Temperature and Thermometers

Step out of an air-conditioned Delhi Metro carriage onto a station platform in May and your body reacts in less than a second: a wall of heat wraps around you, your skin begins to sweat, and the air feels thick. Step off the train in Leh in January and the opposite happens — the cold air bites your nose before you have walked two metres. You feel something, immediately, and your feeling is reliable enough that you reach for a jacket or a water bottle without thinking. But what exactly is your skin measuring?

It is not sensing "heat" directly. The Delhi platform and your skin both contain the same kind of molecules, jiggling around; your skin's nerve endings do not have tiny thermometers in them. What they detect is the direction and rate at which thermal energy is flowing across your skin's surface. In Delhi, energy is flowing into your skin from the air — so you feel hot. In Leh, energy is flowing out of your skin into the air — so you feel cold. The quantity that decides which way the flow goes, and how fast, is temperature.

This article builds what temperature actually is, starting from what it does: sort objects into "hotter" and "cooler" so that you can predict which way heat will move when they touch. That is a surprisingly deep idea, and once you have it, the three temperature scales, the zeroth law, and the workings of every kind of thermometer — from a clinical mercury one to an infrared gun at airport arrivals — all follow.

What temperature does: the direction of heat flow

Pour hot tea into a cold steel tumbler and wait. The tumbler warms up; the tea cools down. Wait long enough and they reach the same final temperature — somewhere in between. This is not an accident of tea and tumblers: it is a universal rule. Two bodies placed in contact, isolated from everything else, always evolve toward a common temperature. They do not drift apart. They do not settle at different temperatures. They meet in the middle (weighted by their specific heats and masses, which you will meet in calorimetry, but never mind that for now).

What is moving between them? Heat — a flow of energy from the hotter body to the cooler one. Heat is not stuff stored inside bodies (that's a common misconception that textbooks from the 19th century unfortunately cemented). Heat is energy in transit, from one body to another, driven by a temperature difference. No difference, no flow.

The operational definition. Temperature is the property that decides the direction of heat flow. If heat flows spontaneously from A to B, then A is hotter than B. If no heat flows either way, A and B are at the same temperature.

That's it. That is the entire physical content of the word "temperature" before any numerical scale gets involved. Before anyone invented a thermometer, before anyone agreed on degrees or zeros, this is what the word meant: a label that lets you predict which way energy will flow when two bodies meet.

Thermal equilibrium

When two bodies have come to the same temperature and no more heat flows between them, they are in thermal equilibrium. Pour chai into a kulhar and put the kulhar on a wooden table. The chai is hot; the kulhar is warm but cooler; the table is cooler still. Over a few minutes, the chai cools, the kulhar warms up a little, then both cool together as the table warms infinitesimally and the room air carries energy away. After some time — maybe fifteen minutes — everything in the system has reached room temperature, and heat stops flowing. The chai is in thermal equilibrium with the kulhar, with the table, with the air.

Thermal equilibrium is the end state of any isolated collection of bodies. Wherever you start — warm tea and cold milk, an ice cube in warm water, a red-hot iron rod in a bucket of oil — the system will evolve to a single common temperature. Nothing in classical physics is more reliable.

The zeroth law: why one number suffices

Here is a subtle point. The operational definition above only compares pairs of bodies. To say "A is hotter than B" you need to bring them into thermal contact and see which way heat flows. But in everyday life, you assign a single temperature to each body ("the chai is 85 °C", "the room is 25 °C") without needing to pair it with anything. Why is that meaningful?

Because nature is consistent in a specific way, codified by what physicists call the zeroth law of thermodynamics:

Zeroth law. 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.

Put differently: thermal equilibrium is transitive. If A matches C, and B matches C, then A matches B — nothing more to check.

This sounds obvious. It is not. It is an empirical fact about the universe — nothing in mathematics forces it to be true. Imagine a world where three bodies could all pairwise not be in equilibrium, always sending a small heat current around in a loop A \to B \to C \to A forever. Such a world would not allow a single "temperature" to be assigned to anything: you would have to keep a whole table of pairwise thermal relationships. The zeroth law says our universe is not like that. Thermal equilibrium carves all bodies into equivalence classes, and each class gets a single label — a temperature.

This is what makes a thermometer possible. When you stick a thermometer into your dal pot and wait for it to equilibrate, the thermometer does not actually measure the dal's temperature directly — it measures its own temperature after equilibrating with the dal. The zeroth law is what guarantees that the thermometer, the dal, and anything else you touch with the same thermometer now all share one number, read off the device. Without the zeroth law, thermometers would be meaningless.

The name is a historical accident: the first, second, and third laws of thermodynamics were named before physicists realised the transitivity property was an even more basic law, logically prior to all of them. Rather than renumber, they just called it the zeroth law. It is the most taken-for-granted law in physics and the most quietly indispensable.

Hot, cold, and the scales

Once you have "temperature" as a single number per body, you need to pick units. The physical quantity does not care — you could label the ice point of water as 0, or 32, or 273.15, or any other number — as long as you pick a scale and stick to it. Three scales are in common use.

The Celsius scale

Anders Celsius (never mind his biography) picked two easily-reproducible states of water and assigned them nice round numbers:

• 0\,^\circC: the temperature at which pure water freezes at standard atmospheric pressure (the ice point).
• 100\,^\circC: the temperature at which pure water boils at standard atmospheric pressure (the steam point).

Divide the interval between them into 100 equal parts (the fixed points are both reproducible to within a small fraction of a degree in any laboratory on Earth), and you have the Celsius scale. A Delhi summer afternoon touches 48\,^\circC; a Leh winter night can drop to -30\,^\circC. Human body temperature is close to 37\,^\circC. A cup of chai poured from a kettle is about 90\,^\circC. These numbers have become part of how you think about weather, fever, and cooking.

The Fahrenheit scale

The Fahrenheit scale was calibrated against two different reference points (a salt-brine mixture and human body temperature, roughly) and ended up with 32\,^\circF for the ice point and 212\,^\circF for the steam point. The interval between the same two points is therefore divided into 212 - 32 = 180 equal parts on the Fahrenheit scale versus 100 on Celsius — so one Celsius degree equals 180/100 = 9/5 Fahrenheit degrees. To convert:

Why: the factor 9/5 converts the size of a degree (because 180 F-degrees span the same range as 100 C-degrees), and the +32 shifts the zero because the two scales' zeros are in different places (the Fahrenheit ice point is 32\,^\circF, the Celsius ice point is 0\,^\circC).

A quick sanity check: a Delhi summer day at 48\,^\circC corresponds to T_F = (9/5)(48) + 32 = 86.4 + 32 = 118.4\,^\circF — uncomfortably consistent with how heatwaves get reported in American newspapers. The US is the only large economy that still uses Fahrenheit for everyday weather; most of the world, including India, uses Celsius.

The Kelvin scale — the one nature uses

The Celsius and Fahrenheit scales both set their zero points at historically convenient places (freezing water, brine mixtures). Those choices are arbitrary. The Kelvin scale, introduced in the 19th century once the behaviour of gases was properly understood, sets its zero at a physically meaningful place: the temperature at which all thermal motion would cease.

The argument comes from observations of gases, which you will see derived in full in Thermal Expansion of Gases. Here is the shape of it. Take a fixed amount of an ideal gas at constant pressure and plot its volume against Celsius temperature. You get a straight line. Extrapolate the line backwards — past the temperature at which the gas would liquefy, past anything you can actually measure — and the line hits zero volume at exactly -273.15\,^\circC. Repeat the experiment with any gas, at any pressure, and the extrapolated zero comes out the same: -273.15\,^\circC. That cannot be a coincidence. The same number keeps appearing because it is a fact about how thermal motion works, not about any particular gas.

Absolute zero is defined as this temperature: -273.15\,^\circC. The Kelvin scale sets absolute zero as its zero and uses the same size of degree as Celsius (for historical compatibility and for agreement about the triple point of water, fixed at 273.16 K). So:

Why: no multiplicative factor, because a degree is the same size on both scales — the scales differ only by where their zero is placed. Add 273.15 to go from Celsius to Kelvin; subtract it to go the other way.

One delicate point: kelvin is an absolute temperature unit, so you do not write "300\,^\circK" (no degree symbol) and you never say "three hundred degrees Kelvin" — the unit is just kelvin, written 300 K. Room temperature is \approx 300 K. Boiling water is 373.15 K. A Leh winter at -30\,^\circC is 243.15 K. The surface of the sun is about 5800 K. The cosmic microwave background is 2.725 K.

All serious physics uses kelvins. The gas law PV = nRT, the radiation law u \propto T^4, the Boltzmann distribution — every one of them fails catastrophically if you plug in Celsius or Fahrenheit. The reason is the same: those scales have arbitrary zeros. Only Kelvin has the zero that nature put there.

How a thermometer works — the general idea

A thermometer is any device with a physical property X (length of a column, electrical resistance, voltage, brightness, colour) such that:

1. X varies smoothly and reproducibly with temperature.
2. Different thermometers of the same type give the same X at the same temperature.
3. The function X(T) is well enough understood — either from theory or from calibration — that you can invert it to get T from a measurement of X.

The simplest kind of calibration uses two fixed points. Dip the thermometer into a melting-ice bath; record the value of X at 0\,^\circC. Dip it into boiling water (at standard atmospheric pressure, or with a pressure correction); record X at 100\,^\circC. Now assume X varies linearly between the two (a good approximation for mercury in a narrow range; not perfect for every thermometer type — see the going-deeper section). Then the temperature corresponding to any intermediate reading X is

Why: the right-hand fraction is the position of X along the interval from X_0 (ice point) to X_{100} (steam point). Multiplying by 100\,^\circC converts that position into degrees on the Celsius scale. The formula is nothing more than linear interpolation between two known points.

Modern metrology has replaced the steam point with the triple point of water — the unique temperature (273.16 K) at which ice, liquid water, and water vapour coexist in equilibrium. The triple point is more reproducible than the boiling point, which depends on atmospheric pressure. But the logic is the same: anchor to one or two reproducible states, and define the scale by interpolation (or by the theoretical behaviour of a chosen substance).

Kinds of thermometers

Different physical properties give different kinds of thermometers, each with its own range, accuracy, and best use case.

The mercury-in-glass thermometer

A narrow glass capillary sealed at the top, with a bulb of mercury at the bottom. When the bulb warms, the mercury inside expands (because the volumetric expansion coefficient of mercury, \beta \approx 1.8 \times 10^{-4} /K, is much larger than that of glass, so the glass only changes size a little while the mercury pushes up the capillary). The height of the mercury column is the readable property X. Mark the column at 0\,^\circC and 100\,^\circC, divide the interval into 100 equal parts, and you have a Celsius thermometer you can use from about -39\,^\circC (where mercury freezes) up to about 350\,^\circC (where the glass softens).

The narrow capillary is the secret to the device's sensitivity. If V_0 is the mercury volume in the bulb at 0\,^\circC and A is the capillary's cross-sectional area, then a temperature rise \Delta T increases the mercury volume by \Delta V = \beta V_0 \Delta T, and the column height rises by \Delta h = \Delta V / A = \beta V_0 \Delta T / A. Make A small and the same small expansion produces a large, easily-read height change.

Clinical mercury thermometers used to be everywhere in Indian households. (They are being phased out by digital electronic thermometers because mercury is toxic if the glass breaks — an environmental and health concern, not a physics one.) Their range is narrow — about 35–42\,^\circC — because the designer only needs to cover human body temperatures; all the sensitivity is concentrated where it matters. A tiny constriction above the bulb also keeps the mercury column from falling back on its own after you remove the thermometer from the patient's mouth, so you can read the maximum temperature reached.

The resistance thermometer

A metal's electrical resistance rises with temperature (because hotter atoms jiggle more, and jiggling atoms scatter electrons more, increasing resistance). For many pure metals over a useful range, the resistance is nearly linear in temperature:

where R_0 is the resistance at a reference temperature T_0 and \alpha is the temperature coefficient of resistance (about 3.9 \times 10^{-3} /K for platinum). Pass a small known current through the wire, measure the voltage across it, divide to get R, and invert the formula to get T.

Platinum is the standard because it is chemically stable (does not oxidise at high temperatures) and its \alpha is well-characterised over a vast range. A platinum resistance thermometer (PRT) is used as an international temperature reference from about -260\,^\circC up to +960\,^\circC. In labs, schools, and industrial sensors, PT-100 resistance sensors (100 Ω at 0 °C) are extremely common.

The thermocouple

Two wires of different metals joined at both ends form a closed loop with two junctions. If the junctions are at different temperatures, a small voltage (the Seebeck voltage) develops across the loop. The voltage is (approximately) proportional to the temperature difference between the two junctions. Keep one junction at a known reference temperature (say, an ice bath at 0\,^\circC), and measuring the voltage tells you the temperature of the other junction.

Thermocouples are rugged, cheap, have very small response times (they are small and low-mass), and cover huge ranges — a platinum/rhodium thermocouple reads usefully from about 0 to 1700\,^\circC. They are the workhorse of industrial thermometry: inside furnaces, rocket engines, kilns, and the jet turbines of an Air India A320.

The infrared thermometer

Every object warmer than absolute zero emits electromagnetic radiation — a broad spectrum peaking at a wavelength that depends on its temperature. A room-temperature body emits mostly infrared (wavelengths around 10 μm); a red-hot iron emits enough visible red light to glow; the sun is hot enough to emit mostly in the visible band (which is why it looks yellowish-white). By measuring the intensity of the infrared radiation coming off a body, a detector can infer the body's temperature without ever touching it. (The radiation law u \propto T^4 — the Stefan-Boltzmann law — is what makes this quantitative; see Radiation.)

Infrared thermometers were pointed at every forehead in India during the COVID-19 pandemic, at airports, railway stations, and mall entrances. They are fast (sub-second), contact-free, and comfortable to use, at the cost of some accuracy (the reading depends on the target's emissivity, a surface property that varies between materials). Astronomers use the same principle to measure the temperature of distant stars — you cannot stick a mercury thermometer in Betelgeuse, but you can measure the spectrum of its emitted radiation.

Worked examples

Common confusions

• "Heat and temperature are the same thing." No. Temperature is a property of a body — a single number that tells you how hot it is. Heat is a transfer of energy between bodies at different temperatures. A cup of boiling water and a swimming pool at the same temperature have the same temperature but vastly different amounts of thermal energy. Dropping a hot spoon into each raises the water's temperature by almost nothing; the pool absorbs heat with barely any change in temperature because it has so much more material.

• "Temperature is how hot something feels." Your feeling depends on the rate of heat flow between the object and your skin, not the temperature alone. A wooden table and a steel chair at the same room temperature feel very different to the touch — the steel feels colder because it conducts heat out of your skin much faster than the wood does. The temperature is the same.

• "Absolute zero is just really, really cold." It is, but more importantly: absolute zero is the temperature at which thermal motion stops. You cannot go colder because there is no less-than-zero motion. (Quantum mechanics subtly modifies this picture — there is residual zero-point motion even at 0 K — but the classical intuition is correct for any practical purpose.)

• "A thermometer shows the temperature of the object." Technically, it shows the thermometer's own temperature after reaching equilibrium with the object. The zeroth law is what guarantees those two temperatures are then equal. For a small thermometer in a large object, the approximation is excellent. For a cold probe stuck briefly into a hot liquid, the reading might be off — you have to wait.

• "Kelvin uses the degree symbol." It does not. Write 300 K, not 300\,^\circK. Celsius and Fahrenheit use the degree (because their zero points are arbitrary and the unit is a size of a degree); Kelvin is an absolute scale and its unit is just "kelvin."

• "Boiling water is always at 100\,^\circC." Only at standard atmospheric pressure (sea level). Dal cooked over an open fire in Leh (altitude \sim 3500 m, pressure \sim 0.65 atm) boils at about 88\,^\circC, and therefore takes considerably longer to cook through — the reason pressure cookers were invented. The boiling point depends on pressure; this is part of why the modern fixed point is the triple point, not the boiling point.

Where this leads next

• Thermal Expansion of Solids and Liquids — how solids and liquids enlarge with temperature, and why railway tracks have expansion gaps.
• Thermal Expansion of Gases — the ideal-gas laws, the Kelvin scale derived from V \propto T and P \propto T, and the argument that absolute zero exists.
• Calorimetry and Specific Heat — how much heat is needed to raise a body's temperature, why water is an unusually good thermal ballast, and the equations that govern heat exchange.
• Conduction — the mechanism by which heat flows through solid materials, and why your tea cools faster in a steel tumbler than in a kulhar.
• Kinetic Theory — Pressure and Temperature — the microscopic meaning of temperature as average molecular kinetic energy, via \tfrac{3}{2}k_B T = \langle \tfrac{1}{2} m v^2 \rangle.