Temperature

• 1. Introduction

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  Temperature is the number we reach for when we ask whether something is hot or cold. It is a measure that turns an everyday feeling into a quantity, one that reflects the average kinetic energy of the particles inside a system. It tells us how energy is spread across the microscopic motions of matter. Yet the lowest temperature ever obtained in a macroscopic system was 20 nanokelvin, achieved in 1995 at NIST. Somewhere below that lies a wall called absolute zero, a point that can be approached but never reached. Why can a body never be cooled to that final degree? Why do three different scales, marked in Celsius, Fahrenheit, and Kelvin, all claim to measure the same single quality? And how did a quantity rooted in the warmth of water and the chill of ice come to be defined by a fixed value of the Boltzmann constant? Those questions sit at the heart of one of science's most universal ideas.

• 2. The Three Scales

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  Zero degrees Celsius was set as the freezing point of water, and one hundred as its boiling point, both at atmospheric pressure at sea level. That hundred-degree interval gave the scale its old name, centigrade. It is an empirical scale, one that grew historically out of convenient reference points rather than deep theory. The Celsius scale is used for most temperature measurements across the world, with the exceptions of Belize, Myanmar, Liberia, and the United States.

  The Fahrenheit scale runs on a different set of marks, with water freezing at 32 and boiling at 212 at sea-level atmospheric pressure. It remains in common use in the United States for non-scientific applications. A shifted version of it, the Rankine scale, is an absolute scale built on the Fahrenheit increment, still used in chemical engineering and in thermodynamic disciplines such as combustion.

  The Kelvin scale stands apart as the one chosen for science. Its symbol is K, its unit name spelled with a lower-case k, and it is one of the seven base units of the International System of Units. A temperature increment of one degree Celsius is exactly the same as an increment of one kelvin. The two scales differ only by an exact offset of 273.15. Other historical scales have come and gone, among them the Delisle, Newton, Reaumur, and Romer scales.

• 3. The Unreachable Floor

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  Absolute zero sits at 0 kelvin, equal to minus 273.15 degrees Celsius, or minus 459.67 degrees Fahrenheit. It is the lowest point on the thermodynamic temperature scale, the temperature at which no energy can be removed from matter as heat. This impossibility is written into the third law of thermodynamics. Experiment can creep toward it but never arrive: the lowest temperature ever attained was 38 picokelvin, or 38 trillionths of a kelvin.

  Classically, a body at absolute zero has had all motion of its particles cease, leaving them at complete rest. But the quantum world refuses that perfect stillness. As predicted by the uncertainty principle, matter at absolute zero still holds quantum-mechanical zero-point energy. Matter there is in its ground state and contains no macroscopic thermal energy, though this zero-point motion does not enter the definition of absolute temperature.

  Negative numbers complicate the picture in a surprising way. On scales not referenced to absolute zero, a negative temperature simply means below the chosen zero point. Dry ice sublimates at minus 78.5 degrees Celsius, which is 194.6 kelvin on the absolute scale. The kinetic theory of temperature forbids a body of matter from taking negative values. Yet a subsystem of particle spins can be driven, by externally imposed force fields, into a strange virtual state. As its energy rises past the point of maximum entropy, its thermodynamic temperature leaps to positive infinity and then flips to negative infinity. Such negative temperatures are hotter than any positive temperature, and energy flows from them into the ordinary positive-temperature world.

• 4. Motion As Heat

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  Maxwell and Boltzmann built a kinetic theory of gases that explains what temperature is at the level of individual particles. It treats a gas as a swarm of many tiny bodies, all alike within a species, and accounts for pressure as the force of atoms striking the walls of their container. In a monatomic ideal gas, all of the energy is translational kinetic energy, the energy of particles moving in straight lines until they collide.

  The equipartition theorem assigns each classical degree of freedom of a freely moving particle an equal share of energy, set by the Boltzmann constant. Translational motion has three degrees of freedom, so the average translational kinetic energy of a particle works out to three-halves of the Boltzmann constant times temperature. This makes temperature directly proportional to the mean molecular kinetic energy, a special case that holds only in the classical limit of a perfect gas.

  Molecules richer than single atoms carry more ways to store energy. Oxygen, written O2, can rotate and vibrate as well as travel through space. Heating raises temperature by raising the average translational energy, but equipartition also pours energy into those vibrational and rotational modes. A diatomic gas therefore needs more energy to warm by a given amount than a monatomic gas, giving it a greater heat capacity. The ideal gas law itself, linking pressure, volume, and temperature, hinted at absolute zero long before kinetic theory arrived, because the relationship only holds on an absolute scale like Kelvin's.

• 5. Kelvin And Carnot

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  Kelvin's original work postulating an absolute temperature was published in 1848, built on the earlier work of Carnot and written before the first law of thermodynamics had been formulated. Carnot had no sound understanding of heat and no concept of entropy; he wrote of caloric, imagining that all of it passing from the hot reservoir flowed unchanged into the cold one. Kelvin claimed his scale was absolute because it was defined, in his words, "independently of the properties of any particular kind of matter". His definitive publication appeared in 1853, from a paper read in 1851.

  The heart of the definition is an idealized device, the Carnot engine, run in a fictive cycle so slow that the working body stays in equilibrium at every step. The engine draws heat from a hot reservoir, passes a lesser amount of waste heat to a cold reservoir, and delivers the difference as work. Because the cycle is imagined to run reversibly, the entropy taken in from the hot side equals the entropy given to the cold side, and that balance defines the ratio of the two absolute temperatures.

  For a long stretch of history the numbers were pinned down by making one reservoir a cell at the triple point of water, defined as exactly 273.16 kelvin. An analysis of the same Carnot engine shows that for a cold reservoir at 0 kelvin the efficiency would reach 100 percent, and below 0 kelvin it would exceed 100 percent, which violates the first law. That impossibility marks 0 kelvin as the minimum possible temperature.

• 6. Redefining The Kelvin

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  Since May 2019, the kelvin has been defined through particle kinetic theory and statistical mechanics, with its magnitude fixed by a conventionally chosen value of the Boltzmann constant. Before that, but since 1954, the International System of Units defined the kelvin as a thermodynamic temperature using two anchors: absolute zero, and the reliably reproducible triple point of water. The triple point was once defined as exactly 273.16 kelvin; today it is an empirically measured quantity.

  Until that 2019 change, the fixed point was not ordinary water but the triple point of Vienna Standard Mean Ocean Water, prepared with a specified blend of hydrogen and oxygen isotopes. That definition fixed the kelvin as precisely one part in 273.16 of the gap between absolute zero and the triple point. It made one kelvin exactly equal in size to one Celsius degree, and set the offset between their null points at 273.15. Since the 2019 redefinition, based on the Boltzmann constant, the scales are scarcely changed.

  The modern definition relies on physical effects with precise theoretical explanations. The speed of sound in a gas can be computed from its molecular character, temperature, pressure, and the Boltzmann constant, often more precisely than the thermodynamic state of water at its triple point can be measured. Black-body radiation offers another route, since the frequency of its maximum spectral radiance is directly proportional to temperature, a fact known as Wien's displacement law. Even the noise-power of an electrical resistor, called Johnson noise, rises in direct proportion to temperature and yields a reading.

• 7. Heat And Its Laws

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  Raising the temperature of water by one kelvin requires 4186 joules per kilogram. That figure is its specific heat, the heat needed to warm a unit quantity of substance by one unit of temperature. Heat capacity in general is the ratio of heat transferred to the size of the change it causes, whether that change is a rise in pressure, a chemical reaction, or a shift in temperature at constant volume.

  The zeroth law of thermodynamics, named that way only since axiomatic treatments became customary in the 1930s, ties these ideas together. If two systems are each in thermal equilibrium with a third, then they are also in equilibrium with each other. Galileo and Newton found that there are indefinitely many empirical temperature scales, and the zeroth law says they all measure the same quality. Every correctly calibrated thermometer reads one and the same temperature for a body in internal equilibrium.

  The second law selects a single preferred scale from that crowd. It states that any process results in either no change or a net increase in the entropy of the universe, a tendency understood through probability. In a long run of coin tosses, perfectly ordered outcomes of all heads or all tails are rare, while combinations near fifty-fifty dominate and grow more likely with each toss. Systems drift toward maximum disorder for the same reason. From this, statistical mechanics defines entropy as the Boltzmann constant times the logarithm of the number of microstates, making temperature the quantity that governs how heat flows from a system of higher temperature to one of lower.

• 8. Where The Picture Breaks

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  Fang and Ward were among the first to report temperature discontinuities as large as 7.8 kelvin at the surface of evaporating water droplets. Their measurement struck at a common belief that temperature must vary continuously across space and time. That assumption holds only in equilibrium; outside it, at interfaces such as a metal and non-metal boundary or a liquid-vapour surface, temperature can jump abruptly.

  The scale of these jumps reaches down to the mean free path of molecules, typically a few micrometers in gases at room temperature. At such interfaces, the abrupt change in vibrational and thermal properties prevents heat from transferring instantly, so a uniform equilibrium temperature never forms. Coarse macro-scale measurements average out this microscopic information and can hide the discontinuities entirely, sometimes producing misleading results. Far from being mere anomalies, these discontinuities have sharpened the understanding of heat transfer at small scales.

  At the opposite extreme, plasma physics handles temperatures so high they are expressed as energy, in units tied to the electronvolt, where 1 eV/kB equals 11605. The table of comparisons stretches from the Sun's visible surface at 5778 kelvin to the gamma-ray heat of CERN's proton-on-nucleus collisions at 10 trillion degrees Celsius. Between those poles sits the cosmic microwave background, measured in 2013 at 2.7260 kelvin, the faint warmth left over and spread across the whole sky.