Particle number fraction

The Teilchenzahlanteil ( Symbol : X ) according to DIN 1310 , a physico-chemical quantity for quantitative description of the composition of mixtures / mixed phase , a so-called content size . It indicates the relative proportion of the number of particles in a particular mixture component in relation to the total number of particles in the mixture.

Definition and characteristics

The following table is in the size equations distinguish between

• the simple case of a binary mixture ( Z = 2, two-substance mixture of components i and j , for example the solution of a single substance i in a solvent j ) and
• the generally applicable formulation for a mixture of substances made up of a total of Z components (index z as a general index for the sums , includes i and possibly j ).

	binary mixture ( Z = 2)	general mixture ( Z components)
definition	${\displaystyle X_{i}={\frac {N_{i}}{N_{i}+N_{j}}}}$	${\displaystyle X_{i}={\frac {N_{i}}{N}}\ \ {\text{mit}}\ \ N=\sum _{z=1}^{Z}N_{z}}$
Range of values	${\displaystyle 0\leq X_{i}\leq 1}$
Sum criterion	${\displaystyle X_{i}+X_{j}=1\ \Rightarrow \ X_{j}=1-X_{i}}$	${\displaystyle \sum _{z=1}^{Z}X_{z}=1\ \Rightarrow \ X_{Z}=1-\sum _{z=1}^{Z-1}X_{z}}$

The Teilchenzahlanteil X _i is defined as the value of the quotient from the particle number N _i of the considered mixture component i and the total particle number N of the mixture. The latter is the sum of the particle numbers of all components ( including i ) of the mixture. “ Particles ” can be material elementary objects such as atoms , molecules , ions or formula units and must be specified for all mixture components.

As the quotient of two dimensionless quantities , the number of particles is itself also a dimensionless quantity and can be specified as a pure decimal number as in the table above or with auxiliary units such as percent (% = 1/100), per mil (‰ = 1 / 1,000) or parts per million (1 ppm = 1 / 1,000,000).

The particle number fraction X _{i of} a considered mixture component i can assume numerical values ​​between 0 = 0% (component i is not contained in the mixture) and 1 = 100% (component i is present as a pure substance ).

The particle number fractions of all components of a mixture add up to 1 = 100%. From this it follows that the knowledge or determination of the particle number fractions of Z  - 1 components is sufficient (in the case of a binary mixture, the particle number fraction of a component), since the particle number fraction of the remaining component can be calculated simply by calculating the difference to 1 = 100%.

The values ​​of the particle number fractions for a substance mixture of a given composition are - in contrast to the volume-related content quantities ( concentrations , volume fraction , volume ratio ) - independent of temperature and pressure , since the particle numbers of the mixture components, in contrast to the volumes, do not vary with temperature or pressure change, provided no material conversions occur.

Relationships with other salary levels

Because of the proportionality between the number of particles N and the amount of substance n (based on the same type of particle; the conversion factor is Avogadro's constant N _A  ≈ 6.022 · 10 ²³  mol ⁻¹ ) the value of the number of particles X _{i is} equal to the value of the amount of substance x _i :

${\displaystyle X_{i}={\frac {N_{i}}{N}}={\frac {n_{i}\cdot N_{\mathrm {A} }}{n\cdot N_{\mathrm {A} }}}={\frac {n_{i}}{n}}=x_{i}}$

The following table shows the relationships between the particle number fraction X _i and the other content values ​​defined in DIN 1310 in the form of size equations . M stands for the molar mass , ρ for the density of the respective pure substance (at the same pressure and temperature as in the substance mixture). The index z in turn serves as a general index for the sums and includes i .

Relationship between the number of particles X _i and other content quantities

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

Since the molar volume V _{m of} a pure substance is equal to the quotient of its molar mass M and density ρ (at a given temperature and a given pressure), the reciprocal terms in some equations in the table above can be replaced accordingly:

${\displaystyle {\frac {M_{i}}{\rho _{i}}}=V_{\mathrm {m} ,i}\ \Leftrightarrow \ {\frac {\rho _{i}}{M_{i}}}={\frac {1}{V_{\mathrm {m} ,i}}}}$

In the case of mixtures of ideal gases, not only do the values ​​of the particle number fraction X _i and the molar fraction x _i match, but because of the uniform molar volumes and the ideal mixture character, there is also equality with the volume fraction φ _i and the volume concentration σ _i :

${\displaystyle X_{i}=x_{i}=\varphi _{i}=\sigma _{i}\ \ {\text{für Mischungen idealer Gase}}}$

Examples

Nitrogen and oxygen in 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). When viewed approximately as a mixture of ideal gases , the usually tabulated mean volume fractions of the individual gases in dry air at sea level (N ₂ : approx. 78.1%; O ₂ : approx. 20.9%) are to be equated with the mole proportions, which in turn equal the Particle number fractions are (see above):

${\displaystyle X_{\mathrm {N_{2}} }=x_{\mathrm {N_{2}} }\approx 0{,}781=78{,}1\ \%\qquad X_{\mathrm {O_{2}} }=x_{\mathrm {O_{2}} }\approx 0{,}209=20{,}9\ \%}$

Solution of glucose in water

A solution of glucose (Glc) in water (H ₂ O) with the mass fractions w _Glc = 0.01 = 1% and correspondingly w _{H 2 O} = 1 - w _Glc = 0.99 = 99% is considered. Taking into account the molar masses , the particle number fractions of glucose molecules and water molecules are:

${\displaystyle X_{\mathrm {Glc} }={\frac {w_{\mathrm {Glc} }/M_{\mathrm {Glc} }}{w_{\mathrm {Glc} }/M_{\mathrm {Glc} }+w_{\mathrm {H_{2}O} }/M_{\mathrm {H_{2}O} }}}=\mathrm {\frac {0{,}01/(180{,}16\ g\cdot mol^{-1})}{0{,}01/(180{,}16\ g\cdot mol^{-1})+0{,}99/(18{,}02\ g\cdot mol^{-1})}} \approx 0{,}001=0{,}1\ \%}$

${\displaystyle X_{\mathrm {H_{2}O} }=1-X_{\mathrm {Glc} }\approx 0{,}999=99{,}9\ \%}$

Individual evidence

1. ↑ ^{a b c} Standard DIN 1310 : Composition of mixed phases (gas mixtures, solutions, mixed crystals); Terms, symbols. February 1984.
2. ↑ ^{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. 34, 224, 225, 444 , 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).