Look at a telephone keypad, and you will usually find the numbers arranged from one to nine, with zero underneath. Compare it with a computer’s number pad, however, and you will notice that zero is also at the bottom, while one sits in the lowest row of nonzero digits rather than at the top. The telephone’s rows read 1–2–3, 4–5–6, and 7–8–9 from top to bottom, whereas the computer number pad reverses that row order.
Since both layouts let us enter numbers, why arrange them differently? The comparison suggests that numerical order alone does not determine where the buttons belong, because designers also need to consider the task and how people will use the layout. Zero’s position is therefore part of a broader design choice, which makes the placement of the other digits just as relevant to the question.
Although the keypad and the laws acquired their arrangements for different reasons, both remind us that numbering can serve a purpose beyond putting numbers in numerical order. For keypads, the arrangement needs to support entering numbers, whereas the order of the thermodynamic laws needs to reflect how their ideas connect.
Physics seems to have its own numbering problem, because thermodynamics has a first, second, and third law, yet also includes a zeroth law. If that law belongs before the first, why wasn’t it called the first law in the first place?
Zero’s position depends on the keypad: it sits below nine on a telephone, but below one on a calculator or computer number pad. These layouts reflect different design histories rather than a universal rule for arranging numbers.
The zeroth law received its name after the other three laws were established. However, its role in defining equal temperatures places it logically before the other three.
A keypad needs to help people enter numbers easily and accurately, so its design involves more than numerical order. But the idea that zero comes “after nine” is not universally true, even as a description of the layout. Telephone keypads place one to three at the top and seven to nine above zero, while calculators and computer number pads place one to three directly above zero. Reading that arrangement upward, zero comes before one rather than after nine.
The calculator-style layout comes from earlier adding machines, which used a compact three-by-three arrangement with a zero key underneath. Computer number pads continued that established arrangement, allowing people familiar with calculators to use a similar layout for entering numbers.
On the other hand, Bell Labs tested different telephone layouts and found that several allowed similar performance, while users showed clearer preferences between them. Those results helped inform the choice of an arrangement in which one to nine fill three rows and zero sits underneath.
The two layouts therefore reflect different design histories. Computer number pads continued the calculator arrangement, while telephone designers tested alternatives for dialing. Zero sits below nine on a telephone keypad but below one on a calculator or computer number pad, so describing it as “after nine” depends on which layout and reading direction we choose. Its numerical value remains unchanged.
Arrangements can serve a purpose other than the order that they came in. They can reflect preference, order of discovery, or logical sequence. A keypad is arranged to help us enter numbers, while the thermodynamic laws are ordered to help us understand how their ideas connect.
To understand why the zeroth law belongs at the beginning, we can look at what happens when we measure temperature with a thermometer.
The first and second laws developed during the nineteenth century as scientists studied heat, work, and energy, while the third law followed in the early twentieth century through work on behavior near absolute zero.
The zeroth law, also called the thermal-equilibrium principle, appeared under that name in Ralph Fowler and Edward Guggenheim’s book Statistical Thermodynamics, published in 1939, well after the other three laws were established. But it provides a foundation for temperature measurement, which we need when applying the other laws. Calling it “zeroth” therefore placed it logically before them without changing their established names: it received its numbered name fourth, but comes first in the explanation.
A thermometer does not directly measure the temperature of the object we place it against. Rather, it tells us its own temperature. We can use that reading to find the object’s temperature only when the thermometer has reached approximately the same temperature as the object.
If a cooler thermometer is placed in warmer water, energy transfers from the water to the thermometer, causing its reading to rise. We therefore wait for the reading to stabilize, so that the thermometer’s temperature can represent the water’s temperature.
This condition is called thermal equilibrium. At this point, the thermometer and water have the same temperature, so there is no net heat transfer between them due to a temperature difference.
The thermometer must also respond to temperature changes without exchanging enough energy to measurably change the object’s temperature. Otherwise, it would alter the temperature we wanted to measure. We will discuss this measurement problem further in another post. For now though, this is the reason why all thermometers are small devices; being small means less mass, and therefore less of an effect on its surroundings.
We can begin with a simple gas model. For the same kind of particles, a higher temperature means greater average kinetic energy and a higher average speed. If two samples start at different temperatures, their particles therefore have different average speeds, although each sample contains a range of speeds.
When the samples can exchange energy through collisions, energy is redistributed between their particles. On average, the hotter sample loses energy and the cooler sample gains it, so their average kinetic energies and, for identical particles, average speeds become closer as their temperatures approach each other.
The particles do not all settle at one speed. Instead, collisions continue to change individual speeds, so some particles remain faster than average and others slower. Nevertheless, thermal equilibrium means the samples have the same temperature, rather than every particle having the same motion.
Different kinds of gas particles at the same temperature have the same average translational kinetic energy. But their masses can differ, so their average speeds need not be equal: heavier particles generally move more slowly than lighter particles at that temperature.
A microstate describes one particular microscopic arrangement of the particles and their energy. Movement and collisions continually change that arrangement, while there are overwhelmingly more available microstates corresponding to the combined system being near equal temperatures than to a large temperature difference. Thermal equilibrium is therefore the most likely overall condition, even though individual particles keep moving and exchanging energy.
If a thermometer reaches thermal equilibrium with one cup of water, its reading represents that cup’s temperature. If it then reaches thermal equilibrium with another cup at the same reading, we conclude that the two cups have the same temperature, even though the cups have not touched each other.
The zeroth law of thermodynamics states:
If two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other.
Let the cups be A and B, and let the thermometer be C. We can represent their temperatures as:
\(T_A = T_C\)
\(T_B = T_C\)
Therefore:
\(T_A = T_B\)
If both cups give an equilibrium reading of 30 °C, their temperature difference is zero:
\(T_A − T_B = 30 − 30\)
\(ΔT = 0 °C\)
Bringing them into thermal contact would therefore produce no net heat transfer.
The thermometer thus provides a common reference, allowing us to compare systems without requiring every pair to be brought together.
Because the zeroth law establishes a consistent meaning for having the same temperature, it gives temperature measurement a physical foundation. Once we can compare thermal states consistently, temperature becomes useful in applying the other laws.
The first law accounts for changes in internal energy through heating and work. Temperature measurements help us connect that energy accounting to observable changes, although the first law itself is a broader statement of energy conservation.
The second law describes the direction of spontaneous processes and places limits on converting heat into work. Temperature is essential here because it helps determine the direction of heat transfer and appears in relationships involving entropy.
The third law concerns behavior as temperature approaches absolute zero, so its statement also requires a well-defined temperature concept.
The zeroth law therefore supplies a basic idea used throughout thermodynamics: how to compare temperatures consistently. But it does not produce the other laws by itself; each adds a different principle about energy, the direction of processes, or behavior near absolute zero.
Equal temperature does not mean equal internal energy. For example, two samples of the same monatomic ideal gas at the same temperature have equal average kinetic energy per particle, but a sample containing twice as many particles has twice the internal energy. The zeroth law concerns equal temperatures, not equal total energies; we will explore that distinction further in another post.
Whenever we use a thermometer, we rely on this thermal-equilibrium relationship.
Measure the temperatures of two containers of water with the same thermometer, allowing enough time for the reading to stabilize. If their equilibrium readings match within the thermometer’s resolution, the measurements indicate approximately equal temperatures.
The waiting matters because a thermometer that is still warming or cooling has not yet reached thermal equilibrium with the water. Its reading may therefore differ from the temperature we intend to measure.
The thermometer also exchanges energy with the sample, so a useful measurement requires that this exchange does not appreciably change the sample’s temperature. This is why measuring a large container of water is generally less affected by the thermometer than measuring a tiny droplet.
Now that we know how the zeroth law explains how thermometers work, we can then use this idea to calibrate them. Of course, calibration needs something that has a constant temperature.
Good thing there’s this video that shows how we can get one such temperature.
