Thermodynamics · Chapter 5

Absolute Zero and the Third Law

How cold can anything get? There is a bottom: absolute zero, 0 kelvin, or −273.15 °C. Labs have come within 38 trillionths of a degree of it, but no one can ever quite arrive. Here you’ll climb down a cooling ladder one rung at a time and see why the last rung never comes. You’ll cool a cloud of atoms by letting the hottest ones escape, until they merge into a new state of matter. And you’ll explore a thermometer that runs from the coldest place ever made to the heart of a fusion reactor.

Start from

Live numbers

Missions

    What you’re seeing

    The sample. A crystal of a paramagnetic salt: each arrow is an atom that acts like a tiny compass needle, a spin. When it’s warm, the spins point every which way. A magnet (the coil) pulls them into line, and so does the spins’ pull on each other, shown by the setting b.

    Rung 1: magnetize. With the heat switch closed, the sample touches a cold bath. The magnet turns on and the spins line up. Lining up is a drop in entropy, and it releases heat, which flows into the bath, so the temperature stays the same. On the graph, the dot drops straight down from the pink curve to the green one.

    Rung 2: demagnetize. Now the switch opens, so no heat can get in or out, and the magnet turns off slowly. The spins’ entropy can’t change, so they stay just as lined up as before, and with a weaker field that’s only possible at a lower temperature. The crystal gets colder. On the graph, the dot slides left to the pink curve.

    The ladder. Each round divides the temperature by the same number, set by the magnet and by b. Here, the cold sample then becomes the bath for the next round. (A real lab uses separate stages, each cooled by the one before.) On the log thermometer at the bottom, every rung is the same size, and there are always more rungs below.

    The third law

    In 1906 the German chemist Walther Nernst noticed, from careful measurements of chemical reactions, that the entropy changes in a reaction get smaller and smaller as the temperature drops toward zero. That became the third law of thermodynamics:

    S → 0 as T → 0the entropy of a perfect crystal approaches zero (or a constant minimum) as its temperature approaches absolute zero

    In 1912 Nernst put it another way, which turns out to say the same thing. In the words of OpenStax: absolute zero temperature cannot be reached through any finite number of cooling steps. The graph shows why. Both entropy curves, magnet on and magnet off, sink to zero as the temperature does. Each rung of the staircase fits between them, and as the curves come together the rungs must shrink. You can take as many as you like, and there are always more.

    If the spins didn’t pull on each other at all (b = 0), the magnet-off curve would stay at ln 2 all the way down, and one demagnetization would land exactly at zero. Real matter never behaves like that: something always makes the entropy fall to zero first. That’s the third law.

    Worked example: counting rungs

    With the starting settings (B = 1 T, b = 0.1 T), each round multiplies the temperature by √(0.1² ÷ (1² + 0.1²)) = 0.0995, almost exactly ÷10.

    1. From 4.2 K: 0.42 K, then 42 mK, 4.2 mK, 0.42 mK. Below 1 mK after 4 rounds.
    2. Rounds to reach 1 μK: you need 4.2 K × 0.0995n < 10−6 K, so n > ln(4.2 × 106) ÷ ln(10.05) = 15.25 ÷ 2.31 = 6.6, which means 7 rounds.
    3. Rounds to reach 0 K: 0.0995n is never zero for any n. Infinitely many.
    Try this: watch the “heat squeezed out last round” number. It shrinks fast: by the time the sample is at a few millikelvin, the spins are already almost perfectly lined up, so there’s almost no entropy left to squeeze out. That’s the third law at work. In a real lab the tiny heat leaking in through wires and supports wins long before the model’s numbers get absurd.

    Why heat capacities shrink toward zero

    The heat capacity says how much energy it takes to warm something by one degree. Near absolute zero it shrinks toward zero, which the third law requires. Think of the Einstein solid from Chapter 3: energy comes in packets, and when the temperature is far below one packet’s worth, almost no atom can take one. Adding a little heat makes almost nothing happen. In 1912 Peter Debye showed that for ordinary solids the heat capacity near zero falls like T3. A smaller heat capacity sounds helpful (less heat to remove), but the entropy you must remove shrinks too, so every method of cooling loses its grip as zero approaches.

    Cooling atoms by evaporation

    Blow on a spoonful of hot soup and it cools: the fastest molecules escape as vapor, carrying away more than their share of energy, and the ones left behind settle to a lower temperature. Physicists cool atoms to a billionth of a degree the same way. Laser light first slows a cloud of atoms to about ten millionths of a degree; then a magnetic trap holds them like marbles in a bowl, and the walls of the bowl are lowered bit by bit so the most energetic atoms spill over the rim.

    The cloud starts with 300 million rubidium-87 atoms at 50 μK in a trap that makes them swing back and forth 100 times a second. Lower the rim slowly and the atoms that escape take lots of energy with them; collisions spread out what’s left, and the cloud cools while keeping enough atoms. Drop the rim too fast and you just throw atoms away without giving the rest time to settle. Too slowly, and stray atoms of leftover gas in the vacuum chamber knock your atoms out first. Below a temperature Tc that depends on how many atoms remain, they pile into the lowest energy state together: a Bose–Einstein condensate.

    Worked example: how slow is slow?

    The typical (root-mean-square) speed of an atom is v = √(3kT/m). A rubidium-87 atom has mass m = 86.909 u = 1.443 × 10−25 kg.

    1. At room temperature, 293 K: v = √(3 × 1.380649 × 10−23 × 293 ÷ 1.443 × 10−25) ≈ 290 m/s, faster than a jet airliner.
    2. At 100 nK: v = √(3 × 1.380649 × 10−23 × 10−7 ÷ 1.443 × 10−25) ≈ 5.4 mm/s, slower than a crawling ant.

    The first one. At 10:54 a.m. on June 5, 1995, at JILA in Boulder, Colorado, Eric Cornell and Carl Wieman’s team made the first Bose–Einstein condensate: about 2,000 rubidium atoms below 170 billionths of a degree above absolute zero, made exactly this way, with laser cooling first and evaporation second. Satyendra Nath Bose and Albert Einstein had predicted this state of matter in 1924–25. Cornell and Wieman shared the 2001 Nobel Prize in Physics with Wolfgang Ketterle of MIT, who made condensates of sodium soon after. In 2018 NASA’s Cold Atom Lab made them on the International Space Station.

    What’s stylized here: the dots are classical particles, each standing for many atoms. A condensate is a quantum effect that this model can’t produce, so when the cloud is colder than Tc = 0.94 ħωN1/3/k (the formula for an ideal gas in this kind of trap), the page draws a bright peak whose share of the atoms is 1 − (T/Tc)3. Its shape is only a picture.

    Strange things near zero

    Superconductors

    In 1908 Heike Kamerlingh Onnes in Leiden became the first to liquefy helium, at about 4.2 K. In 1911 he found that mercury cooled to 4.2 K loses all electrical resistance: a current can flow in a loop with no battery at all. He won the 1913 Nobel Prize in Physics. Some materials found later, such as YBCO, superconduct below about 92 K, warm enough for liquid nitrogen (77 K).

    Superfluid helium

    Below 2.17 K, liquid helium-4 becomes a superfluid: it flows with no friction at all, through gaps too small for any ordinary liquid. Pyotr Kapitsa, and separately John Allen and Don Misener, discovered it in 1937.

    Bose–Einstein condensates

    Thousands to millions of atoms in one single quantum state, behaving like one giant “super-atom”. You can make one in the trap above (in stylized form).

    The cosmic thermometer

    Every mark is a real temperature with its source. Drag to slide, use the buttons, a mouse wheel or two fingers to zoom, and tap a mark to read about it. Each step along the scale is ten times hotter than the last: from 38 trillionths of a kelvin to 150 million degrees, the scale covers 19 powers of ten.

    All the marks as a list
    TemperatureWhatSource
    38 pKColdest ever made: a falling rubidium condensate, Bremen, 2021 (an effective temperature)APS Physics
    100 nKNASA’s Cold Atom Lab on the space station, 2018NASA
    170 nKFirst Bose–Einstein condensate, JILA, 1995NIST
    10 μKLaser cooling of rubidium, before evaporationNIST
    6 mKCUORE’s cubic meter of copper, 2014Wikipedia
    0.25 KFirst magnetic cooling, Giauque and MacDougall, 1933Wikipedia
    1 KBoomerang Nebula, coldest natural place knownWikipedia
    2.17 KHelium-4 becomes superfluidWikipedia
    2.725 KCosmic microwave backgroundNASA
    4.2 KHelium boils; mercury superconductsWikipedia
    48 KPluto’s mean surface (−225 °C)NASA
    77 KLiquid nitrogen boilsWikipedia
    92 KYBCO superconducts below thisWikipedia
    184 KColdest air on Earth, Vostok, 1983 (−89.2 °C)NASA
    194.6 KDry ice (−78.5 °C)Wikipedia
    273.15 KWater freezes (0 °C)NIST SP 330
    310 KHuman body (37 °C)Wikipedia
    373 KWater boils at sea levelWikipedia
    737 KVenus’s surface (464 °C)NASA
    5,800 KThe Sun’s surface (5,500 °C)NASA
    2 million KThe Sun’s coronaNASA
    15 million KThe Sun’s coreNASA
    150 million KITER’s fusion plasma (see Magnetic Bottles for Fusion)ITER
    Common mistake: “At absolute zero, all motion stops”

    It’s a natural picture: temperature measures jiggling, so zero temperature should mean no jiggling. But atoms obey quantum mechanics, and a quantum particle held in a small space can never sit perfectly still. Its lowest possible energy, the zero-point energy, is not zero. Even at absolute zero, atoms keep this leftover motion. You can see the result in helium: its atoms are so light, and pull on each other so weakly, that their zero-point motion keeps it liquid all the way down to absolute zero. It only freezes if you squeeze it at about 25 times normal air pressure. Absolute zero means the lowest possible energy, not no motion.

    The kelvin scale

    In 1848 William Thomson, later Lord Kelvin, proposed a temperature scale that starts at the true bottom, absolute zero, instead of at freezing water. Today’s unit is named for him. One kelvin is the same size as one degree Celsius, so 0 K = −273.15 °C = −459.67 °F, and water freezes at 273.15 K.

    From 1954 the kelvin was defined by fixing the temperature of water’s triple point (where ice, liquid water and vapor coexist) at exactly 273.16 K. Since May 20, 2019, it has been defined instead by giving Boltzmann’s constant an exact value:

    k = 1.380649 × 10−23 J/Kexactly, by definition: one kelvin is the temperature change that changes kT by 1.380649 × 10−23 joule

    So a temperature is now tied directly to an energy, the energy of jiggling atoms, and not to any one substance. That matters most at the extremes, like the very cold temperatures on this page.

    Worked example: converting

    Liquid nitrogen boils at 77 K. In Celsius: 77 − 273.15 = −196 °C. In Fahrenheit: −196.15 × 9/5 + 32 = −321 °F. The first Bose–Einstein condensate, 170 nK, was 0.000000170 K: in Celsius, −273.149999830 °C.

    A little history

    Nernst. Walther Nernst stated his heat theorem in 1906 and his unattainability principle in 1912. He won the 1920 Nobel Prize in Chemistry for his work in thermochemistry.

    Magnetic cooling. Peter Debye (1926) and William Giauque (1927) each proposed cooling by demagnetizing a paramagnetic salt, the trick in the ladder above. Giauque and D. P. MacDougall did it in 1933 and reached 0.25 K. Salts reach about 1 to 100 millikelvin; demagnetizing the much weaker magnets inside atomic nuclei reaches about a microkelvin or below.

    Atoms. Laser cooling and evaporative cooling, developed in the 1980s and 1990s, opened up the nanokelvin world, and the first condensate followed in 1995.

    Where you see it

    • MRI scanners in hospitals use superconducting magnets kept cold by liquid helium.
    • Cold atom clocks and sensors use laser-cooled atoms. NASA’s Cold Atom Lab studies them in orbit, where they can float freely.
    • Magnetic refrigerators, like the ladder above, cool instruments in labs, including detectors on some space telescopes.
    • Space itself is not at absolute zero: the leftover glow of the Big Bang keeps it at 2.725 K, as the thermometer shows.
    • Engines: absolute zero is also why no engine can be 100% efficient. See Chapter 4.

    How the simulation works

    The ladder. The sample is an ideal crystal of spin-½ atoms, each with a magnetic moment of one Bohr magneton (μB = 9.2740100657 × 10−24 J/T, CODATA 2022). The spins’ pull on each other is modeled as an extra field b, so each spin feels Beff = √(B² + b²). With x = μBBeff/kT, the entropy per spin is S/Nk = ln(2 cosh x) − x tanh x. Since S depends only on Beff/T, demagnetizing with S fixed gives Tafter = Tbefore × b/√(B² + b²). The heat given to the bath is TΔS. Leaks of heat from outside, the crystal’s own vibrations and the limits of real salts are left out, which is why the model can go absurdly cold.

    The trap. 3,000 model atoms, each standing for 100,000 real ones at the start, move exactly along their orbits in a round harmonic trap. Random pairs collide, keeping momentum and energy; the collision rate grows with density and speed, like N/T in this trap, up to a limit. Any atom whose energy is above the rim leaves. Each frame, an atom also has a 1-in-4,000 chance of being knocked out by leftover gas. When fewer than 900 model atoms remain, each is split in two, with the copy turned by random rotations that keep its energy, so the picture stays full while each model atom stands for fewer real ones.

    How it was checked (Node.js tests):

    • Entropy formula: S/Nk = ln 2 at x = 0 and matches ln(2 cosh 1) − tanh 1 to 12 digits; written so it never overflows (at x = 50 it gives 4 × 10−42).
    • Ladder: after each demagnetization the entropy matched the magnetized value to 10−16. With B = 1 T and b = 0.05 T each round multiplies T by 0.0499; from 4.2 K: 0.21 K, 10.5 mK, 0.52 mK, … ; 1 mK after 3 rounds, 1 μK after 6; still above zero after 30 rounds. The heat capacity at b = 0.05 T fell from 0.44Nk at 30 mK to 3 × 10−26 Nk at 1 mK.
    • Trap, with no escape and no losses: total energy kept to 8 × 10−16 over 400 frames of collisions; kinetic and potential temperatures agreed (0.995 and 1.000), and ⟨v4⟩/⟨v2⟩2 = 1.670, against 5/3 for a Maxwell distribution. Splitting kept the mean energy per atom exactly.
    • Evaporation: lowering the rim from 500 μK to 500 nK over 300 frames (5 s) left 381 atoms and no condensate; over 900 frames, 3 × 105 atoms condensed; over 1,800 frames, 3 × 106; over 7,200 frames, fewer again (2.5 × 106) because of the background losses.

    Check yourself

    1. Each round of a cooling ladder multiplies the temperature by 0.1. You start at 1 K. How many rounds does it take to reach exactly 0 K?

      Show the answer

      After n rounds the temperature is 0.1n K, which is never zero. That’s the unattainability form of the third law.

    2. What does the third law say happens to the entropy of a perfect crystal as it approaches absolute zero?

      Show the answer

      It approaches zero (or a constant minimum). Both curves on the graph sink to zero, which is why the cooling steps keep shrinking.

    3. Why does blowing on hot soup cool it, and how does the same idea cool atoms?

      Show the answer

      Evaporation removes the most energetic particles. Lowering the walls of an atom trap lets the hottest atoms spill out in just the same way.

    4. At absolute zero, what happens to the motion of atoms?

      Show the answer

      Quantum particles can never be perfectly still. That’s why helium stays liquid at absolute zero unless it’s squeezed hard.

    5. Since 2019, what defines the size of one kelvin?

      Show the answer

      k = 1.380649 × 10−23 J/K exactly. Before that, the kelvin was tied to the triple point of water.

    6. Which is colder: deep space (the cosmic microwave background) or the first Bose–Einstein condensate?

      Show the answer

      2.725 K ÷ 170 nK ≈ 16 million. Labs on Earth (and in orbit) are the coldest places we know of.

    Sources

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