Sidebar. The absolute temperature scale

We have an intuitive comprehension of temperature, with a prime basis being that heat only flows from a hotter body to a cooler body. The question is, What is the scale? Temperature is some measure of a quantity of heat. However, some bodies at a given temperature transfer heat faster to a given body than do other bodies. Some bodies of the same mass and same initial temperature can transfer different amounts of heat to a given reservoir. We need a rigorous definition that’s independent of thermal conductivity, independent of heat capacity, independent of the nature of the material.

We have practical measures of temperature for everyday use – the expansion of a liquid in a clinical thermometer, the voltage generated by thermocouple junctions, the deflection of bimetallic strips. All of these have limited ranges. They also do not show linearity with purported temperature measured by another method. They all have to be calibrated against some measure of absolute temperature. We also use different scales of the true or thermodynamic temperature, which is what we are reaching for. The common Fahrenheit and Celsius scales have their zeroes, but the zeroes do not signify the absence of heat, for the readings can all go very negative.

Several key realizations and building into a theoretical framework gave us thermodynamic temperature on an absolute scale, on which zero means the absence of heat that that be sensed or transferred. Guillaume Amontons in 1702 and again Johann Lambert (of Lambertian reflectance fame, and more) in the late 1700s considered the volume of a noncondensable gas such as air at constant pressure. Using the temperature measures available as if they were linear in the true thermodynamic temperature (a concept not yet devised), they extrapolated the gas volume to zero at the absolute zero of temperature. In our current refined system that’s at -273.16°C, also defined as zero kelvin. The actual scale involves some corrections for gases behaving other than ideally because the molecules have finite volumes and also have some attractive forces among themselves.

Another key realization came from atomic theory, considering the motion of molecules, particularly in gases. A very interesting set of observations and theoretical considerations led to the determination that the average energy of translational motion (linear motion, not rotational or vibrational) defines the temperature. In a gas in a steady state, molecules are moving in all directions and with a distribution of speeds:

Simulation of molecules in a gas in random positions and with random

velocities indicated by length and direction of arrows

The probability distribution of molecular speeds in an ideal gas (arbitrary temperature)

For any molecule of mass m and speed v, the energy of translational motion in any one of the three Cartesian directions in space is ½ mv2. Averaging over all the molecules the energy is taken as ½ kT per direction or (3/2)kT for the total translational motion in three directions. With the choice of the value of Boltzmann’s constant, k = kB, we have the definition of absolute temperature.

So, in translational movement energy is directly proportional to temperature. The direct proportionality is not generally true for other ways that atoms and molecules move. They can rotate, they can vibrate, they can be bound to each other in solids to exhibit collective vibrations called phonons. The amount of energy, or heat, stored in these other motions varies nonlinearly with temperature. However, these different modes of motion, or degrees of freedom, can exchange energy with translational motion. The net exchange ceases when the different modes reach a temperature in common. We still have absolute temperature defined. It is independent of the material of a body (gas, liquid, solid) and of the motions that any body can undergo, other than translational motion of its atoms and molecules. Absolute temperature defines, in turn, what states of motion are present (“occupied”) at any given temperature. When there are different states of motion with different energies E the probability of a state being occupied is proportional to e-E/(kT).  This expression is the negative exponential, noted in another page. It is a strongly declining function as its “argument,” E/kT, rises. Here we have all the intricacies of heat capacities the vary with temperature as motions become more available at higher temperatures – e.g., molecular vibrations are quantized in energy, unlike translational motions, so they substantially “freeze out” at temperatures below the first energy gap from no vibration to the first allowed vibration. Here we have as well the intricacies of phase changes such as from ice to water or in reverse. In first-order phase transitions such as freezing or boiling we have large changes in energy with no change in temperature until the change is complete. The physical origin of such behavior is intriguing but that’s not the scope of this sidebar.

The exchange of heat between bodies at two different temperatures can result in a simple equilibration at a common temperature. This increases the entropy or disorder. The exchanges can also generate mechanical (or electrical) work, as in an automobile engine or in the Earth’s weather system with its hot and cold masses of air and water that generate winds and ocean circulation. Temperatures define the maximal work that can be generated using reversible (basically, not wasteful), by a formula discovered by engineer Sadi Carnot in studying locomotives. (He is honored in the town of Aigues Mortes, France, with rue Sadi Carnot; why can’t we name streets and such for scientists, engineers, artists, and such in the US instead of for political hacks?) The relation is that the fraction of heat exchanged that can be converted into work is (T2 – T1)/T2, where T1 is the lower temperature and T2 is the higher, and both are as absolute temperatures. For the weather system where the temperature differences are generally up to a few tens of degrees at most, and the upper temperature is of the order of 27°C = 300K, that restricts the efficiency of weather processes to 5-10%. The average efficiency for the Earth in generating wind from solar energy has been quoted at 5%. That still leaves an enormous supply for humans to tap.

Natural systems display a vast range of absolute temperatures. The lowest is that of interstellar space, which is limited to the temperature of the Cosmic Microwave Background left from the expansion of he Big Bang, at 2.725K (with lots of tiny variations by direction that tell us of the expansion dynamics!). Among the highest thermal temperatures are in the cores of stars, in tens of millions of kelvin. Exploding stars, the supernovae, reach much higher temperatures near 100 billion kelvin for limited times. Some people quote the effective temperature of radiation such as cosmic rays, but it is inappropriate for such a nonthermal system – the particles are not in thermal equilibrium with some body but are individually accelerated on vast spatial scales by electrical and magnetic fields.

There is a rich body of technology for measuring absolute temperature across various ranges. There are semiconductor junctions for some microkelvin up to about 420K. There are metal-pair thermocouples for T in the range of about 100K to 1800K. Very high temperatures are often measured by the blackbody temperature as intensity (flux density) and wavelength distribution. Pick up an Omega handbook for copious information and have fun.