Conversion of units

Conversion of unitsis the conversion of theunit of measurementin which aquantityis expressed, typically through a multiplicativeconversion factorthat changes the unit without changing the quantity. This is also often loosely taken to include replacement of a quantity with a corresponding quantity that describes the same physical property.

Unit conversion is often easier within ametric systemsuch as theSIthan in others, due to the system'scoherenceand itsmetric prefixesthat act as power-of-10 multipliers.

Overview

The definition and choice of units in which to express a quantity may depend on the specific situation and the intended purpose. This may be governed by regulation,contract,technical specificationsor other publishedstandards.Engineering judgment may include such factors as:

theprecision and accuracyof measurement and the associateduncertainty of measurement
the statisticalconfidence intervalortolerance intervalof the initial measurement
the number ofsignificant figuresof the measurement
the intended use of the measurement, including theengineering tolerances
historical definitions of the units and their derivatives used in old measurements; e.g.,international footvs. USsurvey foot.

For some purposes, conversions from one system of units to another are needed to be exact, without increasing or decreasing the precision of the expressed quantity. Anadaptive conversionmay not produce an exactly equivalent expression.Nominal valuesare sometimes allowed and used.

Factor–label method

Thefactor–label method,also known as theunit–factor methodor theunity bracket method,^[1]is a widely used technique for unit conversions that uses the rules ofalgebra.^[2]^[3]^[4]

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, 10miles per hourcan be converted tometres per secondby using a sequence of conversion factors as shown below: ${\frac {\mathrm {10~{\cancel {mi}}} }{\mathrm {1~{\cancel {h}}} }}\times {\frac {\mathrm {1609.344~m} }{\mathrm {1~{\cancel {mi}}} }}\times {\frac {\mathrm {1~{\cancel {h}}} }{\mathrm {3600~s} }}=\mathrm {4.4704~{\frac {m}{s}}}.$

Each conversion factor is chosen based on the relationship between one of the original units and one of the desired units (or some intermediary unit), before being rearranged to create a factor that cancels out the original unit. For example, as "mile" is the numerator in the original fraction and⁠ $\mathrm {1~mi} =\mathrm {1609.344~m}$ ⁠,"mile" will need to be the denominator in the conversion factor. Dividing both sides of the equation by 1 mile yields⁠ ${\frac {\mathrm {1~mi} }{\mathrm {1~mi} }}={\frac {\mathrm {1609.344~m} }{\mathrm {1~mi} }}$ ⁠,which when simplified results in the dimensionless⁠ $1={\frac {\mathrm {1609.344~m} }{\mathrm {1~mi} }}$ ⁠.Because of the identity property of multiplication, multiplying any quantity (physical or not) by the dimensionless 1 does not change that quantity.^[5]Once this and the conversion factor for seconds per hour have been multiplied by the original fraction to cancel out the unitsmileandhour,10 miles per hour converts to 4.4704 metres per second.

As a more complex example, theconcentrationofnitrogen oxides(NO_x) in theflue gasfrom an industrialfurnacecan be converted to amass flow rateexpressed in grams per hour (g/h) of NO_xby using the following information as shown below:

NO_xconcentration: = 10parts per millionby volume = 10 ppmv = 10 volumes/10⁶volumes
NO_xmolar mass: = 46 kg/kmol = 46 g/mol
Flow rate of flue gas: = 20 cubic metres per minute = 20 m³/min; The flue gas exits the furnace at 0 °C temperature and 101.325 kPa absolute pressure.; Themolar volumeof a gas at 0 °C temperature and 101.325 kPa is 22.414 m³/kmol.

{\frac {1000\ {\ce {g\ NO}}_{x}}{1{\cancel {{\ce {kg\ NO}}_{x}}}}}\times {\frac {46\ {\cancel {{\ce {kg\ NO}}_{x}}}}{1\ {\cancel {{\ce {kmol\ NO}}_{x}}}}}\times {\frac {1\ {\cancel {{\ce {kmol\ NO}}_{x}}}}{22.414\ {\cancel {{\ce {m}}^{3}\ {\ce {NO}}_{x}}}}}\times {\frac {10\ {\cancel {{\ce {m}}^{3}\ {\ce {NO}}_{x}}}}{10^{6}\ {\cancel {{\ce {m}}^{3}\ {\ce {gas}}}}}}\times {\frac {20\ {\cancel {{\ce {m}}^{3}\ {\ce {gas}}}}}{1\ {\cancel {\ce {minute}}}}}\times {\frac {60\ {\cancel {\ce {minute}}}}{1\ {\ce {hour}}}}=24.63\ {\frac {{\ce {g\ NO}}_{x}}{\ce {hour}}}

After cancelling any dimensional units that appear both in the numerators and the denominators of the fractions in the above equation, the NO_xconcentration of 10 ppm_vconverts to mass flow rate of 24.63 grams per hour.

Checking equations that involve dimensions

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 ensure that the equation is correct, but having different units on the two sides (when expressed in terms of base units) of an equation implies that the equation is wrong.

For example, check theuniversal gas lawequation ofPV=nRT,when:

the pressurePis in pascals (Pa)
the volumeVis in cubic metres (m³)
the amount of substancenis in moles (mol)
theuniversal gas constantRis 8.3145 Pa⋅m³/(mol⋅K)
the temperatureTis in kelvins (K)

$\mathrm {Pa{\cdot }m^{3}} ={\frac {\cancel {\mathrm {mol} }}{1}}\times {\frac {\mathrm {Pa{\cdot }m^{3}} }{{\cancel {\mathrm {mol} }}\ {\cancel {\mathrm {K} }}}}\times {\frac {\cancel {\mathrm {K} }}{1}}$

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. Dimensional analysis can be used as a tool to construct equations that relate non-associated physico-chemical properties. The equations may reveal undiscovered or overlooked properties of matter, in the form of left-over dimensions – dimensional adjusters – that can then be assigned physical significance. It is important to point out that such 'mathematical manipulation' is neither without prior precedent, nor without considerable scientific significance. Indeed, thePlanck constant,a fundamental physical constant, was 'discovered' as a purely mathematical abstraction or representation that built on theRayleigh–Jeans lawfor preventing theultraviolet catastrophe.It was assigned and ascended to its quantum physical significance either in tandem or post mathematical dimensional adjustment – not earlier.

Limitations

The factor–label method can convert only unit quantities for which the units are in a linear relationship intersecting at 0 (ratio scalein Stevens's typology). Most conversions fit this paradigm. An example for which it cannot be used is the conversion between theCelsius scaleand theKelvin scale(or theFahrenheit scale). Between degrees Celsius and kelvins, there is a constant difference rather than a constant ratio, while between degrees Celsius and degrees Fahrenheit there is neither a constant difference nor a constant ratio. There is, however, anaffine transform(⁠ $x\mapsto ax+b$ ⁠,rather than alinear transform⁠ $x\mapsto ax$ ⁠) between them.

For example, the freezing point of water is 0 °C and 32 °F, and a 5 °C change is the same as a 9 °F change. Thus, to convert from units of Fahrenheit to units of Celsius, one subtracts 32 °F (the offset from the point of reference), divides by 9 °F and multiplies by 5 °C (scales by the ratio of units), and adds 0 °C (the offset from the point of reference). Reversing this yields the formula for obtaining a quantity in units of Celsius from units of Fahrenheit; one could have started with the equivalence between 100 °C and 212 °F, which yields the same formula.

Hence, to convert the numerical quantity value of a temperatureT[F] in degrees Fahrenheit to a numerical quantity valueT[C] in degrees Celsius, this formula may be used:

T[C] = (T[F] − 32) × 5/9.

To convertT[C] in degrees Celsius toT[F] in degrees Fahrenheit, this formula may be used:

T[F] = (T[C] × 9/5) + 32.

Example

Starting with:

Z=n_{i}\times [Z]_{i}

replace the original unit⁠ $[Z]_{i}$ ⁠with its meaning in terms of the desired unit⁠ $[Z]_{j}$ ⁠,e.g. if⁠ $[Z]_{i}=c_{ij}\times [Z]_{j}$ ⁠,then:

Z=n_{i}\times (c_{ij}\times [Z]_{j})=(n_{i}\times c_{ij})\times [Z]_{j}

Now⁠ $n_{i}$ ⁠and⁠ $c_{ij}$ ⁠are both numerical values, so just calculate their product.

Or, which is just mathematically the same thing, multiplyZby unity, the product is stillZ:

Z=n_{i}\times [Z]_{i}\times (c_{ij}\times [Z]_{j}/[Z]_{i})

For example, you have an expression for a physical valueZinvolving the unitfeet per second(⁠ $[Z]_{i}$ ⁠) and you want it in terms of the unitmiles per hour(⁠ $[Z]_{j}$ ⁠):

Find facts relating the original unit to the desired unit:
1 mile = 5280 feet and 1 hour = 3600 seconds
Next use the above equations to construct a fraction that has a value of unity and that contains units such that, when it is multiplied with the original physical value, will cancel the original units:
$1={\frac {1\,\mathrm {mi} }{5280\,\mathrm {ft} }}\quad \mathrm {and} \quad 1={\frac {3600\,\mathrm {s} }{1\,\mathrm {h} }}$
Last, multiply the original expression of the physical value by the fraction, called aconversion factor,to obtain the same physical value expressed in terms of a different unit. Note: since valid conversion factors aredimensionlessand have a numerical value ofone,multiplying any physical quantity by such a conversion factor (which is 1) does not change that physical quantity.
$52.8\,{\frac {\mathrm {ft} }{\mathrm {s} }}=52.8\,{\frac {\mathrm {ft} }{\mathrm {s} }}{\frac {1\,\mathrm {mi} }{5280\,\mathrm {ft} }}{\frac {3600\,\mathrm {s} }{1\,\mathrm {h} }}={\frac {52.8\times 3600}{5280}}\,\mathrm {mi/h} =36\,\mathrm {mi/h}$

Or as an example using the metric system, you have a value of fuel economy in the unitlitres per 100 kilometresand you want it in terms of the unitmicrolitres per metre:

\mathrm {\frac {9\,{\rm {L}}}{100\,{\rm {km}}}} =\mathrm {\frac {9\,{\rm {L}}}{100\,{\rm {km}}}} \mathrm {\frac {1000000\,{\rm {\mu L}}}{1\,{\rm {L}}}} \mathrm {\frac {1\,{\rm {km}}}{1000\,{\rm {m}}}} ={\frac {9\times 1000000}{100\times 1000}}\,\mathrm {\mu L/m} =90\,\mathrm {\mu L/m}

Calculation involving non-SI Units

In the cases where non-SI unitsare used, the numerical calculation of a formula can be done by first working out the factor, and then plug in the numerical values of the given/known quantities.

For example, in the study ofBose–Einstein condensate,^[6]atomic mass $m$ is usually given indaltons,instead ofkilograms,andchemical potential $μ$ is often given in theBoltzmann constanttimesnanokelvin.The condensate'shealing lengthis given by: $\xi ={\frac {\hbar }{\sqrt {2m\mu }}}\,.$

For a²³Na condensate with chemical potential of (the Boltzmann constant times) 128 nK, the calculation of healing length (in micrometres) can be done in two steps:

Calculate the factor

Assume that⁠ $m=1\,{\text{Da}},\mu =k_{\text{B}}\cdot 1\,{\text{nK}}$ ⁠,this gives $\xi ={\frac {\hbar }{\sqrt {2m\mu }}}=15.574\,\mathrm {\mu m} \,,$ which is our factor.

Calculate the numbers

Now, make use of the fact that⁠ $\xi \propto {\frac {1}{\sqrt {m\mu }}}$ ⁠.With⁠ $m=23\,{\text{Da}},\mu =128\,k_{\text{B}}\cdot {\text{nK}}$ ⁠,⁠ $\xi ={\frac {15.574}{\sqrt {23\cdot 128}}}\,{\text{μm}}=0.287\,{\text{μm}}$ ⁠.

This method is especially useful for programming and/or making aworksheet,where input quantities are taking multiple different values; For example, with the factor calculated above, it is very easy to see that the healing length of¹⁷⁴Yb with chemical potential 20.3 nK is

⁠

\xi ={\frac {15.574}{\sqrt {174\cdot 20.3}}}\,{\text{μm}}=0.262\,{\text{μm}}

⁠.

Software tools

There are many conversion tools. They are found in the function libraries of applications such as spreadsheets databases, in calculators, and in macro packages and plugins for many other applications such as the mathematical, scientific and technical applications.

There are many standalone applications that offer the thousands of the various units with conversions. For example, thefree software movementoffers a command line utilityGNU unitsfor GNU and Windows.^[7]TheUnified Code for Units of Measureis also a popular option.

Notes and references

^Béla Bodó; Colin Jones (26 June 2013).Introduction to Soil Mechanics.John Wiley & Sons. pp. 9–.ISBN 978-1-118-55388-6.
^Goldberg, David (2006).Fundamentals of Chemistry(5th ed.). McGraw-Hill.ISBN 978-0-07-322104-5.
^Ogden, James (1999).The Handbook of Chemical Engineering.Research & Education Association.ISBN 978-0-87891-982-6.
^"Dimensional Analysis or the Factor Label Method".Mr Kent's Chemistry Page.
^"Identity property of multiplication".Retrieved2015-09-09.
^Foot, C. J. (2005).Atomic physics.Oxford University Press.ISBN 978-0-19-850695-9.
^"GNU Units".Retrieved2024-09-24.

Notes

External links

Statutory Instrument 1995 No. 1804Units of measurement regulations 1995Fromlegislation.gov.uk
"NIST: Fundamental physical constants – Non-SI units"(PDF).Archived fromthe original(PDF)on 2016-12-27.Retrieved2004-03-15.
NIST Guide to SI UnitsMany conversion factors listed.
The Unified Code for Units of Measure
Units, Symbols, and Conversions XML Dictionary Archived2023-05-02 at theWayback Machine
"Instruction sur les poids et mesures républicaines – déduites de la grandeur de la terre, uniformes pour toute la République, et sur les calculs relatifs à leur division décimale"(in French)
Math Skills Review
A Discussion of Units
Short Guide to Unit Conversions
Canceling Units Lesson
Chapter 11: Behavior of GasesChemistry: Concepts and Applications,Denton independent school District

[BodóJones2013-1] Béla Bodó; Colin Jones (26 June 2013).Introduction to Soil Mechanics.John Wiley & Sons. pp. 9–.ISBN 978-1-118-55388-6.

[2] Goldberg, David (2006).Fundamentals of Chemistry(5th ed.). McGraw-Hill.ISBN 978-0-07-322104-5.

[3] Ogden, James (1999).The Handbook of Chemical Engineering.Research & Education Association.ISBN 978-0-87891-982-6.

[4] "Dimensional Analysis or the Factor Label Method".Mr Kent's Chemistry Page.

[5] "Identity property of multiplication".Retrieved2015-09-09.

[6] Foot, C. J. (2005).Atomic physics.Oxford University Press.ISBN 978-0-19-850695-9.

[7] "GNU Units".Retrieved2024-09-24.

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