Temperature scales: Celsius, Fahrenheit and Kelvin
Hot and cold can be felt but not seen. How do you put numbers on temperature so that every thermometer, anywhere, gives the same reading?
Where you are 4 in Temperature
Why this exists
A thermometer is a thin tube of liquid that climbs when it warms and sinks when it cools. On its own it only says “higher” or “lower”. Two tubes made in two places climb by different amounts, so a scratch on one glass means nothing to the owner of the other.
To say how hot an oven should be, or how cold tonight will get, every tube has to be marked the same way. That is what a temperature scale does. It is a unit for warmth: an agreed step, the degree, counted from an agreed mark. This lesson builds the three scales in use today, Celsius, Fahrenheit and Kelvin, and shows why each is pinned to two temperatures that nature repeats exactly.
Definition
A temperature scale is made by choosing two fixed points, temperatures that come out the same every time and everywhere, giving each a number, and splitting the gap between them into equal steps called degrees.
The Celsius scale puts 0 at the temperature at which pure water freezes and 100 at the temperature at which it boils, both at the air pressure found at sea level. The gap is split into 100 equal steps, and one step is one degree Celsius, written 1^\circ\text{C}. The steps carry on past both fixed points: above 100 for anything hotter than boiling water, and below 0 for anything colder than ice, where the readings are negative numbers. Eight degrees below freezing is -8^\circ\text{C}, read “negative eight degrees Celsius” or “eight degrees below zero”.
The Fahrenheit scale, used for everyday temperatures in the United States, puts the freezing point of water at 32^\circ\text{F} and the boiling point at 212^\circ\text{F}, again at sea level. Between them lie 212 - 32 = 180 equal steps. Its zero is not tied to water at all: 0^\circ\text{F} is 32 Fahrenheit degrees colder than freezing.
The Kelvin scale starts at absolute zero, the coldest temperature there can be, where the tiny particles a substance is made of have the least motion they can have. Absolute zero is -273.15^\circ\text{C}. A step on the Kelvin scale is exactly the size of a Celsius degree and is called a kelvin, written \text{K} with no degree sign. Water freezes at 273.15\text{ K} and boils at 373.15\text{ K}. Nothing is colder than 0\text{ K}, so no Kelvin reading is ever negative.
Intuition
A thermometer is a number line stood on end, with the numbers increasing upwards. Three things follow from that picture.
Reading. First work out what one small mark is worth, as in reading a thermometer. Below zero, count down from zero: there the numbers grow in size going down, the reverse of above.
Comparing. Higher on the scale is warmer, and it is also the greater number. So -2^\circ\text{C}, which sits above -15^\circ\text{C}, is the warmer of the two, and -15 < -2. The size of a negative reading, its absolute value, is how far below zero it is, so among readings below zero the larger size is the colder one.
Change. Because the steps are equal, a change in temperature is the number of steps between two readings, found by counting along the scale. A move across zero is counted in two pieces, to zero and then past it: from -10^\circ\text{C} up to 15^\circ\text{C} is 10 steps to zero and 15 more, a rise of 25 degrees. Later the same changes are found by subtracting integers.
Results
| {}^\circ\text{C} | {}^\circ\text{F} | \text{K} | |
|---|---|---|---|
| Water boils | 100 | 212 | 373.15 |
| Water freezes | 0 | 32 | 273.15 |
| Absolute zero | -273.15 | -459.67 | 0 |
| Freezing to boiling | 100 steps | 180 steps | 100 steps |
A kelvin and a Celsius degree are the same size. Both scales take 100 steps from freezing to boiling, so a rise of some number of Celsius degrees is a rise of the same number of kelvins. The two scales differ only in where zero sits, 273.15 steps apart.
A Celsius degree is bigger than a Fahrenheit degree. The same gap takes 100 Celsius steps and 180 Fahrenheit steps, so fewer, larger steps on the Celsius side. Since 100 : 180 = 5 : 9, every 5 Celsius degrees span the same as 9 Fahrenheit degrees.
The three zeros are in different places. 0^\circ\text{C} is freezing water, 0^\circ\text{F} is colder than that, and 0\text{ K} is the coldest anything can be. A reading means nothing without its scale, and a positive Fahrenheit reading can still be below freezing.
Turning any Celsius reading into Fahrenheit, or back, takes a rule that both stretches and shifts, and that is its own lesson.
Why it is true
One fixed point is not enough. Suppose two makers both mark 0 where their liquid stands in melting ice, and stop there. Each must still choose how far apart the degree marks go. If one spaces them 2 mm apart and the other 3 mm, then when both liquids climb 30 mm, one tube says 15 and the other 10. They agree at freezing and nowhere else. One point fixes where the scale starts, not how big a step is.
Two fixed points pin both. Mark melting ice and boiling water on each tube and split the gap into 100 equal parts. Now the size of a degree is set by nature, the gap between two temperatures, and not by the maker, so the tubes agree at every reading.
The fixed points must repeat. Melting ice stays at one temperature until all of it has melted, and boiling water at one air pressure stays at one temperature while it boils. A warm bath or a cold morning would make a poor fixed point, since no two are alike.
Kelvin fits the same pattern. Its zero, absolute zero, is set by nature, and its step is given the size of a Celsius degree, which places the freezing point of water 273.15 steps up.
Worked examples
Reading a thermometer below zero
At the Antarctic research station, Iris reads the outdoor thermometer in Figure 3. What is the temperature, and how many degrees below freezing is it?
The numbered marks go up in tens: -40, -30, -20, -10, 0, 10. Between two numbered marks there are 5 spaces, so each small mark is worth 10 \div 5 = 2 degrees.
The liquid stops between -30 and -20. Counting down from the -20 mark it is 3 small marks lower, which is 3 \times 2 = 6 degrees further down, so 20 + 6 = 26 degrees below zero. The reading is -26^\circ\text{C}.
Freezing is 0^\circ\text{C}, so the reading is 26 degrees below freezing: the distance from 0 to -26, which is |-26| = 26.
Check. From the -30 mark the liquid is 2 small marks up, 4 degrees, and 4 degrees above thirty below zero is 30 - 4 = 26 below zero.
Which days were colder than the freezer?
At noon on five days the station’s readings were -14^\circ\text{C}, -31^\circ\text{C}, -9^\circ\text{C}, -24^\circ\text{C} and 2^\circ\text{C}. (a) Put them in order from coldest to warmest. (b) The station’s food freezer is kept at -20^\circ\text{C}. On which days was it colder outside than in the freezer?
Solution(a) Only 2 is above zero, so it is the warmest. The four negatives sit below zero, and the larger the size, the lower on the scale:
(b) Colder means lower on the scale. -31 is 31 degrees below zero and -24 is 24 below, both further down than 20 below, so -31 < -20 and -24 < -20. The readings -14 and -9 are less far below zero than -20, so they sit above it and are warmer. Two days were colder than the freezer.
Counting a change
(a) On the space station, Joss takes a sample out of a freezer at -18^\circ\text{C} and leaves it until it reads 7^\circ\text{C}. By how many degrees did it warm? (b) At the Antarctic station the temperature is 4^\circ\text{C} at noon and -15^\circ\text{C} at midnight. How far did it fall? (c) One morning the reading rises from -31^\circ\text{C} to -14^\circ\text{C}. By how much?
Solution(a) The move crosses zero, so count in two pieces. From -18 up to 0 is 18 steps, and from 0 up to 7 is 7 more: 18 + 7 = 25. The sample warmed 25 degrees.
(b) Down from 4 to 0 is 4 steps, and down from 0 to -15 is 15 more: 4 + 15 = 19. The temperature fell 19 degrees.
(c) Both readings are below zero, so there is no crossing. -31 is 31 below zero and -14 is 14 below zero, so the steps between them number 31 - 14 = 17. A rise of 17 degrees.
Freezing and boiling on two scales
At a hot spring near the coast, close to sea level, Lotte has a thermometer marked in both Celsius and Fahrenheit. (a) She puts it in a pot of spring water at a full boil. What do the two scales read? (b) She pushes it into melting snow beside the pool. What do they read? (c) Which is bigger, a Celsius degree or a Fahrenheit degree? (d) After dark the Fahrenheit side reads 25^\circ\text{F}. Is that above or below freezing?
Solution(a) Boiling water at sea level is a fixed point of both scales: 100^\circ\text{C} and 212^\circ\text{F}.
(b) Melting snow is water at its freezing point: 0^\circ\text{C} and 32^\circ\text{F}.
(c) Between those two marks lie 100 Celsius steps but 212 - 32 = 180 Fahrenheit steps. The same gap is cut into fewer pieces on the Celsius side, so each piece is bigger. A Celsius degree is the bigger one.
(d) Freezing on the Fahrenheit side is 32^\circ\text{F}, and 25 < 32, so the reading is 32 - 25 = 7 Fahrenheit degrees below freezing, although the reading is positive.
Celsius and Kelvin
(a) On a winter night the station records -40^\circ\text{C}. What is that in kelvin? (b) Water at the vent of the hot spring is 65^\circ\text{C}. What is that in kelvin? (c) On the space station, Milo’s experiment cools a sample to 3\text{ K}. What is that in degrees Celsius?
SolutionKelvin steps are the same size as Celsius degrees, and freezing is 0^\circ\text{C} and 273.15\text{ K}. So count steps from the freezing point.
(a) -40^\circ\text{C} is 40 steps below freezing, and 40 steps below 273.15 is 273.15 - 40 = 233.15. The reading is 233.15\text{ K}.
(b) 65^\circ\text{C} is 65 steps above freezing: 273.15 + 65 = 338.15, so 338.15\text{ K}.
(c) 3\text{ K} is 3 steps above absolute zero, which is 273.15 degrees below zero Celsius. Climbing 3 steps from there leaves 273.15 - 3 = 270.15 degrees below zero, so the sample is at -270.15^\circ\text{C}.
Check. From 3\text{ K} up to freezing at 273.15\text{ K} is 270.15 steps, and freezing is 0^\circ\text{C}, so the sample is 270.15 degrees below zero.
Marking a blank thermometer
Iris has a glass tube of liquid with no marks. At sea level she puts it in melting ice and marks where the liquid stops, then in boiling water and marks again. The two marks are 150 mm apart. (a) How far apart are the degree marks of a Celsius scale on this tube? (b) Indoors the liquid stands 33 mm above the ice mark. What is the temperature? (c) Outside it stands 30 mm below the ice mark. What is the temperature?
Solution(a) The two marks are 0^\circ\text{C} and 100^\circ\text{C}, and the scale splits the gap into 100 equal steps: 150 \div 100 = 1.5 mm per degree.
(b) Each degree is 1.5 mm, so 33 mm is 33 \div 1.5 = 22 degrees above freezing: a reading of 22^\circ\text{C}.
(c) 30 \div 1.5 = 20, so the liquid is 20 degrees below freezing: -20^\circ\text{C}.
The ice mark alone could not answer any of these. It says where 0 is, but only the second fixed point says that a degree is 1.5 mm on this tube.
Boundaries
The fixed points need pure water at sea-level air pressure. Higher up the air presses less and water boils below 100^\circ\text{C}; salt water freezes below 0^\circ\text{C}. The scale’s numbers do not move. It is the water that changes.
Nothing is colder than absolute zero. The Integers go down forever, but temperatures stop at -273.15^\circ\text{C}, which is -459.67^\circ\text{F} and 0\text{ K}. A reading of -300^\circ\text{C} is impossible.
Compare readings on one scale. 20^\circ\text{F} is a larger number than 0^\circ\text{C} but the colder temperature. Higher means warmer only on the same scale.
Differences mean something; ratios on Celsius and Fahrenheit do not. Their zeros are chosen marks, so 20^\circ\text{C} is not twice as warm as 10^\circ\text{C}: in kelvin those are 293.15\text{ K} and 283.15\text{ K}, nowhere near double.
Today’s definitions are finer. Scientists now fix the kelvin with a constant of physics, and the Celsius scale is the Kelvin scale with its zero moved up 273.15 steps. On that definition water at sea level boils a few hundredths of a degree below 100^\circ\text{C}, a difference no kitchen thermometer can show.
Where this goes next
Reading, ordering and counting on a thermometer carry over to integers on a number line and to readings like -273.15, which are negative decimals. Counting steps becomes finding a temperature change by subtraction. And because Celsius and Fahrenheit differ in step size and in where zero sits, moving between them needs a stretch and a shift: converting between Celsius and Fahrenheit.
Exercises
Reading and comparing
- Read thermometers A, B and C in Figure 4.
- Which of the three readings in Figure 4 are below the freezing point of water? Of A and B, which is colder?
- (a) Write “twenty-eight degrees below zero Celsius” as a number with its unit. (b) Write -41^\circ\text{C} in words.
- Order from coldest to warmest: -19^\circ\text{C}, 6^\circ\text{C}, -27^\circ\text{C}, -2^\circ\text{C}, 0^\circ\text{C}, -33^\circ\text{C}.
- Write < or > between each pair. (a) -16 and -21 (b) -1 and 1 (c) -40 and -4
- On a Celsius thermometer marked every 2 degrees, the liquid stops 2 small marks above the -50 mark. What is the reading?
Counting changes
Count on a thermometer to find each change, and say whether it is a rise or a fall.
- From -23^\circ\text{C} to 5^\circ\text{C}.
- From 3^\circ\text{C} to -18^\circ\text{C}.
- From -27^\circ\text{C} to -11^\circ\text{C}.
- The temperature is -12^\circ\text{C} and falls 17 degrees. What is it now?
- The temperature is -21^\circ\text{C} and rises 26 degrees. What is it now?
The three scales
- How many Fahrenheit degrees lie between the freezing and boiling points of water? How many kelvins?
- (a) Is 30^\circ\text{F} above or below the freezing point of water, and by how many Fahrenheit degrees? (b) Which is colder, -5^\circ\text{F} or 0^\circ\text{C}?
- A hot spring pool is at 42^\circ\text{C}. Write that in kelvin.
- A winter night at the station reaches -50^\circ\text{C}. Write that in kelvin.
- An instrument on the space station is cooled to 20\text{ K}. What is that in degrees Celsius?
- Is 280\text{ K} above or below the freezing point of water, and by how many degrees?
- Which is warmer, 260\text{ K} or -10^\circ\text{C}?
Building a scale
- Kian marks a blank tube in melting ice and in boiling water at sea level, and the marks are 200 mm apart. (a) How far apart are the Celsius degree marks? (b) What is the temperature when the liquid is 74 mm above the ice mark? (c) And when it is 34 mm below it?
- On the space station, Joss makes up a scale with water freezing at 0 and boiling at 50. (a) What does it read halfway between freezing and boiling? (b) Is one of its degrees bigger or smaller than a Celsius degree, and how many Celsius degrees does it span? (c) What does it read at 10^\circ\text{C}?
Reasoning
- Two thermometers are each marked with 0 at melting ice and have no other fixed point. Explain why they can disagree at every other temperature.
- Explain why a Kelvin reading is never negative, while Celsius and Fahrenheit readings can be.
- 15^\circ\text{F} is a larger number than 0^\circ\text{C}. Explain why it is nevertheless the colder temperature.
- Why would “the temperature of a warm bath” make a poor fixed point? Why does the boiling point in the definition of the Celsius scale say “at sea level”?
- Explain why a rise of 10 Celsius degrees is a rise of 10 kelvins but not of 10 Fahrenheit degrees. How many Fahrenheit degrees is it?
Answers
- A: -22^\circ\text{C}, 2 marks of 1 degree below -20. B: -35^\circ\text{C}, since 20 \div 4 = 5 degrees a mark and the liquid is 1 mark above -40. C: 24^\circ\text{F}, 2 marks of 2 degrees above 20.
- All three. A and B are below 0^\circ\text{C}, and C is below 32^\circ\text{F}, since 24 < 32. B is colder than A: -35 < -22.
- (a) -28^\circ\text{C}. (b) “Forty-one degrees below zero Celsius”, or “negative forty-one degrees Celsius”.
- -33, -27, -19, -2, 0, 6. The negatives come first, the largest size lowest.
- (a) -16 > -21 (b) -1 < 1 (c) -40 < -4. The number higher on the scale is the greater.
- -46^\circ\text{C}. Two marks are 4 degrees, and 4 above fifty below zero is 50 - 4 = 46 below zero.
- A rise of 28: 23 steps up to zero, then 5 more.
- A fall of 21: 3 steps down to zero, then 18 more.
- A rise of 16: no crossing, and from 27 below zero to 11 below is 27 - 11 = 16 steps.
- -29^\circ\text{C}: 12 below zero and 17 further down is 12 + 17 = 29 below zero.
- 5^\circ\text{C}: the first 21 degrees of the rise reach zero, and 26 - 21 = 5 remain.
- 180 Fahrenheit degrees, since 212 - 32 = 180, and 100 kelvins, since 373.15 - 273.15 = 100.
- (a) Below, by 2 Fahrenheit degrees: freezing is 32^\circ\text{F} and 32 - 30 = 2. (b) -5^\circ\text{F} is colder. 0^\circ\text{C} is the freezing point, which is 32^\circ\text{F}, and -5 < 32.
- 315.15\text{ K}: 42 steps above freezing, 273.15 + 42 = 315.15.
- 223.15\text{ K}: 50 steps below freezing, 273.15 - 50 = 223.15.
- -253.15^\circ\text{C}: 20 steps above absolute zero leaves 273.15 - 20 = 253.15 steps below freezing.
- Above, by 6.85 degrees: 280 - 273.15 = 6.85, so it is 6.85^\circ\text{C}.
- -10^\circ\text{C} is warmer. It is 10 steps below freezing, 273.15 - 10 = 263.15, so 263.15\text{ K}, and 263.15 > 260.
- (a) 2 mm, since 200 \div 100 = 2. (b) 37^\circ\text{C}, since 74 \div 2 = 37. (c) -17^\circ\text{C}, since 34 \div 2 = 17 degrees below freezing.
- (a) 25, half of 50. (b) Bigger: 50 of its steps cover the same gap as 100 Celsius degrees, so each spans 100 \div 50 = 2 Celsius degrees. (c) 5, since 10 Celsius degrees is 10 \div 2 = 5 of its steps.
- A single fixed point says where 0 goes but not how far apart the degree marks are. Each maker chooses a different spacing, so the same climb of the liquid gives different numbers on the two tubes. Only at the shared mark, 0, must they agree.
- Zero on the Kelvin scale is absolute zero, the coldest temperature there is, so every temperature is at or above it and its reading is 0 or more. The zeros of Celsius and Fahrenheit are chosen marks with colder temperatures below them, and those get negative numbers.
- The two readings are on different scales. 0^\circ\text{C} is the freezing point, which on the Fahrenheit scale is 32^\circ\text{F}. Since 15 < 32, 15^\circ\text{F} is 17 Fahrenheit degrees below freezing, colder than 0^\circ\text{C}.
- A fixed point must come out the same every time. Warm baths differ from one to the next, so two makers using one would mark different places. Water’s boiling point depends on air pressure: up a mountain water boils at a lower temperature, so the fixed point names the pressure, the one at sea level.
- A kelvin is the same size as a Celsius degree, so a rise covers the same number of steps on both. A Fahrenheit degree is smaller: 180 of them span what 100 Celsius degrees span, 9 for every 5. So a rise of 10 Celsius degrees is 10 \times 180 \div 100 = 18 Fahrenheit degrees.