Factor-label conversion of units
Factor-label conversion of units, also known as the unit-factor method or dimensional analysis, is a widely used method for converting one set of dimensional units to another set of equivalent units.
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 set of units to another is often somewhat complex and being able to perform such conversions is an important skill to acquire.
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 metres per second by using a sequence of conversion factors as shown below:
Thus, when the units mile and hour are cancelled out and the arithmetic is done, 10 miles per hour converts to 4.47 metres per second.
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:
- NOx concentration
- = 10 parts per million by volume = 10 ppmv = (10 volumes of NOx) / (106 volumes of flue gas)
- NOx molar mass
- = 46 g/mol = 46 kg/kmol
- Volumetric flow rate of flue gas (expressed at 0 °C and 101.325 kPa)
- = 20 cubic metres per minute = 20 m³/min
- The molar volume of a gas at 0 °C and 101.325 kPa
- = 22.414 m³/kmol
Using the factor-label method:
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 ppmv converts 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 guarantee that the equation is correct, but having different units on the two sides of an equation does guarantee that the equation is wrong.
For example, check the ideal gas law equation of P·V = n·R·T, when:
- The pressure P is in pascals (Pa)
- The volume V is in cubic meters (m³)
- The amount of substance n is in moles (mol)
- The universal gas law constant R is in Pa·m³/(mol·K)
- The temperature T is in kelvins (K)
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.
The factor-label method can convert only unit quantities which have a constant ratio to each other. Most units fit this paradigm.
An example for which it cannot be used is the conversion between kelvins and degrees Celsius or between degrees Celsius and degrees Fahrenheit. Between kelvins and degrees Celsius, there is a constant difference rather than a constant ratio, while between degrees Celsius and degrees Fahrenheit there is both a constant difference and a constant ratio.