Mole fraction




The proportion of a constituent to the total amount of all constituents in a mixture, expressed in mol/mol

In chemistry, the mole fraction or molar fraction (xi) is defined as the amount of a constituent (expressed in moles), ni, divided by the total amount of all constituents in a mixture (also expressed in moles), ntot:[1]


xi=nintot{displaystyle x_{i}={frac {n_{i}}{n_{mathrm {tot} }}}}{displaystyle x_{i}={frac {n_{i}}{n_{mathrm {tot} }}}}

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


∑i=1Nni=ntot;∑i=1Nxi=1{displaystyle sum _{i=1}^{N}n_{i}=n_{mathrm {tot} };;sum _{i=1}^{N}x_{i}=1}{displaystyle sum _{i=1}^{N}n_{i}=n_{mathrm {tot} };;sum _{i=1}^{N}x_{i}=1}

The same concept expressed with a denominator of 100 is the mole percent or molar percentage or molar proportion (mol%).


The mole fraction is also called the amount fraction.[1] It is identical to the number fraction, which is defined as the number of molecules of a constituent Ni divided by the total number of all molecules Ntot. The mole fraction is sometimes denoted by the lowercase Greek letter χ (chi) instead of a Roman x.[2][3] For mixtures of gases, IUPAC recommends the letter y.[1]


The National Institute of Standards and Technology of the United States prefers the term amount-of-substance fraction over mole fraction because it does not contain the name of the unit mole.[4]


Whereas mole fraction is a ratio of moles to moles, molar concentration is a quotient of moles to volume.


The mole fraction is one way of expressing the composition of a mixture with a dimensionless quantity; mass fraction (percentage by weight, wt%) and volume fraction (percentage by volume, vol%) are others.




Contents






  • 1 Properties


  • 2 Related quantities


    • 2.1 Mass fraction


    • 2.2 Molar mixing ratio


      • 2.2.1 Mixing binary mixtures with a common component to form ternary mixtures




    • 2.3 Mole percentage


    • 2.4 Mass concentration


    • 2.5 Molar concentration


    • 2.6 Mass and molar mass




  • 3 Spatial variation and gradient


  • 4 References





Properties


Mole fraction is used very frequently in the construction of phase diagrams. 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 gases, the mole fraction can be expressed as the ratio of partial pressure to total pressure of the mixture

  • In a ternary mixture one can express mole fractions of a component as functions of other components mole fraction and binary mole ratios:


x1=1−x21+x3x1{displaystyle x_{1}={frac {1-x_{2}}{1+{frac {x_{3}}{x_{1}}}}}}{displaystyle x_{1}={frac {1-x_{2}}{1+{frac {x_{3}}{x_{1}}}}}}


x3=1−x21+x1x3{displaystyle x_{3}={frac {1-x_{2}}{1+{frac {x_{1}}{x_{3}}}}}}{displaystyle x_{3}={frac {1-x_{2}}{1+{frac {x_{1}}{x_{3}}}}}}


Differential quotients can be formed at constant ratios like those above:


(∂x1∂x2)x1x3=−x11−x2{displaystyle left({frac {partial x_{1}}{partial x_{2}}}right)_{frac {x_{1}}{x_{3}}}=-{frac {x_{1}}{1-x_{2}}}}{displaystyle left({frac {partial x_{1}}{partial x_{2}}}right)_{frac {x_{1}}{x_{3}}}=-{frac {x_{1}}{1-x_{2}}}}


or


(∂x3∂x2)x1x3=−x31−x2{displaystyle left({frac {partial x_{3}}{partial x_{2}}}right)_{frac {x_{1}}{x_{3}}}=-{frac {x_{3}}{1-x_{2}}}}{displaystyle left({frac {partial x_{3}}{partial x_{2}}}right)_{frac {x_{1}}{x_{3}}}=-{frac {x_{3}}{1-x_{2}}}}


Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems:


X=x3x1+x3{displaystyle X={frac {x_{3}}{x_{1}+x_{3}}}}{displaystyle X={frac {x_{3}}{x_{1}+x_{3}}}}


Y=x3x2+x3{displaystyle Y={frac {x_{3}}{x_{2}+x_{3}}}}{displaystyle Y={frac {x_{3}}{x_{2}+x_{3}}}}


Z=x2x1+x2{displaystyle Z={frac {x_{2}}{x_{1}+x_{2}}}}{displaystyle Z={frac {x_{2}}{x_{1}+x_{2}}}}


These can be used for solving pde like:


(∂μ2∂n1)n2,n3=(∂μ1∂n2)n1,n3{displaystyle left({frac {partial mu _{2}}{partial n_{1}}}right)_{n_{2},n_{3}}=left({frac {partial mu _{1}}{partial n_{2}}}right)_{n_{1},n_{3}}}{displaystyle left({frac {partial mu _{2}}{partial n_{1}}}right)_{n_{2},n_{3}}=left({frac {partial mu _{1}}{partial n_{2}}}right)_{n_{1},n_{3}}}


or


(∂μ2∂n1)n2,n3,n4...,ni=(∂μ1∂n2)n1,n3,n4...,ni{displaystyle left({frac {partial mu _{2}}{partial n_{1}}}right)_{n_{2},n_{3},n_{4}...,n_{i}}=left({frac {partial mu _{1}}{partial n_{2}}}right)_{n_{1},n_{3},n_{4}...,n_{i}}}{displaystyle left({frac {partial mu _{2}}{partial n_{1}}}right)_{n_{2},n_{3},n_{4}...,n_{i}}=left({frac {partial mu _{1}}{partial n_{2}}}right)_{n_{1},n_{3},n_{4}...,n_{i}}}


This equality can be rearranged to have differential quotient of mole amounts or fractions on one side.


(∂μ2∂μ1)n2,n3=−(∂n1∂n2)μ1,n3=−(∂x1∂x2)μ1,n3{displaystyle left({frac {partial mu _{2}}{partial mu _{1}}}right)_{n_{2},n_{3}}=-left({frac {partial n_{1}}{partial n_{2}}}right)_{mu _{1},n_{3}}=-left({frac {partial x_{1}}{partial x_{2}}}right)_{mu _{1},n_{3}}}{displaystyle left({frac {partial mu _{2}}{partial mu _{1}}}right)_{n_{2},n_{3}}=-left({frac {partial n_{1}}{partial n_{2}}}right)_{mu _{1},n_{3}}=-left({frac {partial x_{1}}{partial x_{2}}}right)_{mu _{1},n_{3}}}


or


(∂μ2∂μ1)n2,n3,n4,...ni=−(∂n1∂n2)μ1,n2,n4...,ni{displaystyle left({frac {partial mu _{2}}{partial mu _{1}}}right)_{n_{2},n_{3},n_{4},...n_{i}}=-left({frac {partial n_{1}}{partial n_{2}}}right)_{mu _{1},n_{2},n_{4}...,n_{i}}}{displaystyle left({frac {partial mu _{2}}{partial mu _{1}}}right)_{n_{2},n_{3},n_{4},...n_{i}}=-left({frac {partial n_{1}}{partial n_{2}}}right)_{mu _{1},n_{2},n_{4}...,n_{i}}}


Mole amounts can be eliminated by forming ratios:


(∂n1∂n2)n3=(∂n1n3∂n2n3)n3=(∂x1x3∂x2x3)n3{displaystyle left({frac {partial n_{1}}{partial n_{2}}}right)_{n_{3}}=left({frac {partial {frac {n_{1}}{n_{3}}}}{partial {frac {n_{2}}{n_{3}}}}}right)_{n_{3}}=left({frac {partial {frac {x_{1}}{x_{3}}}}{partial {frac {x_{2}}{x_{3}}}}}right)_{n_{3}}}{displaystyle left({frac {partial n_{1}}{partial n_{2}}}right)_{n_{3}}=left({frac {partial {frac {n_{1}}{n_{3}}}}{partial {frac {n_{2}}{n_{3}}}}}right)_{n_{3}}=left({frac {partial {frac {x_{1}}{x_{3}}}}{partial {frac {x_{2}}{x_{3}}}}}right)_{n_{3}}}


Thus the ratio of chemical potentials becomes:


(∂μ2∂μ1)n2n3=−(∂x1x3∂x2x3)μ1{displaystyle left({frac {partial mu _{2}}{partial mu _{1}}}right)_{frac {n_{2}}{n_{3}}}=-left({frac {partial {frac {x_{1}}{x_{3}}}}{partial {frac {x_{2}}{x_{3}}}}}right)_{mu _{1}}}{displaystyle left({frac {partial mu _{2}}{partial mu _{1}}}right)_{frac {n_{2}}{n_{3}}}=-left({frac {partial {frac {x_{1}}{x_{3}}}}{partial {frac {x_{2}}{x_{3}}}}}right)_{mu _{1}}}


Similarly the ratio for the multicomponents system becomes


(∂μ2∂μ1)n2n3,n3n4,...ni−1ni=−(∂x1x3∂x2x3)μ1,n3n4,...ni−1ni{displaystyle left({frac {partial mu _{2}}{partial mu _{1}}}right)_{{frac {n_{2}}{n_{3}}},{frac {n_{3}}{n_{4}}},...{frac {n_{i-1}}{n_{i}}}}=-left({frac {partial {frac {x_{1}}{x_{3}}}}{partial {frac {x_{2}}{x_{3}}}}}right)_{mu _{1},{frac {n_{3}}{n_{4}}},...{frac {n_{i-1}}{n_{i}}}}}{displaystyle left({frac {partial mu _{2}}{partial mu _{1}}}right)_{{frac {n_{2}}{n_{3}}},{frac {n_{3}}{n_{4}}},...{frac {n_{i-1}}{n_{i}}}}=-left({frac {partial {frac {x_{1}}{x_{3}}}}{partial {frac {x_{2}}{x_{3}}}}}right)_{mu _{1},{frac {n_{3}}{n_{4}}},...{frac {n_{i-1}}{n_{i}}}}}



Related quantities



Mass fraction


The mass fraction wi can be calculated using the formula


wi=xi⋅MiM{displaystyle w_{i}=x_{i}cdot {frac {M_{i}}{M}}}w_{i}=x_{i}cdot {frac {M_{i}}{M}}

where Mi is the molar mass of the component i and M is the average molar mass of the mixture.


Replacing the expression of the molar mass:


wi=xi⋅Mi∑ixiMi{displaystyle w_{i}=x_{i}cdot {frac {M_{i}}{sum _{i}x_{i}M_{i}}}}w_{i}=x_{i}cdot {frac {M_{i}}{sum _{i}x_{i}M_{i}}}


Molar mixing ratio


The mixing of two pure components can be expressed introducing the amount or molar mixing ratio of them rn=n2n1{displaystyle r_{n}={frac {n_{2}}{n_{1}}}}{displaystyle r_{n}={frac {n_{2}}{n_{1}}}}. Then the mole fractions of the components will be:



x1=11+rn{displaystyle x_{1}={frac {1}{1+r_{n}}}}{displaystyle x_{1}={frac {1}{1+r_{n}}}}

x2=rn1+rn{displaystyle x_{2}={frac {r_{n}}{1+r_{n}}}}{displaystyle x_{2}={frac {r_{n}}{1+r_{n}}}}


The amount ratio equals the ratio of mole fractions of components:


n2n1=x2x1{displaystyle {frac {n_{2}}{n_{1}}}={frac {x_{2}}{x_{1}}}}{displaystyle {frac {n_{2}}{n_{1}}}={frac {x_{2}}{x_{1}}}}

due to division of both numerator and denominator by the sum of molar amounts of components. This property has consequences for representations of phase diagrams using, for instance, ternary plots.



Mixing binary mixtures with a common component to form ternary mixtures


Mixing binary mixtures with a common component gives a ternary mixture with certain mixing ratios between the three components. These mixing ratios from the ternary and the corresponding mole fractions of the ternary mixture x1(123), x2(123), x3(123) can be expressed as a function of several mixing ratios involved, the mixing ratios between the components of the binary mixtures and the mixing ratio of the binary mixtures to form the ternary one.


x1(123)=n(12)x1(12)+n13x1(13)n(12)+n(13){displaystyle x_{1(123)}={frac {n_{(12)}x_{1(12)}+n_{13}x_{1(13)}}{n_{(12)}+n_{(13)}}}}{displaystyle x_{1(123)}={frac {n_{(12)}x_{1(12)}+n_{13}x_{1(13)}}{n_{(12)}+n_{(13)}}}}


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 ρi is given by:


xi=ρiρ⋅MMi{displaystyle x_{i}={frac {rho _{i}}{rho }}cdot {frac {M}{M_{i}}}}x_{i}={frac {rho _{i}}{rho }}cdot {frac {M}{M_{i}}}

where M is the average molar mass of the mixture.


ρi=xiρ⋅MiM{displaystyle rho _{i}=x_{i}rho cdot {frac {M_{i}}{M}}}rho _{i}=x_{i}rho cdot {frac {M_{i}}{M}}


Molar concentration


The conversion to molar concentration ci is given by:


ci=xi⋅ρM=xic{displaystyle c_{i}={frac {x_{i}cdot rho }{M}}=x_{i}c}{displaystyle c_{i}={frac {x_{i}cdot rho }{M}}=x_{i}c}

or


ci=xi⋅ρ∑ixiMi{displaystyle c_{i}={frac {x_{i}cdot rho }{sum _{i}x_{i}M_{i}}}}{displaystyle c_{i}={frac {x_{i}cdot rho }{sum _{i}x_{i}M_{i}}}}

where M is the average molar mass of the solution, c is the total molar concentration and ρ is the density of the solution.



Mass and molar mass


The mole fraction can be calculated from the masses mi and molar masses Mi of the components:


xi=miMi∑imiMi{displaystyle x_{i}={frac {frac {m_{i}}{M_{i}}}{sum _{i}{frac {m_{i}}{M_{i}}}}}}{displaystyle x_{i}={frac {frac {m_{i}}{M_{i}}}{sum _{i}{frac {m_{i}}{M_{i}}}}}}


Spatial variation and gradient


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



References





  1. ^ abc IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "amount fraction". doi:10.1351/goldbook.A00296


  2. ^ Zumdahl, Steven S. (2008). Chemistry (8th ed.). Cengage Learning. p. 201. ISBN 0-547-12532-1..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  3. ^ Rickard, James N.; Spencer, George M.; Bodner, Lyman H. (2010). Chemistry: Structure and Dynamics (5th ed.). Hoboken, N.J.: Wiley. p. 357. ISBN 978-0-470-58711-9.


  4. ^ Thompson, A.; Taylor, B. N. "The NIST Guide for the use of the International System of Units". National Institute of Standards and Technology. Retrieved 5 July 2014.











Comments

Post a Comment

Popular posts from this blog

Information security

Image
This article is part of a series on Information security Related security categories Internet security Cyberwarfare Computer security Mobile security Network security Threats Computer crime Vulnerability Eavesdropping Malware Spyware Ransomware Trojans Viruses Worms Rootkits Bootkits Keyloggers Screen scrapers Exploits Backdoors Logic bombs Payloads Denial of service Defenses Computer access control Application security Antivirus software Secure coding Secure by default Secure by design Secure operating systems Authentication Multi-factor authentication Authorization Data-centric security Encryption Firewall Intrusion detection system Mobile secure gateway Runtime application self-protection (RASP) v t e Information security , sometimes shortened to InfoSec , is the practice of preventing unauthorized access, use, disclosure, disruption, modification, inspection, recording or destruction of information. Th
Read more

章鱼与海女图

Image
本条目 可参照英語維基百科相應條目来扩充 。 （2014/5/25） 若您熟悉来源语言和主题，请协助参考外语维基扩充条目。请勿直接提交机械翻译，也不要翻译不可靠、低品质内容。依版权协议，译文需在编辑摘要注明来源，或于讨论页顶部标记 {{Translated page}} 标签。 本条目 需要擴充。 （2013年8月15日） 请協助改善这篇條目，更進一步的信息可能會在討論頁或扩充请求中找到。请在擴充條目後將此模板移除。 《 章鱼与海女 》（日语： 蛸と海女 ／ たことあま   Tako to ama ）是一幅由日本畫家葛饰北斋创作的浮世绘，出自1814年（文化4年）出版的艳本《喜能会之故真通》（日语： 喜能会之故真通 ／ きのえのこまつ   Kinoe no Komatsu ）。畫中有兩隻章鱼以觸手攻擊一名採珠的海女，並進行擬似戀獸癖的交媾行為。作品尺寸为6½" × 8¾" （16.51 cm × 22.23 cm）。 [1] 在此之前的类似作品还有北尾重政的艳本《謡曲色番組》及勝川春潮的艳本《艳本千夜多女志》等，有被北斋作为参考的可能性。 《章鱼与海女图》有不同英文翻译，一般称The Dream of the Fisherman's Wife，又作Girl Diver and Octopi、Diver and Two Octopi，Richard Douglas Lane称其为 Girl Diver and Octopi [2] ，Mathi Forrer 称其为 Pearl Diver and Two Octopi [3] ，Danielle Talerico 称其为 Diver and Two Octopi 。 [4] 目录 1 作品介绍 1.1 画中对话 1.2 解释 2 评价 3 影响 3.1 模仿作品 4 另见 5 参考来源 6 外部連結 作品介绍 画中对话 原文 译文 大蛸： いつぞハいつぞハとねらいすましてゐたかいがあつて、けふといふけふ、とうとうとらまへたア。てもむつくりとしたいいぼぼだ。いもよりハなをこうぶつだ。サアサア、すつてすつてすいつくして
Read more

Farm Security Administration

Image
Farm Security Administration Farm Security Administration logo Agency overview Formed September 1, 1937  ( 1937-09-01 ) Preceding agencies Resettlement Administration Federal Emergency Relief Administration Dissolved 1946 Superseding agency Farmers Home Administration Key documents Farm Security Act Bankhead-Jones Farm Tenant Act The Farm Security Administration ( FSA ) was a New Deal agency created in 1937 to combat rural poverty during the Great Depression in the United States. It succeeded the Resettlement Administration (1935–1937). [1] The FSA stressed "rural rehabilitation" efforts to improve the lifestyle of sharecroppers, tenants, very poor landowning farmers, and a program to purchase submarginal land owned by poor farmers and resettle them in group farms on land more suitable for efficient farming. Critics, including the Farm Bureau, strongly opposed the FSA as an experiment in collectivizing agriculture—that is, in bringing
Read more