Temperature Measurement and Physical Properties of Matter

Lessons

Senior 1 · Physics

Temperature Measurement and Physical Properties of Matter

5 lessons

Lesson 2 of 5

Temperature Scales and Converting Between Them

25 min

98.6 or 37?

A nurse at a health centre in Rwamagana writes a patient's temperature as 37 °C. A thermometer imported from the United States reads 98.6 for the same patient, a scientist's report says 310 K, and an old French book on a library shelf says 29.6. All four are correct: they are the same temperature on four different temperature scales. In this lesson you will learn how each scale is built and how to change from one to another.

Definition

A fixed point is a temperature that can be reproduced exactly, and is used to mark a thermometer's scale.

The lower fixed point (ice point) is the temperature of pure melting ice at standard atmospheric pressure.
The upper fixed point (steam point) is the temperature of steam from pure water boiling at standard atmospheric pressure.

The gap between them is the fundamental interval. Each scale divides it into a different number of equal parts called degrees.

Experiment: marking the fixed points

Put an unmarked mercury or alcohol thermometer in a funnel of pure crushed melting ice. When the thread stops moving, mark its level: the ice point.
Hang the thermometer in the steam just above boiling water in a flask (a hypsometer), not in the water itself. Mark the level: the steam point.
Divide the length between the marks into 100 equal parts to make a Celsius scale.

The thermometer is held in the steam, not the water, because dissolved impurities raise the boiling temperature of the water but not the temperature of the steam. On a mountain the steam point is lower, which is why it is defined at standard atmospheric pressure.

The four scales

Figure 2.1: The same three temperatures on four scales. The mercury columns all stand at the same height: it is the same temperature, only the numbers differ.

Scale	Symbol	Ice point	Steam point	Divisions between them	Absolute zero
Celsius (Anders Celsius, 1742)	°C	0	100	100	−273
Fahrenheit (Daniel Fahrenheit, 1724)	°F	32	212	180	about −459
Kelvin (Lord Kelvin, 1848)	K	273	373	100	0
Réaumur (René de Réaumur, 1730)	°Ré (°R in some books)	0	80	80	about −218

Definition

Absolute zero is the lowest possible temperature, at which the particles of a substance have the least possible energy. It is 0 K, or about −273 °C. The Kelvin scale, the SI scale, starts at absolute zero, so it has no negative temperatures, and a kelvin is the same size as a degree Celsius:

T(\mathrm{K}) = \theta(^\circ\mathrm{C}) + 273

One rule for every conversion

Any temperature is the same fraction of the way from the ice point to the steam point on every scale. That fraction is (reading − ice point) divided by (steam point − ice point):

\frac{\theta_C - 0}{100} = \frac{\theta_F - 32}{180} = \frac{T - 273}{100} = \frac{\theta_R - 0}{80}

Picking two parts of this equation at a time gives the everyday formulas:

\theta_F = \frac{9}{5}\theta_C + 32 \qquad \theta_C = \frac{5}{9}(\theta_F - 32)

T = \theta_C + 273 \qquad \theta_R = \frac{4}{5}\theta_C \qquad \theta_C = \frac{5}{4}\theta_R

Worked example 1: body temperature on all four scales

Convert normal body temperature, 37 °C, to kelvin, degrees Fahrenheit and degrees Réaumur.

T = 37 + 273 = 310\,\mathrm{K}

\theta_F = \frac{9}{5} \times 37 + 32 = 66.6 + 32 = 98.6\,^\circ\mathrm{F}

\theta_R = \frac{4}{5} \times 37 = 29.6\,^\circ\mathrm{R\acute{e}}

Worked example 2: an oven from a foreign recipe

A cake recipe says “bake at 350 °F”. Find the oven temperature in °C.

\theta_C = \frac{5}{9}(\theta_F - 32) = \frac{5}{9}(350 - 32) = \frac{5}{9} \times 318 = 176.7\,^\circ\mathrm{C} \approx 177\,^\circ\mathrm{C}

Subtract 32 FIRST, then multiply by $\frac{5}{9}$ .

Worked example 3: from Réaumur to kelvin

An old record gives the temperature of a hot spring as 60 °Ré. Find it in °C and in K.

\theta_C = \frac{5}{4}\theta_R = \frac{5}{4} \times 60 = 75\,^\circ\mathrm{C}

T = 75 + 273 = 348\,\mathrm{K}

Worked example 4: the one temperature where two scales agree

At what temperature do the Celsius and Fahrenheit scales give the same number?

Put $\theta_F = \theta_C = x$ in $\theta_F = \frac{9}{5}\theta_C + 32$ :

x = \frac{9}{5}x + 32 \quad\Rightarrow\quad -\frac{4}{5}x = 32 \quad\Rightarrow\quad x = -40

So $-40\,^\circ\mathrm{C} = -40\,^\circ\mathrm{F}$ .

Warning

Common conversion mistakes

Kelvin has no degree sign: write 310 K, not 310 °K.
To go from °C to K, ADD 273; to go from K to °C, subtract 273.
From °F to °C, subtract 32 before multiplying by $\frac{5}{9}$ ; from °C to °F, multiply by $\frac{9}{5}$ before adding 32.
A temperature change is different: a rise of 10 °C is a rise of 10 K, but a rise of 18 °F. Never add or subtract 32 or 273 to a change.

Which scale, where?

Rwanda, like most of the world, uses the Celsius scale for weather, health and cooking: Meteo Rwanda forecasts give temperatures in °C, and Kigali's afternoons are usually around 25 to 28 °C because the city lies about 1560 m above sea level. Scientists use kelvin. The United States still uses Fahrenheit, so imported ovens, fridges and clinical thermometers sometimes show °F. Réaumur is no longer used, but you will meet it in older books and exam questions.

Tip

Check with the fixed points. After converting, test your formula on 0 °C and 100 °C: it must give 32 and 212 °F, 273 and 373 K, 0 and 80 °Ré. If it does not, the formula is wrong.

Temperature (CrashCourse Physics). Watch for how the Celsius, Fahrenheit and Kelvin scales were set up, and why absolute zero is the natural starting point.

✦

Practice

Define the lower and upper fixed points of a thermometer.
Convert 25 °C to kelvin, °F and °Ré.
Convert 350 K to °C.
Convert 104 °F (a high fever) to °C.
Convert 40 °Ré to °C and °F.
The temperature of a room rises by 15 °C. What is the rise in K and in °F?
Explain why the Kelvin scale has no negative temperatures.
A pupil converts 50 °F to °C like this: $\frac{5}{9} \times 50 - 32 = -4.2\,^\circ\mathrm{C}$ . Find the mistake and the correct answer.

Show answersHide answers

Lower: the temperature of pure melting ice at standard pressure. Upper: the temperature of steam from pure water boiling at standard pressure.
298 K; $\frac{9}{5} \times 25 + 32 = 77\,^\circ\mathrm{F}$ ; $\frac{4}{5} \times 25 = 20\,^\circ\mathrm{R\acute{e}}$
$350 - 273 = 77\,^\circ\mathrm{C}$
$\frac{5}{9}(104 - 32) = \frac{5}{9} \times 72 = 40\,^\circ\mathrm{C}$
$\frac{5}{4} \times 40 = 50\,^\circ\mathrm{C}$ ; $\frac{9}{5} \times 50 + 32 = 122\,^\circ\mathrm{F}$
15 K; $15 \times \frac{9}{5} = 27\,^\circ\mathrm{F}$
It starts at absolute zero (0 K), the lowest possible temperature, so no temperature can be below 0 K.
She multiplied before subtracting 32. Correct: $\frac{5}{9}(50 - 32) = \frac{5}{9} \times 18 = 10\,^\circ\mathrm{C}$ .

Key points

Fixed points: ice point (melting pure ice) and steam point (steam over pure boiling water), at standard pressure.
Ice and steam points: 0 and 100 °C; 32 and 212 °F; 273 and 373 K; 0 and 80 °Ré.
$T = \theta_C + 273$ ; $\theta_F = \frac{9}{5}\theta_C + 32$ ; $\theta_R = \frac{4}{5}\theta_C$ .
Absolute zero, 0 K or about −273 °C, is the lowest possible temperature.

Sources and credits

13.1: Temperature, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Consulted for the physics; no text was copied.
Photo: CelsiusKelvinThermometer.jpg by Martinvl, Wikimedia Commons, licensed under CC BY-SA 3.0.
Video: Temperature: Crash Course Physics #20, CrashCourse, embedded from YouTube.

Written by Ganzaa: the lesson text, the diagram, tables, worked examples and practice exercises.

198.6 or 37?

2Experiment: marking the fixed points

3The four scales

4One rule for every conversion

5Which scale, where?

98.6 or 37?

Experiment: marking the fixed points

The four scales

One rule for every conversion

Which scale, where?