The Third Law of Thermodynamics is concerned with the limiting behavior of systems as the temperature approaches absolute zero. Most thermodynamics calculations use only entropy *differences*, so the zero point of the entropy scale is often not important. However, we discuss the Third Law for purposes of completeness because it describes the condition of zero entropy.

The Third Law states, “The entropy of a perfect crystal is zero when the temperature of the crystal is equal to absolute zero (0 K).” According to Purdue University, “The crystal must be perfect, or else there will be some inherent disorder. It also must be at 0 K; otherwise there will be thermal motion within the crystal, which leads to disorder.”

Siabal Mitra, a professor of physics at Missouri State University, provides another implication of this law. “One version of the Third Law states that it would require an infinite number of steps to reach absolute zero, which means you will never get there. If you could get to absolute zero, it would violate the Second Law, because if you had a heat sink at absolute zero, then you could build a machine that was 100 percent efficient.”

In theory it would be possible to grow a perfect crystal in which all of the lattice spaces are occupied by identical atoms. However, it is generally believed that it is impossible to achieve a temperature of absolute zero (although scientists have come quite close). Therefore, all matter contains at least some entropy owing to the presence of some heat energy.

## History

The Third Law of Thermodynamics was first formulated by German chemist and physicist Walther Nernst. In his book, "A Survey of Thermodynamics" (American Institute of Physics, 1994), Martin Bailyn quotes Nernst’s statement of the Third Law as, “It is impossible for any procedure to lead to the isotherm *T* = 0 in a finite number of steps.” This essentially establishes a temperature absolute zero as being unattainable in somewhat the same way as the speed of light *c*. Theory states and experiments have shown that no matter how fast something is moving, it can always be made to go faster, but it can never reach the speed of light. Similarly, no matter how cold a system is, it can always be made colder, but it can never reach absolute zero.

In her book, "The Story of Physics" (Arcturus, 2012), Anne Rooney wrote, “The third law of thermodynamics requires the concept of a minimum temperature below which no temperature can ever fall — known as absolute zero.” She continued, “Robert Boyle first discussed the concept of a minimum possible temperature in 1665, in "New Experiments and Observations Touching Cold," in which he referred to the idea as *primum frigidum*.”

Absolute zero is believed to have been first calculated with reasonable precision in 1779 by Johann Heinrich Lambert. He based this calculation on the linear relationship between the pressure and temperature of a gas. When a gas is heated in a confined space, its pressure increases. This is because the temperature of a gas is a measure of the average speed of the molecules in the gas. The hotter it gets, the faster the molecules move, and the greater the pressure they exert when they collide with the walls of the container. It was reasonable for Lambert to assume that if the temperature of the gas could be brought to absolute zero, the motion of the gas molecules could be brought to a complete stop so they could no longer exert any pressure on the walls of the chamber.

If one were to plot the temperature-pressure relationship of the gas on a graph with temperature on the *x* (horizontal) axis and pressure on the *y* (vertical) axis, the points form an upward-sloping straight line, indicating a linear relationship between temperature and pressure. It should be rather simple, then, to extend the line backward and read the temperature where the line crosses the *x*axis, i.e., where *y* = 0, indicating zero pressure. Using this technique, Lambert calculated absolute zero to be minus 270 degrees Celsius (minus 454 Fahrenheit), which was remarkably close to the modern accepted value of minus 273.15 C (minus 459.67 F).

## The Kelvin temperature scale

The person most associated with the concept of absolute zero is William Thomson, 1st Baron Kelvin. The temperature unit bearing his name, the kelvin (K), is the one most commonly used by scientists worldwide. Temperature increments in the Kelvin scale are the same size as in the Celsius scale, but because it starts at absolute zero, rather than the freezing point of water, it can be used directly in mathematical calculations, particularly in multiplication and division. For example, 100 K actually is twice as hot as 50 K. A sample of confined gas at 100 K also contains twice as much thermal energy, and it has twice the pressure as it would have at 50 K. Such calculations cannot be done using the Celsius or Fahrenheit scales, i.e., 100 C is *not* twice as hot as 50 C, nor is 100 F twice as hot as 50 F.

## Implications of the Third Law

Because a temperature of absolute zero is physically unattainable, the Third Law may be restated to apply to the real world as: the entropy of a perfect crystal approaches zero as its temperature approaches absolute zero. We can extrapolate from experimental data that the entropy of a perfect crystal reaches zero at absolute zero, but we can never demonstrate this empirically.

According to David McKee, a professor of physics at Missouri Southern State University, “There’s a field of ultra-low-temperature research, and every time you turn around there’s a new record low. These days, nanokelvin (nK = 10^{−9} K) temperatures are reasonably easy to achieve, and everyone’s now working on picokelvins (pK =, 10^{−12} K).” As of this writing, the record-low temperature was achieved 1999 by the YKI-group of the Low Temperature Laboratory at Aalto University in Finland. They cooled a piece of rhodium metal to 100 pK, or 100 trillionths of a degree Celsius above absolute zero besting the previous record of 280 pK set by them in 1993.

While a temperature of absolute zero does not exist in nature, and we cannot achieve it in the laboratory, the concept of absolute zero is critical for calculations involving temperature and entropy. Many measurements imply a relationship to some starting point. When we state a distance, we have to ask, distance from what? When we state a time, we have to ask, time since when? Defining the zero value on the temperature scale gives meaning to positive values on that scale. When a temperature is stated as 100 K, it means that the temperature is 100 K above absolute zero, which is twice as far above absolute zero as 50 K and half as far as 200 K.

On first reading, the Third Law seems rather simple and obvious. However, it serves and the final period at the end of a long and consequential story that fully describes the nature of heat and thermal energy.

**Additional resources**

- The University of California, Davis' ChemWiki Dynamic Textbook describes the 3rd law and entropy.
- Purdue University has a lesson on "Entropy and the 2nd and 3rd Laws of Thermodynamics."
- Cornell University: "Teaching the Third Law of Thermodynamics"