User:Milton Beychok/Sandbox: Difference between revisions

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The '''molar volume''' (symbol ''V''<sub>m</sub>) of a substance is the [[volume (science)|volume]] occupied by one [[mole (unit)|mole]] of the substance at a given [[temperature]] and [[pressure]].<ref name="GreenBook">[http://www.iupac.org/publications/books/gbook/green_book_2ed.pdf International Union of Pure and Applied chemistry (IUPAC): Quantities, Units and Symbols in Physical Chemistry] 2nd Edition, 1993</ref><ref name="NIST">[http://physics.nist.gov/Pubs/SP811/sec08.html NIST Guide to SI] Item 8.6.3 in Section 8</ref> It is equal to the [[molecular mass]] (''M'') of the substance divided by its [[density (chemistry)|density]] (''ρ'') at the given temperature and pressure:


::<math>V_{\rm m} = {M\over\rho}</math>
It has an [[SI unit]] of cubic [[metre]]s per mole (m<sup>3</sup>/mol).<ref name="GreenBook"/><ref name="NIST"/> However, molar volumes are often expressed as cubic metres per 1,000 moles (m<sup>3</sup>/kmol) or cubic decimetres per mol (dm<sup>3</sup>/mol) for gases and as centimetres per mole (cm<sup>3</sup>/mol) for liquids and solids.
If a substance is a mixture containing ''N'' components, the molar volume is calculated using:
::<math>V_{\rm m} = \frac{\displaystyle\sum_{i=1}^{N}x_{i}M_{i}}{\rho_{mixture}}</math>
where ''x<sub> i</sub>'' is the [[mole fraction]] of the ith component, ''M<sub> i</sub>'' is the molecular mass of the ith component and ''ρ<sub>mixture</sub> is the mixture density at the given temperature and pressure.
When stating molar volume numerical values, it is important to also state the given conditions of temperature and pressure. Otherwise, the numerical values are meaningless.
== Ideal gases ==
The [[ideal gas law]] equation can be rearranged to give this expression for the molar volume of an ideal gas:
::<math>V_{\rm m} = {V\over{n}} = {{RT}\over{P}}</math>
'''Where in [[SI unit]]s:'''
{| border="0" cellpadding="2"
|-
!align=right| ''P''
|align=left|= the gas absolute pressure, in [[pascal (unit)|Pa]]
|-
!align=right|''n''
|align=left|= number of moles, in [[mole (unit)|mol]]
|-
!align=right| ''V''<sub>m</sub> 
|align=left|= the gas molar volume, in m<sup>3</sup>/mol
|-
!align=right| ''T''
|align=left|= the gas absolute temperature, in [[kelvin|K]]
|-
!align=right| ''R''
|align=left|= the [[molar gas constant|universal gas law constant]] of  8.314472 m<sup>3</sup>·Pa·mol<sup>-1</sup>·K<sup>-1</sup>
|}
'''Where in [[U.S. customary units]]:'''
{| border="0" cellpadding="2"
|-
!align=right| ''P''
|align=left|= the gas absolute pressure, in [[pound-force per square inch|psia]]
|-
!align=right|''n''
|align=left|= number of moles, in [[mole (unit)|lb-mol]]
|-
!align=right| ''V''<sub>m</sub>
|align=left|= the gas molar volume, in ft<sup>3</sup>/lb-mol
|-
!align=right| ''T''
|align=left|= the gas absolute temperature, in [[Rankine (unit)|degrees Rankine]] (°R)
|-
!align=right| ''R''
|align=left|= the universal gas law constant of 10.7316 ft<sup>3</sup>·psia·lb-mol<sup>-l</sup>·°R<sup>-1</sup>
|}
'''Example calculations of ideal gas molar volumes:'''
* In SI metric units:
:'''''V'''''<sub>'''m'''</sub> = 8.314472 × 273.15 / 101,325 = 0.022414 m<sup>3</sup>/mol at 0 °C and 101,325 Pa absolute pressure = 22.414 m<sup>3</sup>/kmol at 0 °C (273.15 K) and 101.325 kPa absolute pressure
:'''''V'''''<sub>'''m'''</sub>  = 8.314472 × 273.15 / 100,000 = 0.022711 m<sup>3</sup>/kmol at 0 °C and 100,000 Pa  absolute pressure = 22.711 m<sup>3</sup>/kmol at 0 °C (273.15 K) and 100 kPa  absolute pressure
* In customary USA units:
:'''''V'''''<sub>'''m'''</sub> = 10.7316 × 519.67 / 14.696 = 379.48  ft<sup>3</sup>/lb-mol at 60 °F (519.67 °R) and 14.696 psia
'''Notes:'''
* lb-mol is an abbreviation for [[Mole (unit)|pound-mol]]
* °R is [[Rankine (unit)|degrees Rankine]] (an absolute temperature scale) and °F is [[Fahrenheit (unit)|degrees Fahrenheit]] (a temperature scale).
* °R = °F + 459.67
* The technical literature can be confusing because some authors often fail to explain whether they are using the universal gas law constant '''''R''''', which applies to any ideal gas, or whether they are using the specific gas law constant '''''R<sub>s</sub>''''', which only applies to a specific individual gas.  The relationship between the two constants is '''''R<sub>s</sub>''''' = '''''R / M''''',  where '''''M''''' is the molecular mass of the gas.
== Real gases ==
Real gases are those gases that do not exhibit ideal gas behavior. For such gases, the simplest method of determining molar volumes is by using [[Compressibility factor (gases)|compressibility factors]] as in the following expression:
::<math>V_{\rm m} = {V\over{n}} = {{ZRT}\over{P}}</math>
where '''''Z''''' is the gas compressibility factor, which is a useful thermodynamic property for modifying the ideal gas law to account for behavior of real gases.<ref name=Compressibility>[http://en.citizendium.org/wiki/Compressibility_factor#Determination_of_gas_compressibility_values Determination of gas compressibility values] Information on how to determine gas compressibility factors and molar volumes.</ref> The above equation is basically a simple [[equation of state]] (EOS). The major limitation of this equation of state is that the gas compressibility factor, '''''Z''''', is not a constant but varies from one gas to another as well as with the temperature and pressure of the gas under consideration.
More accurate values of real gas molar volumes may be obtained by using equations of state such as the [[Van der Waals equation|van der Waals equation]] developed in 1873, the [[Redlich-Kwong equation]] developed in 1949, the [[Soave-Redlich-Kwong equation]] developed in 1972 and the [[Peng-Robinson equation]] developed in 1976.<ref name=Compressibility/>
== References ==
{{reflist}}

Revision as of 01:20, 12 January 2010

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