Units conversion by factor-label

# Units conversion by factor-label

Overview
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 and being able to perform such conversions is an important skill to acquire. The factor-label method, also known as the unit-factor method or dimensional analysis
Dimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...

, is a widely used approach for performing such conversions.
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Encyclopedia
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 and being able to perform such conversions is an important skill to acquire. The factor-label method, also known as the unit-factor method or dimensional analysis
Dimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...

, is a widely used approach for performing such conversions. It is also used for determining whether the two sides of a mathematical equation involving dimensions have the same dimensional units.

## The factor-label method for converting units

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
Miles per hour
Miles per hour is an imperial unit of speed expressing the number of statute miles covered in one hour. It is currently the standard unit used for speed limits, and to express speeds generally, on roads in the United Kingdom and the United States. It is also often used to express the speed of...

can be converted to meters per second by using a sequence of conversion factors as shown below:
10 mile 1609 meter 1 hour meter
-- ---- × ---- ----- × ---- ------ = 4.47 ------
1 hour 1 mile 3600 second second

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.

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.

As a more complex example, the concentration
Concentration
In chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...

of nitrogen oxides (i.e., NOx) in the flue gas
Flue gas
Flue gas is the gas exiting to the atmosphere via a flue, which is a pipe or channel for conveying exhaust gases from a fireplace, oven, furnace, boiler or steam generator. Quite often, the flue gas refers to the combustion exhaust gas produced at power plants...

from an industrial furnace
Furnace
A furnace is a device used for heating. The name derives from Latin fornax, oven.In American English and Canadian English, the term furnace on its own is generally used to describe household heating systems based on a central furnace , and sometimes as a synonym for kiln, a device used in the...

can be converted to a mass flow rate
Mass flow rate
Mass flow rate is the mass of substance which passes through a given surface per unit time. Its unit is mass divided by time, so kilogram per second in SI units, and slug per second or pound per second in US customary units...

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/106 volumes
NOx molar mass := 46 kg/kgmol (sometimes also expressed as 46 kg/kmol)
Flow rate of flue gas := 20 cubic meters per minute = 20 m³/min
The flue gas exits the furnace at 0 °C temperature and 101.325 kPa absolute pressure.
The molar volume of a gas at 0 °C temperature and 101.325 kPa is 22.414 m³/kgmol.

10 m³ NOx 20 m³ gas 60 minute 1 kgmol NOx 46 kg NOx 1000 g g NOx
--- ------ × -- ------ × -- ------ × ------ --------- × -- --------- × ---- -- = 24.63 -----
106 m³ gas 1 minute 1 hour 22.414 m³ NOx 1 kgmol NOx 1 kg hour

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 Universal Gas Law
Gas laws
The early gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between the pressure, volume and temperature of a sample of gas could be obtained which would hold for all gases...

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
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

is 8.3145 Pa·m³/(mol·K)
• the temperature T is in kelvins (K)

mol (Pa)(m³) K
(Pa)(m³) = ----- × ---------- × ---
1 (mol)(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.

## Limitations

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
Fahrenheit
Fahrenheit is the temperature scale proposed in 1724 by, and named after, the German physicist Daniel Gabriel Fahrenheit . Within this scale, the freezing of water into ice is defined at 32 degrees, while the boiling point of water is defined to be 212 degrees...

). 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 (), not a linear transform (). 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.

For instance, the freezing point of water is 0° in Celsius and 32° in Fahrenheit, and a 5° change in Celsius correspond 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.
[°F = 1.8(°C) + 32°]

To convert Celsius to Fahrenheit, simply plug in the known numbers in the above formula.

[°C = (°F-32°) ÷ 1.8]

To convert Fahrenheit to Celsius (Centigrade), plug the known temperature into the above formula.

EX. °F = 1.8(-40°C) + 32° = -40°F (Identical temperature point in °C and °F)

EX. °C = (98.6°F-32°) ÷ 1.8 = 37°C (Known standard body temperature in °C and °F)

• Conversion of units
Conversion of units
Conversion of units is the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors.- Process :...

• Dimensional analysis
Dimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...

• ISO 31
ISO 31
International Standard ISO 31 was the most widely respected style guide for the use of physical quantities and units of measurement, and formulas involving them, in scientific and educational documents worldwide...

• Units of measurement
Units of measurement
A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of...