Conversion of units

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 the universal gas law equation of PV = nRT, when:

• the pressure P is in pascals (Pa)
• the volume V is in cubic metres (m³)
• the amount of substance n is in moles (mol)
• the universal gas constant R is 8.3145 Pa⋅m³/(mol⋅K)
• the temperature T is in kelvins (K)

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

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 hitherto unknown 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, the Planck constant, a fundamental physical constant, was 'discovered' as a purely mathematical abstraction or representation that built on the Rayleigh–Jeans law for preventing the ultraviolet catastrophe. It was assigned and ascended to its quantum physical significance either in tandem or post mathematical dimensional adjustment – not earlier.

The factor-label method can convert only unit quantities for which the units are in a linear relationship intersecting at 0. (Ratio scale in Stevens's typology) Most units fit this paradigm. An example for which it cannot be used is the conversion between degrees Celsius and kelvins (or degrees Fahrenheit). 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, an affine transform (${\displaystyle x\mapsto ax+b}$, rather than a linear transform ${\displaystyle x\mapsto ax}$) 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, though this would yield the same formula at the end.

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

T[C] = (T[F] − 32) × 5/9.

To convert T[C] in degrees Celsius to T[F] in degrees Fahrenheit, this formula may be used:

T[F] = (T[C] × 9/5) + 32.

Starting with:

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

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

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

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

Or, which is just mathematically the same thing, multiply Z by unity, the product is still Z:

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

For example, you have an expression for a physical value Z involving the unit feet per second (${\displaystyle [Z]_{i}}$) and you want it in terms of the unit miles per hour (${\displaystyle [Z]_{j}}$):

Or as an example using the metric system, you have a value of fuel economy in the unit litres per 100 kilometres and you want it in terms of the unit microlitres per metre:

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

Assume that ${\displaystyle m=1\,{\text{Da}},\mu =k_{\text{B}}\cdot 1\,{\text{nK}}\,,}$ this gives

$${\displaystyle \xi ={\frac {\hbar }{\sqrt {2m\mu }}}=15.574\,\mathrm {\mu m} \,,}$$

which is our pre-factor.

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

This method is especially useful for programming and/or making a worksheet, where input quantities are taking multiple different values; For example, with the pre-factor calculated above, it is very easy to see that the healing length of ¹⁷⁴Yb with chemical potential 20.3 nK is ${\displaystyle \xi ={\frac {15.574}{\sqrt {174\cdot 20.3}}}\,{\text{μm}}=0.262\,{\text{μm}}}$.