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| {{merge to|Dimensional analysis|date=June 2013}}
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| {{merge to|Units of measurement#Calculations with units of measurements|date=June 2013|section=yes}}
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| Many, if not most, parameters and measurements in the physical sciences and engineering are expressed as a numerical quantity and a corresponding dimensional unit; for example: 1000 kg/m³, 100 kPa/bar, 50 miles per hour, 1000 Btu/lb. Converting from one dimensional unit to another is often somewhat complex. The '''factor-label method''', also known as the '''unit-factor method''' or [[dimensional analysis]], is a widely used technique for performing such conversions.<ref>{{cite book|author=David Goldberg|title=Fundamentals of Chemistry|edition=5th Edition|publisher=McGraw-Hill|year=2006|isbn=0-07-322104-X}}</ref><ref>{{cite book|author=James Ogden|title=The Handbook of Chemical Engineering|publisher=Research & Education Association|year=1999|isbn=0-87891-982-1}}</ref><ref>[http://www.kentchemistry.com/links/Measurements/dimensionalanalysis.htm Dimensional Analysis or the Factor Label Method]</ref> It is also used for determining whether the two sides of a mathematical equation involving dimensions have the same dimensional units.
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| ==The factor-label method for converting units==
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| The factor-label method is the sequential application of conversion factors expressed as fractions and arranged so that any dimensional unit appearing in both the numerator and denominator of any of the fractions can be cancelled out until only the desired set of dimensional units is obtained. For example, 10 [[miles per hour]] can be converted to [[meters per second]] by using a sequence of conversion factors as shown below:
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| : <math> \frac{10\text{ mile}}{1\text{ hour}} \times \frac{1609\text{ meter}}{1\text{ mile}} \times \frac{1\text{ hour}}{3600\text{ second}} = \frac{4.47\text{ meter}}{\text{second}}. </math> -->
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| 10 <s>mile</s> 1609 meter 1 <s>hour</s> meter
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| -- ---- × ---- ----- × ---- ------ = 4.47 ------
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| 1 <s>hour</s> 1 <s>mile</s> 3600 second second
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| It can be seen that each conversion factor is equivalent to the value of one. For example, starting with 1 mile = 1609 meters and dividing both sides of the equation by 1 mile yields 1 mile / 1 mile = 1609 meters / 1 mile, which when simplified yields 1 = 1609 meters / 1 mile.
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| So, when the units ''mile'' and ''hour'' are cancelled out and the arithmetic is done, 10 miles per hour converts to 4.47 meters per second.
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| As a more complex example, the [[concentration]] of [[nitrogen oxides]] (i.e., NOx) in the [[flue gas]] from an industrial [[furnace]] can be converted to a [[mass flow rate]] expressed in grams per hour (i.e., g/h) of NOx by using the following information as shown below:
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| ; NOx concentration := 10 [[parts per notation|parts per million]] by volume = 10 ppmv = 10 volumes/10<sup>6</sup> volumes
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| ; NOx molar mass := 46 kg/kgmol (sometimes also expressed as 46 kg/kmol)
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| ; Flow rate of flue gas := 20 cubic meters per minute = 20 m³/min
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| : The flue gas exits the furnace at 0 °C temperature and 101.325 kPa absolute pressure.
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| : The [[standard conditions of temperature and pressure#Molar volume of a gas|molar volume]] of a gas at 0 °C temperature and 101.325 kPa is 22.414 m³/kgmol.
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| 10 <s>m³ NOx</s> 20 <s>m³ gas</s> 60 <s>minute</s> 1 <s>[[kgmol]] NOx</s> 46 <s>kg</s> NOx 1000 g g NOx
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| --- ------ × -- ------ × -- ------ × ------ --------- × -- --------- × ---- -- = 24.63 -----
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| 10<sup>6</sup> <s>m³ gas</s> 1 <s>minute</s> 1 hour 22.414 <s>m³ NOx</s> 1 <s>kgmol NOx</s> 1 <s>kg</s> hour
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| After cancelling out any dimensional units that appear both in the numerators and denominators of the fractions in the above equation, the NOx concentration of 10 ppm<sub>v</sub> converts to mass flow rate of 24.63 grams per hour.
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| == Checking equations that involve dimensions ==
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| The factor-label method can also be used on any mathematical equation to check whether or not the dimensional units on the left hand side of the equation are the same as the dimensional units on the right hand side of the equation. Having the same units on both sides of an equation does not guarantee that the equation is correct, but having different units on the two sides of an equation does guarantee that the equation is wrong.
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| For example, check the [[Gas Laws|Universal Gas Law]] equation of ''P·V = n·R·T'', when:
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| * the pressure ''P'' is in pascals (Pa)
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| * the volume ''V'' is in cubic meters (m³)
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| * the amount of substance ''n'' is in moles (mol)
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| * the [[Gas constant|universal gas law constant ''R'']] is 8.3145 Pa·m³/(mol·K)
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| * the temperature ''T'' is in kelvins (K)
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| <s>mol</s> (Pa)(m³) <s>K</s>
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| (Pa)(m³) = ----- × ---------- × ---
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| 1 (<s>mol</s>)(<s>K</s>) 1
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| As can be seen, when the dimensional units appearing in the numerator and denominator of the equation's right hand side are cancelled out, both sides of the equation have the same dimensional units.
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| ==Limitations==
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| The factor-label method can convert only unit quantities for which the units are in a linear relationship intersecting at 0. Most units fit this paradigm. An example for which it cannot be used is the conversion between [[degrees Celsius]] and [[kelvins]] (or [[Fahrenheit]]). Between degrees Celsius and kelvins, there is a constant difference rather than a constant ratio, while between Celsius and Fahrenheit there is both a constant difference and a constant ratio. Instead of multiplying the given quantity by a single conversion factor to obtain the converted quantity, it is more logical to think of the original quantity being divided by its unit, being added or subtracted by the constant difference, and the entire operation being multiplied by the new unit. Mathematically, this is an [[affine transform]] (<math>ax+b</math>), not a [[linear transform]] (<math>ax</math>). Formally, one starts with a displacement (in some units) from one point, and ends with a displacement (in some other units) from some other point.
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| For instance, the freezing point of water is 0° in Celsius and 32° in Fahrenheit, and a 5° change in Celsius corresponds to a 9° change in Fahrenheit. Thus, to convert from Fahrenheit to Celsius one subtracts 32° (displacement from one point), multiplies by 5 and divides by 9 (scales by the ratio of units), and adds 0 (displacement from new point). Reversing this yields the formula for Celsius; one could have started with the equivalence between 100° Celsius and 212° Fahrenheit, though this would yield the same formula at the end.
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| Hence, to convert Fahrenheit to Celsius, enter the Fahrenheit value into this formula:
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| :°C = (°F-32°) ÷ 1.8
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| Note that dividing by 1.8 is the same as multiplying by 5 and dividing by 9.
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| And, to convert Celsius to Fahrenheit, enter the Celsius value into this formula:
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| :°F = 1.8(°C) + 32°
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| Note that multiplying by 1.8 is the same as multiplying by 9 and dividing by 5.
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| == See also ==
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| * [[Conversion of units]]
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| * [[Dimensional analysis]]
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| * [[ISO 31]]
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| * [[Units of measurement]]
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| ==References==
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| {{reflist}}
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| == External links ==
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| * [http://www.calchemy.com/uclive.htm Unicalc Live web calculator doing units conversion by dimensional analysis]
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| * [http://www.chem.tamu.edu/class/fyp/mathrev/mr-da.html Math Skills Review]
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| * [http://www.epa.gov/eogapti1/toc/full_toc.htm U.S. EPA tutorial]
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| * [http://www.felderbooks.com/papers A Discussion of Units]
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| * [http://www.astro.yale.edu/astro120/unitconv.pdf Short Guide to Unit Conversions]
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| * [http://www.purplemath.com/modules/units.htm Cancelling Units Lesson]
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| * [http://www.dentonisd.org/512125919103412/lib/512125919103412/_files/chemChap11.pdf Chapter 11: Behavior of Gases] ''Chemistry: Concepts and Applications'', Denton Independent School District
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| * [http://www.air-dispersion.com/formulas.html Air Dispersion Modeling Conversions and Formulas]
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| {{systems of measurement}}
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| [[Category:Dimensional analysis]]
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| [[Category:Conversion of units of measurement]]
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| [[Category:Chemical engineering]]
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| [[Category:Mechanical engineering]]
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| [[Category:Environmental engineering]]
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Hello. Allow me introduce the author. Her name is Emilia Shroyer but it's not the most feminine title out there. Bookkeeping is my profession. He is really fond of performing ceramics but he is struggling to discover time for it. North Dakota is where me and my spouse reside.
Here is my blog; hard-ass-porn.com