23  mol −1 ). ${\ displaystyle {\ overline {M}}}$

Relationship between the mean molar mass and the content${\ displaystyle {\ overline {M}}}$
Masses - ... Amount of substance - ... Particle number - ... Volume - ...
... - share Mass fraction w Amount of substance fraction x Particle number fraction X Volume fraction φ
${\ displaystyle {\ overline {M}} = {\ frac {1} {\ sum _ {z = 1} ^ {Z} (w_ {z} / M_ {z})}}}$ ${\ displaystyle {\ overline {M}} = \ sum _ {z = 1} ^ {Z} (x_ {z} \ cdot M_ {z})}$ ${\ displaystyle {\ overline {M}} = \ sum _ {z = 1} ^ {Z} (X_ {z} \ cdot M_ {z})}$ ${\ displaystyle {\ overline {M}} = {\ frac {\ sum _ {z = 1} ^ {Z} (\ varphi _ {z} \ cdot \ rho _ {z})} {\ sum _ {z = 1} ^ {Z} (\ varphi _ {z} \ cdot \ rho _ {z} / M_ {z})}}}$
… - concentration Mass concentration β Molar concentration c Particle number concentration C Volume concentration σ
${\ displaystyle {\ overline {M}} = {\ frac {\ rho} {\ sum _ {z = 1} ^ {Z} (\ beta _ {z} / M_ {z})}}}$ ${\ displaystyle {\ overline {M}} = {\ frac {\ rho} {\ sum _ {z = 1} ^ {Z} c_ {z}}}}$ ${\ displaystyle {\ overline {M}} = {\ frac {N _ {\ mathrm {A}} \ cdot \ rho} {\ sum _ {z = 1} ^ {Z} C_ {z}}}}$ ${\ displaystyle {\ overline {M}} = {\ frac {\ rho} {\ sum _ {z = 1} ^ {Z} (\ sigma _ {z} \ cdot \ rho _ {z} / M_ {z })}}}$
... - ratio Mass ratio ζ Molar ratio r Particle number ratio R Volume ratio ψ
${\ displaystyle {\ overline {M}} = {\ frac {\ sum _ {z = 1} ^ {Z} \ zeta _ {zi}} {\ sum _ {z = 1} ^ {Z} (\ zeta _ {zi} / M_ {z})}}}$ ${\ displaystyle {\ overline {M}} = {\ frac {\ sum _ {z = 1} ^ {Z} (r_ {zi} \ cdot M_ {z})} {\ sum _ {z = 1} ^ {Z} r_ {zi}}}}$ ${\ displaystyle {\ overline {M}} = {\ frac {\ sum _ {z = 1} ^ {Z} (R_ {zi} \ cdot M_ {z})} {\ sum _ {z = 1} ^ {Z} R_ {zi}}}}$ ${\ displaystyle {\ overline {M}} = {\ frac {\ sum _ {z = 1} ^ {Z} (\ psi _ {zi} \ cdot \ rho _ {z})} {\ sum _ {z = 1} ^ {Z} (\ psi _ {zi} \ cdot \ rho _ {z} / M_ {z})}}}$
Quotient
amount of substance / mass
Molality b
${\ displaystyle {\ overline {M}} = {\ frac {x_ {i}} {b_ {i} \ cdot w_ {j}}}}$ ( i = solute, j = solvent)
specific amount of partial substances q
${\ displaystyle {\ overline {M}} = {\ frac {1} {\ sum _ {z = 1} ^ {Z} q_ {z}}}}$

## Examples

### air

Air as the gas mixture of the earth's atmosphere contains the two main components nitrogen (particles: N 2 molecules) and oxygen (particles: O 2 molecules), fluctuating amounts of water vapor (particles: H 2 O molecules) and, in addition, above all argon (particles: Ar atoms) and carbon dioxide (particles: CO 2 molecules). When viewed approximately as a mixture of ideal gases , the usually tabulated mean volume fractions of the individual gases in dry air - i.e. without the variable water vapor fraction - are at sea level (N 2 : approx. 78.08%; O 2 : approx. 20.94%; Ar : approx. 0.93%; CO 2 : approx. 0.04%) to be equated with the molar proportions x (equivalent: particle number proportions X ). With the molar masses M of N 2 , O 2 , Ar and CO 2 , the mean molar mass of dry air can be calculated (further trace components of the air such as neon can be approximately neglected):

{\ displaystyle {\ begin {aligned} {\ overline {M}} _ {\ text {dry air}} & \ approx x _ {\ mathrm {N_ {2}}} \ cdot M _ {\ mathrm {N_ {2} }} + x _ {\ mathrm {O_ {2}}} \ cdot M _ {\ mathrm {O_ {2}}} + x _ {\ mathrm {Ar}} \ cdot M _ {\ mathrm {Ar}} + x _ {\ mathrm {CO_ {2}}} \ cdot M _ {\ mathrm {CO_ {2}}} \\ & \ approx \ mathrm {0 {,} 7808 \ cdot 28 {,} 01 \ g \ cdot mol ^ {- 1 } +0 {,} 2094 \ times 32 {,} 00 \ g \ times mol ^ {- 1} +0 {,} 0093 \ times 39 {,} 95 \ g \ times mol ^ {- 1} +0 { ,} 0004 \ cdot 44 {,} 01 \ g \ cdot mol ^ {- 1}} \\ & \ approx \ mathrm {28 {,} 96 \ g \ cdot mol ^ {- 1}} \ end {aligned} }}

In reality the air is not completely dry; Due to the water vapor as an additional mixture component in the mixture of substances, the mean molar mass is somewhat lower - because of the correspondingly lower molar proportions of the gases considered above and the comparatively lower molar mass of H 2 O (18.02 g mol −1 ) .

### Mixing elements

Mixed elements are chemical elements that, in contrast to pure elements , occur in nature as a mixture of several isotopes . It is customary for them to indicate mean molar masses (or mean relative atomic masses with the same numerical value ) and to use these for calculations. Take magnesium Mg as an example , which is found in the earth's envelope as a mixture of the isotopes Mg-24, Mg-25 and Mg-26 with the molar proportions x (equivalent: particle number proportions X ) 78.99%, 10.00% and 11.01% occurs. The mean molar mass of the natural magnesium isotope mixture can be calculated from the molar proportions and the molar masses M of the individual isotopes:

{\ displaystyle {\ begin {aligned} {\ overline {M}} _ {\ text {Mg}} & = x _ {\ text {Mg-24}} \ cdot M _ {\ text {Mg-24}} + x_ {\ text {Mg-25}} \ cdot M _ {\ text {Mg-25}} + x _ {\ text {Mg-26}} \ cdot M _ {\ text {Mg-26}} \\ & = \ mathrm {0 {,} 7899 \ times 23 {,} 9850 \ g \ times mol ^ {- 1} +0 {,} 1000 \ times 24 {,} 9858 \ g \ times mol ^ {- 1} +0 {, } 1101 \ cdot 25 {,} 9826 \ g \ cdot mol ^ {- 1}} \\ & = \ mathrm {24 {,} 305 \ g \ cdot mol ^ {- 1}} \ end {aligned}}}

Since a closer look reveals that the isotopic compositions of mixing elements can be slightly different depending on the origin of the material , value intervals for the mole fractions of the individual isotopes and the resulting value intervals for the mean molar mass (or mean relative atomic mass) of the mixing element have recently been introduced given, for magnesium for example:

{\ displaystyle {\ begin {aligned} x _ {\ text {Mg-24}} & = [0 {,} 7888; \ 0 {,} 7905] \ qquad x _ {\ text {Mg-25}} = [0 {,} 09988; \ 0 {,} 10034] \ qquad x _ {\ text {Mg-26}} = [0 {,} 1096; \ 0 {,} 1109] \\ {\ overline {M}} _ { \ mathrm {Mg}} & = \ mathrm {[24 {,} 304; \ 24 {,} 307] \ g \ cdot mol ^ {- 1}} \ end {aligned}}}

If a chemical compound contains one or more mixed elements, strictly speaking, only an average molar mass (or a value range of the same) can be specified for this chemical compound, even if this is often not specifically mentioned or identified in practice (for the preceding air -Example, all air components are composed exclusively of mixing elements, even the molar masses of the individual substances should therefore actually be referred to as mean molar masses). Exceptions to this are special cases in which specifically produced isotopically pure versions of the elements were used for the synthesis of the chemical compound.

## Individual evidence

1. Norm DIN 1345: Thermodynamics: basic terms. December 1993. Section 7: Mass-related, substance-quantity-related, volume-related and partial quantities.
2. ^ A b PW Atkins, J. de Paula: Physikalische Chemie . 4th edition. Wiley-VCH, Weinheim 2006, ISBN 3-527-31546-2 , pp. 724 ff . ( limited preview in Google Book Search [accessed September 30, 2015]).
3. a b c P. Kurzweil: The Vieweg unit lexicon: terms, formulas and constants from natural sciences, technology and medicine . 2nd Edition. Springer Vieweg, 2013, ISBN 978-3-322-83212-2 , p. 251 f ., doi : 10.1007 / 978-3-322-83211-5 ( limited preview in the Google book search - softcover reprint of the 2nd edition 2000). Lexical part (PDF; 71.3 MB) accessed on September 30, 2016.
4. ^ A b MD Lechner, K. Gehrke, EH Nordmeier: Macromolecular Chemistry: A textbook for chemists, physicists, materials scientists and process engineers . 5th edition. Springer, Berlin / Heidelberg 2014, ISBN 978-3-642-41768-9 , pp. 15th ff ., doi : 10.1007 / 978-3-642-41769-6 ( limited preview in Google Book Search [accessed September 30, 2015]).
5. a b ER Cohen, T. Cvitas, JG Frey, B. Holmström, K. Kuchitsu, R. Marquardt, I. Mills, F. Pavese, M. Quack, J. Stohner, HL Strauss, M. Takami, AJ Thor : Quantities, Units and Symbols in Physical Chemistry (=  IUPAC Green Book ). 3. Edition. IUPAC & RSC Publishing, Cambridge 2007, ISBN 978-0-85404-433-7 , pp. 47 ( limited preview in Google Book Search [accessed September 30, 2015] second corrected print 2008). Website ( Memento of the original from June 24, 2015 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. ; iupac.org ( Memento of the original dated February 11, 2014 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 2.5 MB).
6. Magnesium. IUPAC Inorganic Chemistry Division - Commission on Isotopic Abundances and Atomic Weights [CIAAW], accessed on September 30, 2015 (English, variability of the isotopic composition and the mean relative atomic mass of magnesium).