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3.1 — Temperature and Heat
Put your hand on a metal railing and a wooden bench that have been sitting side by side outdoors all night. The metal feels colder. It is not colder — a thermometer laid on each reads the same number. What your hand is actually reporting is not temperature at all but the rate at which heat is leaving your skin, and metal pulls it away far faster than wood does.
That single mistake is the reason this chapter has to start by separating two words that everyday speech treats as the same thing.
Temperature is a property an object has on its own. You can measure it without reference to anything else.
Heat is not a property of anything. Heat is energy in transit, and it exists only while it is moving from one body to another because of a temperature difference. Once it arrives, it stops being heat and becomes internal energy. Asking "how much heat is in this cup of tea" is like asking how much rain is in a lake — the rain became lake water on arrival.
The zeroth law, and why a thermometer is possible at all
Before you can measure temperature you need to be sure that the idea is even coherent. Here is the thing that has to be true, and it is not obvious:
If body A is in thermal equilibrium with body C, and body B is also in thermal equilibrium with body C, then A and B are in thermal equilibrium with each other.
"Thermal equilibrium" means: put them in contact and nothing changes. No net energy flows either way.
That statement is the zeroth law of thermodynamics. It was named zeroth because it was recognised as a law only in the 1930s, decades after the first and second laws had been numbered, and it was too fundamental to be filed at the end.
Why does it matter? Because it is the entire licence for a thermometer to exist. A thermometer is body C. You touch it to A, wait, read it. You touch it to B, wait, read it. If the readings match you conclude A and B would not exchange heat if you put them together — but that conclusion only follows if the zeroth law is true. Without it, "the same temperature" would not be a transitive relation, thermometers would be meaningless, and every pair of objects would need to be tested directly against every other pair.
The zeroth law is what lets you replace "is in equilibrium with" — a relation between two bodies — by "has temperature T" — a single number attached to one body. That is a large gift and it cost nothing but noticing.
Reading a temperature: the three scales, and the one that is not arbitrary
To attach a number, pick something that changes reliably with hotness and calibrate it. Liquid mercury expands. A resistor's resistance rises. A gas pushes harder. Any of them works.
Celsius was set by Anders Celsius in 1742 using water: 0 at the ice point, 100 at the steam point, both at one atmosphere. Fahrenheit came earlier, from Daniel Gabriel Fahrenheit in 1724, with 0 at the coldest temperature he could reliably reproduce (a brine of ice, water and ammonium chloride) and 96 for roughly human body temperature. The conversion between them is:
T_F = \frac{9}{5}T_C + 32
which reads: to go from Celsius to Fahrenheit, stretch by nine fifths because a Fahrenheit degree is smaller, then shift by 32 because the zeros are in different places.
Both of those scales are agreements about convenient landmarks. Neither has a zero that means anything physically. But there is a scale whose zero is forced by nature, and finding it was a real discovery.
Take a fixed amount of gas in a sealed container of fixed volume, and measure its pressure at several temperatures. Plot pressure against Celsius temperature. You get a straight line. Now extend that line backwards, below the temperatures you actually measured, and ask where it would hit zero pressure.
Every gas gives the same answer: $-273.15\ ^\circ$C. Hydrogen, helium, nitrogen, carbon dioxide, any of them, whatever the pressure, whatever the amount. The lines have different slopes, and they all cross zero at the same place.
That is a strong hint. Pressure comes from molecules hitting the walls. Zero pressure means nothing is hitting the walls, which means the molecular motion has stopped. You cannot have less motion than none, so you cannot go below that temperature — not because it is forbidden by rule, but because there is nothing left to remove.
That point is absolute zero, and the scale that starts there is the kelvin:
T(\text{K}) = T(^\circ\text{C}) + 273.15
A kelvin is exactly the same size as a Celsius degree, so a change of 5 K and a change of 5 °C are identical. Only the zero moved. Notice there is no degree sign: it is "300 kelvin", not "300 degrees kelvin", because the kelvin is a unit like the metre, not a division of a scale.
Since the 2019 redefinition of the SI (Chapter 1.1), the kelvin is defined by fixing the Boltzmann constant at exactly k_B = 1.380649\times10^{-23} J/K. That constant is the exchange rate between temperature and energy, and Chapter 3.2 derives where it comes from.
What temperature is, one chapter early
The full argument is Chapter 3.2's job, but the result is worth having now because it makes everything else in this chapter make sense:
\boxed{\tfrac{1}{2}m\overline{v^2} = \tfrac{3}{2}k_BT}
Read aloud: one half m v-squared-bar equals three halves k-B T. On the left is the average kinetic energy of one molecule — m its mass, \overline{v^2} the average of the square of its speed. On the right is temperature multiplied by that exchange-rate constant, and the three is there because a molecule can move in three independent directions.
So temperature is average molecular kinetic energy, in disguise. Not total energy — average, per molecule. This is why a bathtub at 40 °C contains vastly more energy than a cup of tea at 90 °C while still being the cooler thing: temperature counts the average energy per molecule, and the bath simply has far more molecules.
The animation shows one molecule shaking, tumbling and drifting. Every molecule in the railing you touched is doing this. Warmth is that motion; cold is less of it. There is nothing else going on.
Thermal expansion, derived rather than stated
Warm a metal rod and it gets longer. The reason lives in the shape of the force between two neighbouring atoms.
Atoms in a solid sit in a valley of potential energy — pull them apart and they attract, push them together and they repel hard. The crucial detail is that the valley is not symmetric. The repulsive wall on the close side is steep, the attractive slope on the far side is gentle. So when an atom vibrates with more energy, it swings further out on the easy side than it does inwards on the hard side, and its average position moves outwards. More heat, more vibration, bigger average spacing, longer rod. Expansion is not the atoms getting bigger — it is the gaps getting bigger.
Now the formula. Over a modest range the extra length is proportional both to how much you heated it and to how long the rod was to start with. Double the temperature rise, double the extra length; double the rod, double the extra length. Writing that as an equation:
\Delta L = \alpha L_0 \Delta T
Read aloud: delta L equals alpha L-nought delta T — the change in length equals the coefficient of linear expansion times the original length times the temperature change. The symbol \alpha (alpha) is that coefficient, and it is a property of the material with units of "per kelvin", because the length units cancel. Steel is about 12\times10^{-6} per K, aluminium about 23\times10^{-6}, and Invar — a nickel–iron alloy invented for exactly this reason — about 1.2\times10^{-6}.
Area and volume follow without any new physics. Take a square of side L_0. After heating, each side is L_0(1+\alpha\Delta T), so the area is
A = L_0^2(1+\alpha\Delta T)^2 = A_0\left(1 + 2\alpha\Delta T + \alpha^2\Delta T^2\right)
The last term has \alpha^2 \approx 10^{-10} in it, which is a hundred-thousandth of the middle term for any temperature change you will meet, so it is dropped and we keep:
\Delta A = 2\alpha A_0 \Delta T
The same argument in three dimensions gives (1+\alpha\Delta T)^3 \approx 1 + 3\alpha\Delta T, so:
\Delta V = 3\alpha V_0 \Delta T = \gamma V_0 \Delta T, \qquad \gamma = 3\alpha
The coefficient of volume expansion is exactly three times the linear one. That is not a separate fact to memorise; it is the number 3 from cubing.
Worked example: the hole in the plate
A steel plate has a circular hole of diameter 10.000 cm at 20 °C. The plate is heated to 220 °C. What is the hole's diameter now? Take \alpha_{\text{steel}} = 12\times10^{-6} K⁻¹.
Almost everyone's first instinct is that the metal expands inwards and the hole shrinks. It does not, and here is the clean way to see why. Imagine the disc of metal that you drilled out is still sitting in place. Heat everything. That disc expands by \alpha L_0\Delta T like any other piece of steel, and the hole it occupies must expand with it, because they are in contact the whole time. Now remove the disc. Nothing about the hole changes when you remove it. So a hole expands exactly as if it were made of the surrounding material.
\Delta d = \alpha\, d_0\, \Delta T = (12\times10^{-6})(10.000)(200) = 0.024\ \text{cm}
d = 10.024\ \text{cm}
The hole is bigger. This is why a jar lid that will not budge comes loose under hot water — the metal lid expands more than the glass jar, and the gap opens.
The coil in the picture is two strips of different metals welded along their length. Heated, the higher-\alpha metal wants to be longer than the lower-\alpha one, but they are stuck together, so the only shape that satisfies both is a curve with the longer metal on the outside. That curl is the oldest thermostat there is: mount a contact on the end and the strip physically switches a circuit at a set temperature with no electronics of any kind. Old ovens, irons, kettles and car indicator flashers all worked on this.
Water, the exception that keeps fish alive
Nearly everything expands when heated. Water between 0 °C and 4 °C contracts when heated, so it is at its densest at 4 °C rather than at its freezing point.
The reason is hydrogen bonding, which Chapter 10.3 takes apart properly. In short: ice's crystal structure is unusually open, held that way by hydrogen bonds pointing in fixed directions, and it wastes space. Just above melting, some of that open structure survives in clumps. Warming from 0 to 4 °C breaks those leftover clumps and the molecules pack closer, which shrinks the volume. Above 4 °C ordinary thermal expansion wins and water behaves normally.
The consequence is that a pond freezes from the top down. Cold surface water sinks only until it reaches 4 °C; after that further cooling makes it less dense so it stays on top, freezes there, and the ice — less dense still — floats and insulates everything below. If water behaved like every other liquid, lakes would freeze solid from the bottom and almost nothing living in them would survive a winter.
Heat capacity: why some things are hard to warm up
Give the same amount of energy to a kilogram of water and a kilogram of iron and the iron gets far hotter. The proportionality that describes this is:
Q = mc\,\Delta T
Read aloud: Q equals m c delta T — the heat energy supplied equals mass times specific heat capacity times temperature change. Q is energy in joules. The symbol c is the specific heat capacity, the energy needed to raise one kilogram of the substance by one kelvin, measured in J kg⁻¹ K⁻¹.
Some values worth carrying:
| Substance | c (J kg⁻¹ K⁻¹) |
|---|---|
| Water (liquid) | 4186 |
| Ice | 2100 |
| Aluminium | 900 |
| Iron | 450 |
| Copper | 385 |
| Lead | 128 |
Water's 4186 is enormous — the largest of any common substance. Chapter 10.3 explains it: heating water means partly breaking hydrogen bonds, and that soaks up energy without raising the temperature, because the energy went into breaking bonds rather than into speeding molecules up.
That one number shapes the planet. Oceans absorb vast amounts of solar energy for a small temperature rise, which is why coastal towns have mild winters and mild summers while places far inland swing brutally between the two. It is why the sea is still cold in June after weeks of hot weather, and still warm in October. And it is why water is the working fluid in almost every cooling system ever built, from a car radiator to a nuclear plant.
At the other end, lead's 128 means the same joule raises lead's temperature 33 times more than water's. A lead sinker left in the sun gets painfully hot; a bucket of water beside it barely warms.
Worked example: the metal block in water
A 200 g block of copper at 95 °C is dropped into 500 g of water at 20 °C inside an insulated cup. What is the final temperature?
The whole method is one sentence: the heat leaving the copper equals the heat entering the water, because the cup lets nothing escape. Call the final temperature T. Both objects end at T, since they end in equilibrium.
Heat lost by copper:
Q_{\text{lost}} = m_c c_c (95 - T) = (0.200)(385)(95-T) = 77(95-T)
Heat gained by water:
Q_{\text{gained}} = m_w c_w (T - 20) = (0.500)(4186)(T-20) = 2093(T-20)
Set them equal:
77(95-T) = 2093(T-20)
7315 - 77T = 2093T - 41860
49175 = 2170T
T = 22.66\ ^\circ\text{C}
The water warmed by 2.7 degrees; the copper cooled by 72. That lopsidedness is the whole point — the water's high c and larger mass mean it barely notices. This method is called calorimetry, and it is how specific heats were measured in the first place: put in a known mass at a known temperature, measure the final temperature, solve for the unknown c.
Latent heat: the energy that changes nothing you can see
Put a pan of ice on a burner and watch a thermometer in it. The temperature climbs to 0 °C and then stops — and stays stopped, sometimes for minutes, while the burner pours in energy the whole time and the ice visibly melts. Only when the last of the ice is gone does the number start rising again. The same thing happens at 100 °C, and there the pause is far longer.
Where is the energy going? Into breaking the bonds that hold molecules in place, not into speeding them up. Temperature measures average kinetic energy, and during a phase change the average kinetic energy is not changing — the potential energy of the arrangement is. That hidden energy is why it was named latent, from the Latin for hidden, by Joseph Black in the 1760s.
Q = mL
with no \Delta T anywhere in it, because there is no temperature change. L is the specific latent heat, in joules per kilogram. For water:
- Latent heat of fusion (solid ↔ liquid), L_f = 334{,}000 J/kg
- Latent heat of vaporisation (liquid ↔ gas), L_v = 2{,}260{,}000 J/kg
Compare L_f = 334 kJ/kg against c = 4186 J/kg/K: melting one kilogram of ice at 0 °C takes as much energy as heating one kilogram of liquid water by 334000/4186 = 80 degrees. And boiling it takes another 2260 kJ/kg, which is enough to heat that water by 540 degrees if it were not busy leaving as steam.
That last number is why steam burns are so much worse than boiling-water burns. Water at 100 °C on your skin cools to 37 °C and delivers 4186\times63 = 264 kJ per kilogram. Steam at 100 °C first condenses on your skin, dumping the full 2260 kJ/kg before it starts cooling — nearly nine times as much energy, delivered instantly.
It is also why sweating works. Evaporating sweat pulls its latent heat out of your skin. One gram of sweat evaporated removes 2260 J. Losing half a litre in an hour of exercise removes roughly 1.1 MJ, which is comparable to the entire heat output of hard physical work — and it is why humidity makes heat unbearable, since if the air is already saturated the sweat cannot evaporate and the cooling channel is simply switched off.
Worked example: ice into water, the trap version
How much heat is needed to turn 50 g of ice at −10 °C into steam at 100 °C?
This has to be done in four stages, because the material behaves differently in each. The trap is doing it in one step with a single c.
Stage 1 — warm the ice from −10 °C to 0 °C. Use ice's specific heat, 2100:
Q_1 = (0.050)(2100)(10) = 1050\ \text{J}
Stage 2 — melt it at 0 °C. No temperature change:
Q_2 = (0.050)(334000) = 16{,}700\ \text{J}
Stage 3 — warm the water from 0 °C to 100 °C. Now use water's specific heat, 4186:
Q_3 = (0.050)(4186)(100) = 20{,}930\ \text{J}
Stage 4 — boil it at 100 °C.
Q_4 = (0.050)(2260000) = 113{,}000\ \text{J}
Q_{\text{total}} = 1050 + 16700 + 20930 + 113000 = 151{,}680\ \text{J} \approx 152\ \text{kJ}
Look at the proportions. Stage 4 alone is 74 % of the total. Getting the water from freezing to boiling — the part that feels like the whole job — is only 14 %. This is exactly why a kettle reaches boiling quickly and then takes ages to boil dry, and why leaving a pan on a rolling boil wastes energy: once it is at 100 °C, every extra joule goes into making steam, not into cooking anything faster.
Where this shows up in your life
Bridge expansion joints. A 100 m steel bridge deck swinging between −20 °C and +40 °C changes length by \Delta L = (12\times10^{-6})(100)(60) = 7.2 cm. If both ends were rigidly bolted, that 7.2 cm of growth would have nowhere to go and would appear instead as an enormous compressive stress, which buckles the deck. The finger-like metal joints you drive over are there to swallow it. Railways use the same trick, and before continuous welded rail the gaps between rail lengths were what made the clack-clack of an old train.
Your car's coolant is water with antifreeze precisely because of water's specific heat: nothing cheap carries more heat per kilogram out of an engine.
A frost-free freezer works on latent heat: it periodically warms its evaporator just enough to melt accumulated ice, which absorbs L_f per kilogram and then drains away.
Cooking pasta at a rolling boil versus a gentle simmer makes no difference to cooking time. Both are at 100 °C, and after that all extra power goes into latent heat of vaporisation — into steam leaving the pan, not into the food.
Fire walking is thermal conductivity plus heat capacity, not mysticism. Wood embers are hot but have low heat capacity per unit volume and conduct badly, so in the short time each footfall lasts, very little energy actually crosses into the foot. Try the same walk on a bed of red-hot steel bars and the physics gives an entirely different and much worse answer, for the same reason the railing felt colder than the bench.
What the next chapter fixes
This chapter used Q = mc\Delta T and \frac{1}{2}m\overline{v^2} = \frac{3}{2}k_BT as facts handed over from somewhere else. Chapter 3.2 goes underneath both. It starts with nothing but molecules bouncing off walls and Newton's laws from Part 1, and out of that comes the pressure of a gas, the meaning of temperature, the value of k_B, the whole ideal gas law, and a prediction of what c should be for a gas — a prediction which then fails in an interesting way that quantum mechanics had to fix.