AcademyTemperature and Heat
Academy
Temperature Measurement Scales
Level 1 - Physics topic page in Temperature and Heat.
Principle
A temperature scale is a linear calibration rule that assigns numbers to reproducible thermal states.
Celsius and Fahrenheit use different zero points and different degree sizes, so actual temperatures and temperature intervals must be converted differently.
Notation
\(T_C\)
Celsius temperature
\(\mathrm{deg\,C}\)
\(T_F\)
Fahrenheit temperature
\(\mathrm{deg}\)
\(\Delta T_C\)
Celsius temperature interval
\(\mathrm{K}\)
\(\Delta T_F\)
Fahrenheit temperature interval
\(\mathrm{deg}\)
Method
Celsius and Fahrenheit agree on two fixed reference states for water at standard atmospheric pressure, but assign different numbers to them.
Linear scale
\[T_F=aT_C+b\]
Use a linear map because equal scale divisions represent equal temperature intervals.
Freezing point
\[32=a(0)+b\Rightarrow b=32\]
Boiling point
\[212=a(100)+32\]
Scale slope
\[a=\frac{212-32}{100}=\frac{9}{5}\]
Conversion
\[T_F=\frac{9}{5}T_C+32\]
For an interval, the offset does not apply. A change of \(10\\,\\mathrm\{^\\circ C\}\) is not a change of \(50\\,\\mathrm\{^\\circ F\}\); only the scale factor matters.
Interval conversion
\[\Delta T_F=\frac{9}{5}\Delta T_C\]
Reverse interval
\[\Delta T_C=\frac{5}{9}\Delta T_F\]
Rules
These are the compact temperature-scale conversions.
Celsius to Fahrenheit
\[T_F=\frac{9}{5}T_C+32\]
Fahrenheit to Celsius
\[T_C=\frac{5}{9}(T_F-32)\]
Interval conversion
\[\Delta T_F=\frac{9}{5}\Delta T_C\]
Reverse interval
\[\Delta T_C=\frac{5}{9}\Delta T_F\]
Examples
Question
Convert
\[77\,\mathrm{^\circ F}\]
to Celsius.Answer
\[T_C=\frac{5}{9}(77-32)=25\,\mathrm{^\circ C}\]
Checks
- Use the offset only for an actual temperature, not for a temperature change.
- A Celsius degree and a kelvin have the same interval size.
- A Fahrenheit degree is smaller than a Celsius degree.
- Always check whether the question asks for a temperature or a temperature interval.