The Zeroth Law of Thermodynamics

If I have two bodies and I claim that one of them is longer or shorter than the other, you can prove my claim and either verify it or falsify it by measuring the lengths of the bodies. One of the assumptions that you must include in your reasoning about the measurement says that if both bodies have sides whose lengths match the length of your ruler, then those lengths will match each other. In that assumption we have a statement that conforms perfectly to Euclid’s first common notion: "Things equal to the same thing are also equal to each other."

If I have two bodies and claim that one of them is heavier or lighter than the other, you can prove my claim by measuring the weights of the bodies with a balance scale. In your reasoning you must include an assumption to the effect that if both bodies balance the same set of standard weights, they will balance each other. Again we have Euclidean transitivity of equal measures.

If I have two bodies and claim that one of them is hotter or colder than the other, how can you put that claim to the proof? What comparison can you make that measures relative hotness or coldness?

We start with the basic notions of hotness and coldness.
All people have a primitive idea of degrees of hotness and coldness: some things
are too hot to touch, some things are uncomfortable but touchable, some things
are pleasantly warm, and so on toward things that are too cold to touch. We want
at first to focus our attention on __relative__ degrees of hotness and
coldness. We want to consider the concepts of "hotter than" and "colder than".

Place the palms of your hands against your cheeks and feel their gentle warmth. Now put your hands together, palm to palm, and rub them vigorously back and forth over each other. Soon your palms will feel hot. Then place then against your cheeks and feel how much warmer they have become. By feeling that heat on your cheeks you verify that the feeling of heat on your palms was not an illusion. We say that the second time your palms were hotter than they were the first time or that they were colder the first time than they were the second time: we conceive the phrases "hotter than" and "colder than" as each the inverse of the other.

You can get a similar result by taking up a hammer and vigorously pounding an anvil. After a time the spot that you have been pounding feels warmer than it did before you began hitting it. That experiment, like Count Rumford’s cannon-boring experiment, shows that dissipating work into a body (through impact or through friction, as in the hand-rubbing experiment) produces heat, either evoking it as a kind of ætherial fluid (called caloric or phlogiston) or transforming the work into a melee of scrambled minuscule vibrations. Thus we conceive the idea of measuring the hotness of an object by measuring the work that we dissipate in the object.

But friction and impact do not give us the only ways to make a body hotter. We know that chemical reactions, such as fire, change the hotness of objects. In addition, radiation, such as heat from the sun, can also change the hotness of a body. So attempting to keep an account of the work dissipated in a body as a means of determining the hotness of the body won’t give us any accuracy at all.

We want a more direct way to measure the hotness of a
body. We need to find some property of matter that changes with hotness in a way
that we __can__ measure. We know that heat consists of minuscule vibrations
and we know that fact because we can use friction to generate an unlimited
amount of heat (if heat consisted of a fluid, such as caloric, any given body
would contain a limited amount of it). If we add heat to a body, we make those
vibrations more numerous and/or more vigorous. In at least some materials that
increase in the vibrations will make the particles constituting the body move
farther apart, thereby making the body expand. We expect that identical bodies
made of different materials and exposed to the same degree of hotness will
expand by different amounts, so we conceive a simple device to measure the
hotness of objects: we take two stiff, straight wires of equal length and made
of different substances, put them close together, side by side, and attach their
ends to two long, thin rods near __their__ ends. If we add heat to or
withdraw heat from the wires, the free ends of the rods move farther apart or
closer together by an amount that we can measure with a ruler. We call that
device (and others like it) a thermometer.

Measuring the distance between the tips of the rods gives us a number that we call temperature. That number gives us a measure of the degree of hotness of a body, but only after we have calibrated the scale. For convenience we set two points on the temperature scale by immersing the thermometer into a liquid that gives us two clear, easily reproducible states of hotness; specifically, we immerse the thermometer in ice water and then in water boiling at standard sea-level atmospheric pressure. For the Fahrenheit scale we define the melting temperature of ice as 32 degrees and the standard boiling temperature of water as 212 degrees; for the Celsius scale we define the melting temperature of ice as zero degrees and the standard boiling temperature of water as 100 degrees.

Do the results of the calibration experiments depend upon how much ice water or boiling water we use? We certainly expect the temperature of a body to depend, in some way, on how much heat the body contains. But we also expect the amount of matter containing the heat to have an effect as well. Imagine that we have a body made of a certain amount of matter containing a certain amount of heat. We measure the temperature of that body by putting our thermometer into contact with it in such a way that the wires touch the body. The thermal vibrations in the wires do work upon the particles that make up the body and the thermal vibrations in the body do work upon the particles that make up the wires, thereby redistributing heat until each body does work upon the other at the same rate at which the other body does work upon it. At that time we say that the body and the thermometer have come into thermal equilibrium with each other. We also say that the body and the thermometer have the same temperature, so reading the temperature of the thermometer also tells us the temperature of the body. Of course, we want to ensure that the thermometer does not change the temperature of the body whose temperature we want to measure.

Now suppose that the body under consideration consists of a set of closely-fitting blocks. The vibrations that constitute the heat in the body pass freely across the boundaries where the blocks meet each other, so the blocks all exist in thermal equilibrium with each other. That fact means that the blocks all have the same temperature.

If we separate one group of blocks from the rest of the body, we do no work on the body, so we haven’t changed the body’s heat content. We have also done nothing to move heat between the two pieces that we have created, so the temperatures of the two pieces haven’t changed; they remain equal to each other. That fact means that temperature exists as an intensive parameter, like pressure. We have turned one body into two separate bodies with different extensive parameters (volume and mass) but with the same intensive parameter (temperature).

We know now that if we were to touch the thermometer to each of the two parts thus created, it will show the same temperature. Thus we have the zeroth law of thermodynamics:

If two systems are in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.

That law tells us that equilibrium in intensive parameters is transitive. Paraphrasing Euclid, we say that things in thermal equilibrium with the same thing are in thermal equilibrium with each other. Equivalently, we can say that all three objects have the same temperature.

Appendix: Absolute Temperature

We have defined temperature as a measure of hotness and set up a temperature scale through two easily conducted experiments. One of those experiments, immersing a thermometer in ice water, gives us a Celsius temperature defined as zero. But even ice water contains heat. That fact implies the existence of a true zero of temperature, which would occur in matter that contains no heat at all. How would we locate where that absolute zero lies on the Celsius scale?

We have a simple yet accurate theory of matter due largely to James Clerk Maxwell. We call it the kinetic theory of gases and its description of a dilute monatomic gas conforms closely to measurements made on helium, neon, argon, krypton, xenon, and radon. Even at normal atmospheric pressure each particle has a volume vastly smaller than the part of its container’s volume it has available to its occupancy, so the particles conform to the idea of a dilute gas. In that kinetic theory a gas kept in a container at constant pressure has a volume that stands in direct proportion to the absolute temperature of the gas. We expect that relation to break down as the temperature of the gas approaches absolute zero and the assumption of dilution no longer correctly describes the gas. But at higher temperatures we can assume that the relation holds perfectly and use that assumption to determine how far below any given temperature the absolute zero lies.

We can measure the temperature of a certain amount of gas by enclosing it inside a vertical cylinder fitted with a gas-tight, freely-sliding piston whose weight provides a constant pressure on the gas. As the temperature of the gas changes, the piston rises or falls in the cylinder in accordance with the change in the gas’s volume. If we immerse the cylinder in ice water and then in boiling water and measure the volume of the gas (by measuring the location of the piston and multiplying by the area of the piston’s face) at each temperature, we get the ratios

V_{2}/V_{1}=T_{2}/T_{1}=1.366072409.

We also have T_{2}-T_{1}=100 . Dividing that difference by T_{1}
gives us

(T_{2}/T_{1})-1=100/T_{1}.

A little simple algebraic manipulation gives us

T_{1}=100/0.366072409=273.17,

which means that the melting temperature of ice, the temperature of ice water, comes 273.17 Celsius degrees above absolute zero. We designate the scale of centigrade degrees that begins at absolute zero as the Kelvin scale, so we say that ice water has an absolute temperature of 273.17 Kelvin and that boiling water at standard atmospheric pressure has an absolute temperature of 373.17 Kelvin.

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