Polynomial arithmetic: Difference between revisions

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In [[chemistry]], the '''mass fraction''' <math>w_i</math> is the ratio of one substance with mass <math>m_i</math> to the mass of the total mixture <math>m_{tot}</math>, defined as<ref>{{GoldBookRef | file = M03722 | title = mass fraction}}</ref>
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:<math>w_i = \frac {m_i}{m_{tot}}</math>
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The sum of all the mass fractions is equal to 1:
 
:<math>\sum_{i=1}^{N} m_i = m_{tot} ; \sum_{i=1}^{N} w_i = 1</math>
 
Mass fraction is one way of expressing the composition of a mixture in a dimensionless size ([[mole fraction]] is another).
 
For [[elemental analysis]], mass fraction (or "mass percent composition") can also refer to the ratio of the mass of one element to the total mass of a compound. It can be calculated for any compound using its [[empirical formula]].<ref>[http://khanexercises.appspot.com/video?v=xatVrAh2U0E Formula from Mass Composition]</ref> or its [[chemical formula]]<ref>[http://chemistry.about.com/od/workedchemistryproblems/a/mass-percent-worked-problem.htm]</ref>
 
==Terminology==
"Percent concentration" does not refer to this quantity. This improper name persists, especially in elementary textbooks. In biology, the unit "%" is sometimes (incorrectly) used to denote mass concentration, also called "mass/volume percentage." A solution with 1 g of solute dissolved in a final volume of 100 mL of solution would be labeled as "1 %" or "1 % m/v" (mass/volume). The notation is mathematically flawed because the unit "%" can only be used for dimensionless quantities. "Percent solution" or "percentage solution" are thus terms best reserved for "mass percent solutions" (m/m = m% = mass solute/mass total solution after mixing), or "volume percent solutions" (v/v = v% = volume solute per volume of total solution after mixing). The very ambiguous terms "percent solution" and "percentage solutions" with no other qualifiers continue to occasionally be encountered.
 
In [[thermal engineering]] '''[[vapor quality]]''' is used for the mass fraction of vapor in the steam.
 
In alloys, especially those of noble metals, the term '''[[fineness]]''' is used for the mass fraction of the noble metal in the alloy.
 
==Properties==
The mass fraction is independent of temperature.
 
==Related quantities==
===Mass concentration===
The mass fraction of a component in a solution is the ratio of the [[mass concentration (chemistry)|mass concentration]] of that component <math>\rho_i</math> (density of that component in the mixture) to the [[density]] of solution <math>\rho</math>.
 
:<math>w_i = \frac {\rho_i}{\rho}</math>
 
===Molar concentration===
The relation to molar concentration is like that from above substituting the relation between mass and molar concentration.
 
:<math>w_i = \frac {\rho_i}{\rho}=\frac {c_i M_i}{\rho}</math>
 
===Mass percentage===
 
Multiplying mass fraction by 100 gives the mass percentage. It is sometimes called weight percent (wt%) or weight-weight percentage. However, since
mass and weight are different quantities, this is incorrect (see [[mass versus weight]] for details).
 
===Mole fraction===
The [[mole fraction]] <math>x_i</math> can be calculated using the formula
 
:<math>x_i = w_i \cdot \frac {M}{M_i}</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 produces:
 
: <math>x_i = \frac {\frac{w_i}{M_i}}{\sum_i \frac{w_i}{M_i}}</math>
 
==Spatial variation and gradient==
 
In a [[inhomogeneous|spatially non-uniform]] mixture, the mass fraction [[gradient]] triggers the phenomenon of [[diffusion]].
 
==References==
{{reflist}}
 
{{Chemical solutions}}
 
<!--Categories-->
[[Category:Chemical properties]]
[[Category:Dimensionless numbers of chemistry]]

Latest revision as of 09:25, 18 May 2014

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