Perturbation theory

2014-02-26T13:16:56Z

202.78.172.162: /* Commentary */

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MIPS architecture

2014-01-30T08:57:17Z

202.78.136.254: Undid revision 593083371 by 98.190.133.83 (talk). Reason: It breaks the page formatting

In [[chemistry]], the '''[[Mole (unit)|mole]] fraction''' <math>x_i</math> is defined as the [[amount of substance|amount]] of a constituent <math>n_i</math> divided by the total amount of all constituents in a mixture <math>n_{tot}</math>:<ref name="goldbook">{{GoldBookRef | file = A00296 | title = amount fraction}}</ref>

:<math>x_i = \frac{n_i}{n_{tot}}</math>

The sum of all the mole fractions is equal to 1:

:<math>\sum_{i=1}^{N} n_i = n_{tot} ; \; \sum_{i=1}^{N} x_i = 1</math>

The mole fraction is also called the '''amount fraction'''.<ref name="goldbook"/> It is identical to the '''number fraction''', which is defined as the number of [[molecule]]s of a constituent <math>N_i</math> divided by the total number of all molecules <math>N_{tot}</math>. It is one way of expressing the composition of a mixture with a [[dimensionless quantity]] ([[Mass fraction (chemistry)|mass fraction]] is another). The mole fraction is sometimes denoted by the lowercase [[Greek alphabet|Greek]] letter ''<math alt="χ">\chi</math>'' (''[[Chi (letter)|chi]]'') instead of a [[Latin alphabet|Roman]] <math>x</math>.<ref>{{cite book|last=Zumdahl|first=Steven S.|title=Chemistry|year=2008|publisher=Cengage Learning|isbn=0-547-12532-1|edition=8th ed.|page=201}}</ref><ref>{{cite book|last=Rickard|first=James N. Spencer, George M. Bodner, Lyman H.|title=Chemistry : structure and dynamics.|year=2010|publisher=Wiley|location=Hoboken, N.J.|isbn=978-0-470-58711-9|edition=5th ed.|page=357}}</ref> For mixtures of gases, [[IUPAC]] recommends the letter <math>y</math>.<ref name="goldbook"/>

==Properties==
Mole fraction is used very frequently in the construction of [[phase diagram]]s. It has a number of advantages:

* it is not temperature dependent (such as [[molar concentration]]) and does not require knowledge of the densities of the phase(s) involved
* a mixture of known mole fraction can be prepared by weighing off the appropriate masses of the constituents
* the measure is ''symmetric'': in the mole fractions x=0.1 and x=0.9, the roles of 'solvent' and 'solute' are reversed.
* In a mixture of [[ideal gas]]es, the mole fraction can be expressed as the ratio of [[partial pressure]] to total [[pressure]] of the mixture.

==Related quantities==

===Mass fraction===

The [[mass fraction (chemistry)|mass fraction]] <math>w_i</math> can be calculated using the formula

:<math>w_i = x_i \cdot \frac {M_i}{M}</math>

where <math>M_i</math> is the molar mass of the component <math>i</math> and <math>M</math> is the average [[molar mass]] of the mixture.

Replacing the expression of the molar mass:

:<math>w_i = x_i \cdot \frac {M_i}{\sum_i x_i M_i}</math>

===Mole percentage===

Multiplying mole fraction by 100 gives the mole percentage, also referred as amount/amount percent (abbreviated as n/n%).

===Mass concentration===
The conversion to and from [[mass concentration (chemistry)|mass concentration]] <math>\rho_i</math> is given by:

: <math>x_i = \frac{\rho_i}{\rho} \cdot \frac{M}{M_i}</math>

where <math>M</math> is the average molar mass of the mixture.

: <math>\rho_i = x_i \rho \cdot \frac{M_i}{M}</math>

===Molar concentration===
The conversion to [[molar concentration]] <math>c_i</math> is given by:
:<math>c_i = \frac{{x_i \cdot \rho}}{{M}} = x_i c </math>

or
:<math>c_i = \frac{{x_i \cdot \rho}}{{\sum_i x_i M_i}} </math>

where <math>M</math> is the average molar mass of the solution, c total molar concentration and <math>\rho</math> is the [[density]] of the solution .

===Mass and molar mass===

The mole fraction can be calculated from the [[mass]]es <math>m_i</math> and [[molar mass]]es <math>M_i</math> of the components:

:<math> x_i= \frac{{\frac{{m_i}}{{M_i}}}}{{\sum_i \frac{{m_i}}{{M_i}}}}</math>

==Spatial variation and gradient==

In a [[inhomogeneous|spatially non-uniform]] mixture, the mole fraction [[gradient]] triggers the phenomenon of [[diffusion]].

==References==
{{Reflist}}

{{Chemical solutions}}


[[Category:Chemical properties]]
[[Category:Dimensionless numbers of chemistry]]

Monte Carlo localization

2012-08-09T14:50:24Z

202.78.175.199: /* Theory */ Spelling correction - looses to loses

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Perturbation theory

MIPS architecture

Monte Carlo localization

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