In chemistry, the factor-label method is a technique used to solve complex calculations involving conversions between different units of measurement. This method involves orienting all factors involved in the calculation so that their labels will appropriately cancel and/or combine to yield the desired unit in the result. The basic approach is the same for all calculations, regardless of the details; all factors involved in the calculation must be properly oriented to ensure that their labels will combine correctly to produce the desired unit in the result.

One common application of this method is in solving temperature conversions. Temperature is measured using different scales, such as Celsius and Fahrenheit. To convert between these scales, a linear equation must be derived based on the reference temperatures used to define each scale. For example, the freezing temperature of water on the Celsius scale is 0°C, while its boiling temperature is 100°C. On the Fahrenheit scale, the freezing point of water is defined as 32°F, and the boiling point is 212°F. The space between these two points on a Fahrenheit thermometer is divided into 180 equal parts.

The linear equation relating Celsius and Fahrenheit temperatures can be derived by representing the Celsius temperature as x and the Fahrenheit temperature as y, then computing the slope of the line using the following formula:

m = (y - 32°F) / (100°C - 0°C)

where m is the conversion factor. The equation relating the two scales can then be written as:

T (°F) = + 32 °F + (5 / 9) × T (°C)

Rearranging this equation yields the form useful for converting from Fahrenheit to Celsius:

T (°C) = (9 / 5) × T (°F) - 32

As mentioned earlier in this chapter, the SI unit of temperature is the kelvin. Unlike the Celsius and Fahrenheit scales, the kelvin scale is an absolute temperature scale in which 0 K corresponds to the lowest temperature that can theoretically be achieved. The early 19th-century discovery of the relationship between a gas's volume and temperature suggested that the volume of a gas would be zero at -273.15 °C. In 1848, British physicist William Thompson proposed an absolute temperature scale based on this concept. The freezing temperature of water on this scale is 273.15 K, and its boiling temperature is 373.15 K.

The equations for converting between the kelvin and Celsius temperature scales are:

T (K) = T (°C) + 273.15

T (°C) = T (K) - 273.15

In conclusion, the factor-label method is a useful technique for solving complex calculations involving conversions between different units of measurement. By properly orienting all factors involved in the calculation, this method ensures that the desired unit will be produced in the result. The linear equation relating Celsius and Fahrenheit temperatures can be derived using the reference temperatures used to define each scale, and the kelvin scale is an absolute temperature scale with 0 K corresponding to the lowest temperature theoretically achievable.