Heat (Truly) Explained

(The following requires high school physics)

Classical thermodynamics is the study of the relationship between heat, work, and energy. Naturally, mastering the rudiments of thermodynamics first requires understanding what heat, work, and energy are. Though not difficult to comprehend, many students of science and engineering are unfamiliar with their most important nuances. This and the following post will explain what standard treatises don’t.    

The somatic senses illuded hundreds of generations of scholars into believing heat was a manifestation of a distinct substance. Even today, many educated people intuit it as such, and disillusionment doesn't always follow from the standard explanations.   

Sadly, these illusions are rarely, if ever, addressed in texts. Perhaps heat and temperature are so easily defined professors have tacitly decided to ignore the lingering illusions.

Temperature is a measurement of the average translational kinetic energy of a gas (KE relative to center of mass) or the temperature of a gas that a liquid or solid can remain in thermal equilibrium with. Since liquids and solids on Earth (including humans) are almost always surrounded by gases and that the temperature of gases is easiest to evaluate, its cause is often accepted as the general definition of temperature (though perhaps shouldn't be, given it technically only applies to gases). Heat is a measurement of the sum of all its internal kinetic and potential energies. Sometimes, temperature is loosely dubbed the “intensity” of heat. 

Heat and temperature are entirely mechanical, yet to most, it still feels like a substance.  

Fire in some ways behaves like a substance, even though it’s actually light given off when large sums of electrons go back and forth between atoms, in what’s known as oxidation-reduction reactions. In fact, fire often transforms the air it burns into plasma—an electrified gas that is sometimes called the fourth state of matter. Fire is regularly transferred into liquids to boil them, just as it’s transferred into solids to melt them. For example, when you boil water, fire is shot into a pot and the liquid bubbles and turns to gas. If you come close to the pot, heat stings your skin in a way distinctly different than if you were poked with a needle or hit by a fastball. You also know (correctly) that pipes, heat pumps, air conditioners, and refrigerators transmit heat, just as thermal insulators retard the rate of heat transfer.

Things that can be transmitted in such ways and that don’t feel mechanical are naturally thought not to be mechanical. A prime example is an electric current, the flow of atomic and/or subatomic particles containing electric charge. Like electricity, heat can be conducted, and substances that conduct electricity tend to be effective heat conductors. Yet heat is mechanical—not in the exact same way as other things, but entirely mechanical, nonetheless.

Then what creates this misimpression? Quite simply, there are specific nerves designed to sense the transfer of heat. In the language of biology, thermal-sensing neurons have thermal-gated ion channels. These channels respond not so much to temperature differentials themselves, but to the rate of heat transfer (although temperature differentials are a major factor in the rate of heat transfer). This is an important distinction because certain substances, such as metals, conduct heat very well and will FEEL cold, as a result, even at room temperature. Soda companies exploit this illusion by selling “cool cans” with better heat-conducting exteriors, implying that the soda is also colder.

It can be shown empirically that the product of the pressure and volume of an ideal gas (which most gases approximate) is equal to the product of its temperature (T), number of molecules (N), and a constant (C): PV=NCT. The relationship between these variables and average translational KE can be shown using a little-known thought experiment employed by Daniel Bernoulli in 1738 (note that it had little effect on the history of science, and for the next fifty years or more, most scientists thought heat and temperature were attributable to a “subtle fluid,” usually called caloric).

As you probably know, a gas is made up of billions of molecules zipping around in all directions at a wide range of speeds, constantly undergoing nearly elastic collisions with the walls (real or imaginary) containing them. Experience shows they can be treated as point particles that behave like billiard balls, and their collisions against the walls give rise to the system’s pressure. The densities of ideal gases are small enough that intermolecular collisions can be ignored. Regardless of the wall’s shape, or even if it’s only imaginary, the gas’s temperature and pressure will be the same throughout, provided it’s in thermal equilibrium—that is, at a steady temperature and not exchanging net energy with its surroundings.

The walls containing the gases will be cubes. If you doubt that the container’s shape is irrelevant, this problem is easily remedied in the next paragraph. If not, move on.

Imagine a shoebox-sized cylinder filled with a gas consisting of 100 billion trillion molecules in a given thermodynamic state. Imagine removing the gas, then putting ten thousand tiny cubes inside, each with an equal amount of a different sample of the same gas in the same state. It should be intuitively obvious that if the walls “magically” disappeared, the gas would behave the same as its counterpart. Now imagine the gas were still inside the cubes. It’s easy to see if you carefully reassembled the cubes into one large one, its behavior would remain unaltered. Ultimately, what this means is that if you don’t like the shape of the container for any problem or thought experiment, you can change it to a cube with a side equal to the cubed root of the total volume.  

In your mind’s eye, imagine a cube filled with an ideal gas. Remember that the length and width of the cube are equal. Now double the length of its side, increasing volume by a factor of eight. Though the molecules have the same speed, they will travel, on average, twice the distance between collisions and will be colliding with an area four times as large. Since the number of molecules colliding per unit time is cut in half while the “collision density” is cut by a factor of four, pressure is cut by a factor of eight. But the average translational kinetic energy is the same, so temperature remains constant. Though pressure was decreased by a factor of eight, its volume was increased by the same factor--thus, temperature can remain the same.

Now double the velocity, but hold volume the same. What will this do to the pressure? The collisions will now happen twice as often and hit twice as hard—thus, increasing pressure by a factor of four. Now keep everything the same, but double the mass of the molecules. Now pressure is doubled. Now double the number of molecules, and you get twice the number of collisions per unit time, doubling pressure yet again.

Clearly, PV equals CNmvsquared, where N is the number of molecules, m is the mass of an individual molecule, and C is a constant. Since a constant multiplied by 1/2 is another constant, this will show PV is a function of the average translational KE of the molecules, but I’ll keep the equation as is for the time being. If you were using translational v total, rather than in the x direction, you’d have to cut it by a third, since it’s assumed that the trans KE is distributed equally in all directions. 

Now I can say that PV equals CNm x Vtotal squared/3, where C is a constant.

As you can see, temperature is NOT average translational KE, just a function of it. Upon multiplication by 2, C becomes K, Boltzmann’s constant. I want PV to be a function of average translational KE, so PV= 2/3KNAverage KE.

Since the derivation matches experimental results, it seems that translational KE can alone account for temperature in gases--while vibrational KE clearly plays a greater proportional role in solids. However, the variability of specific heats makes clear that rotational and vibrational KE make a substantial contribution to heat. The involvement of the other forms of KE necessitates that people be precise when defining temperature and heat and not simply explain them as "the speed of the molecules" or the "kinetic energy of the molecules."

Heat always flows from higher to lower temperatures for the simple reason that mechanical energy always flows from higher to lower force/pressure--which, in this case, is a function of v squared. Note, however, that heat's flow is driven by a TEMPERATURE gradient, not a heat one. Substances like water can hold a lot of heat at relatively low temperatures because of how much energy it stores in the other forms of internal KE.

Interestingly, this collection of experiments, along with Joule's work, was the coup de grace for the caloric theory of heat. As demonstrated, a mechanical interpretation works; and if temperature, the “intensity” of heat, were the density of a heat-carrying material, a passive expansion of an ideal gas in vacuum (just increasing volume) should slowly decrease its temperature—but it doesn’t. This experiment was performed by Meyer and Joule. (Note that in real gases, there is a slight drop in temperature in the expansion due to the molecules having to overcome the small electrical attraction between them).     

With this in mind, it might seem contradictory that two substances with different quantities of heat have the same temperature. However, all factors held equal, a substance with high internal KE relative to temperature will exchange more heat with a third substance than a second at the same initial temperature. Suppose the lower internal KE substance exchanges heat with a third substance and equilibrium is reached. If the high internal KE substance exchanged the same quantity of heat with the third, each at the same initial temperature as before, it's temperature would have changed less than the low KE substance; and, therefore, additional heat would need to be exchanged to reach equilibrium.    

The above reveals and explains thermal illusions while giving an easier and more intuitive explanation for temperature than the standard derivation. It’s easier, adds meaning to v squared and its relationship with the relevant variables and, ultimately, invalidates a material theory of heat (though Joule’s weight and pulley experiment--performed independently by Meyer, as well--lends nice reinforcement).

I’m a staunch proponent of derivations, and doing them in different ways, but not all are created equal. Deriving things can become a questionable obsession; but many provide conceptual reinforcement and are easy to derive, sometimes easy enough that you don’t have to memorize them because you can so readily rederive them. This is one such derivation.

I’d love to hear why much of the above isn’t explained in the texts.