 # Temperature is More Complicated Than You Think

Learn enough science and one starts to discover that everything is complicated. There is no clearer example of this than the scientific study of heat. It seems like it should be the simplest thing; everyone knows what hot feels like. As infants we easily learn the practical definitions for hot and cold, but the ability to characterize heat with numerical values required the invention of the thermometer. Even with the thermometer an arbitrary scale had to be developed, and here water came in handy. In the Celsius scale the freezing and boiling points of water make convenient bookends. The freezing temperature is defined as zero degrees Celsius and the boiling point as one hundred degrees Celsius, which allows for the definition of a single degree Celsius by division. Anything less than zero is simply a negative temperature, which betrays the pure arbitrary nature of this scale, but puts everyone on the same page when talking about temperature.

But what is heat? For most of history heat was the opposite of cold. We quickly learned that the Sun gives off heat, but where does cold come from? For a long time there was a theory that cold existed as the opposite of heat. Heat was described as being able to flow from one material into another, and inversely, cold could also flow. Put a hot metal rod into a bucket of water and the rod will cool while the water gets hotter. It makes sense to conclude that heat flowed from the metal into the water and in turn cold flowed from the water into the metal. This is why science dubbed the study of heat as thermodynamics, as how “temperature moves.”

Eventually, scientists realized that the idea of both heat and cold flowing was unnecessary. Instead there was heat and the absence of heat. The science of thermodynamics was built around the concept of heat magically flowing from one material to another. For instance, a fire generates heat which can flow through the air to warm up a room and the people in it. The flow of heat follows a few very simple rules, while observation and experimentation led to mathematically complex rules that describe very intuitive results. The best example of this is the differential equation for heat flow. Differential equations are complex mathematical equations that can describe how a system changes over space and time. The heat flow equation describes the rate that heat can flow from one point in space to another. All of this sounds very complicated, but here’s one thing that the differential equation for heat flow says: the hotter the object in comparison to its surroundings then the faster it gives off heat. Think about that for a second; convince yourself that it makes perfect sense. Remember, this is heat we’re talking about – it’s inherently intuitive.

Despite the great successes of thermodynamics – in particular the invention of the steam engine – scientists kept coming back to the fact that they didn’t understand what heat actually was. Thermodynamics could describe how heat flowed from one point to another while having no idea what was flowing. This happens often in science. Sometimes our ability to predict a thing far outpaces our understanding of its fundamental nature.

While many engineers were satisfied with the practical applications of thermodynamics, and untroubled by the lack of understanding of what heat was, some scientists worked to develop an explanation for heat grounded in atomic theory. Early atomic theory stated that all matter is made of atoms which are separated by relatively massive distances between them. We know from the macroscopic world that moving objects have kinetic energy, which is related to the mass of the object and its velocity. What if atoms behaved like little bouncy balls, constantly in motion and constantly bouncing off one another? If that were true then each atom would have a kinetic energy, and that energy might be related to heat. Though intriguing, this concept was nothing more than that: a concept. There was no mathematical proof that could tie the motion of atoms to a measurable temperature.

In the later half of the 19th century, James Clerk Maxwell (the same Maxwell that had previously revolutionized the study of electromagnetism) and Ludwig Boltzmann mathematically proved that the motion of atoms could result in what we call heat. Their theory, eventually named “statistical mechanics,” explains everything about heat: what it is, how it flows, and that it has an absolute minimum. Their original theory describes gasses, being the simplest state of matter, but the conclusions of the model extend to all states of matter.

Though the kinetic energy of an individual atom is incredibly small at normal temperatures (due to its tiny mass) there are many atoms in any macroscopically meaningful chunk of matter. Because of this, Maxwell and Boltzmann realized that they could describe gasses using the mathematics of statistics. The exact behavior of a single atom (or molecule) doesn’t need to be known because small differences in behavior wash out when millions of atoms are constantly bumping into one another. Maxwell and Boltzmann proved mathematically – starting from simple assumptions that atoms behave like bouncing balls – that the average kinetic energy of all the atoms in a gas could be related to what we call temperature. There are a few amazing things about this. First is that heat, which is one of the most fundamental forms of energy we know of, is actually a form of kinetic energy, which is another fundamental form of energy. Maxwell and Boltzmann not only demystified temperature, they showed that it was a version of something already well understood.

One of the revolutionary things about their discovery is that it connects action on the atomic scale with a phenomenon on the macroscopic scale. This was one of the first instances of such a connection being mathematically proven. If the motion of atoms could result is such a measurable effect as temperature, then who knows how many macroscopic physical effects are due to microscopic features? (The fifty years following Maxwell and Boltzmann’s discovery would find the field of chemistry transformed by modern atomic theory, with all chemical properties discovered to be a result of interactions between individual atoms.)

Maxwell and Boltzmann’s work leads directly to a thought experiment. If atoms are tiny bouncy balls ricocheting off one another and that motion is the source of temperature, then what happens if all the atoms stop moving entirely? If this were to happen then there is absolutely zero thermal energy in the atoms and we have discovered the lowest possible temperature! Appropriately, this temperature is called “absolute zero.” No longer would scientists have to use an arbitrary temperature scale because an absolute scale exists as a consequence of the nature of heat.

There was prior evidence that an absolute zero temperature existed. The absolute temperature scale was created by, and named after, Lord Kelvin (born William Thomson) who in 1848 worked out that absolute zero was at -273.15 degrees Celsius. (Actually, he put it at -273 Celsius, which is so amazingly close to the modern value that I can barely handle it.) Lord Kelvin proposed keeping the Celsius as the size of a degree, thus water freezes at 273.15 Kelvin or zero degrees Celsius. Lord Kelvin derived this value years before Maxwell and Boltzmann’s work by using the previously known relationship between the temperature of a gas and the amount of volume it takes up. It had long been known that the volume of a fixed mass of gas was determined by its temperature; hot gas takes up more volume. This is part of how hot air balloons work. By turning the math around he was able to show the temperature at which the gas would take up the smallest volume possible. Using this concept and the thermometers available at the time, Lord Kelvin worked out the temperature of absolute zero, without understanding why such a state should exist! Like it said earlier, in science it is sometimes possible to take what is known and reach a conclusion ahead of a full understanding of the how and why.

“The determination of temperature has long been recognized as a problem of the greatest importance in physical science.” It was with those words that Lord Kelvin began the paper announcing his discovery of the value of absolute zero. At the time his statement was not hyperbole. With the benefit of hindsight it is still not hyperbole. If something simple and intuitive like hot and cold can puzzle some of the greatest minds in science, just imagine what fascination more complicated topics can bring. #### About Andrew Porwitzky

Dr Andrew Porwitzky is a professional scientist, comic book junkie, and freelance writer. He is also on Twitter way too much.