# Is the third law of thermodynamics a non-essential law?

The third law of thermodynamics is sometimes called as impossibility of reaching absolute zero of temperature.

But somehow I think it can be inferred from the second law of thermodynamics that it is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work (The Kelvin–Planck statement).

Let's firstly assume that there exists a reservoir with absolute zero of temperature, then what will happen if there is a Carnot heat engine working between two reservoirs of absolute zero temperature and finite temperature?

You find that the heat engine works with efficiency as high as $\eta=1-\frac{T_c}{T_H}=1$, and transfers zero heat to the low temperature reservoir.

This implies that the heat engine absorbs energy in the form of heat from a single thermal reservoir and delivers an equivalent amount of work, which definitely violates the second law of thermodynamics as mentioned above.

As a result, the assumption made at first is not true that a reservoir with absolute zero of temperature do not exist, and thus it is impossible to reach absolute zero of temperature, as demonstrated by third law of thermodynamics.

So, what is the essentially of the third law of thermodynamics?

And its relation to the second law of thermodynamics?

• I would see the 3rd law as saying that as temperature goes towards zero Kelvin, the system must go to the state with minimum energy. Then for a perfect crystal this means that the entropy goes to zero as well since the ground state is non degenerate. Apr 25, 2018 at 12:24

The Kelvin-Planck statement of the second law (the one about not being able to build perfectly efficient heat engines) was not originally backed up by a deeply mathematical theory but was a simple statement of observed fact.1

Carnot's theorem looks in some ways like a stronger statement (because it leads a numeric limit lower than unity on efficiency), but that numeric limit is expressed in terms of the temperatures of reservoirs: $$\eta = 1 - \frac{T_C}{T_H} \;.$$ So it allows a violation of the Kelvin-Plank statement if you have a reservoir with zero thermodynamic temperature.2 (Kelvin and Planck didn't have observations of that case, so the loophole would be something new.)

The third laws plugs up the loop-hole in Kelvin-Planck opened by the Carnot theorem.

1 So it makes the people who want words like "law" and "theorem" to have precises denotations happy.

2 Note that it would take infinite time for such an engine to run through a cycle because the low temperature energy transfer stage has a problem. The "very small" temperature difference that we rely on to make quasistatic thermal flows has to approach zero as the low temperature does, making the machine not just tediously slow but actually stopped.

This is, in fact, the time reversal of the last step of a nominal "let's make that zero temperature reservoir" thing that the 3rd law pushes out to infinite time.

One way to formulate the Third Law is to state that entropy is equal to zero at absolute zero temperature. Taking statistical interpretation of entropy $$S =k \ln {W}$$, this can be reformulated as a statement that the ground state of a system must be unique. In other words the ground state is non-degenerate. This is typically true, as according to QM a generic system always tries to lift any degeneracy. This is the essence of the Third Law, which became pretty transparent after the advent of Quantum Theory.

P.S. It is important to mention here that sometimes degeneracy can survive if the equality of energies of different levels is protected e.g. by some sort of symmetry. Systems which sport degenerate ground states are called "topologically ordered" after X.-G. Wen. Such topologically ordered systems can be seen as violating the Third Law.

The heat engine form of the second law mathematically allows for a 100% efficient heat engine given a theoretical thermal reservoir at 0 absolute temperature (or infinite absolute temperature). However, you are on an interesting path. The real question is whether we can make a region of 0 absolute temperature, since we know we have never found one.

The thermodynamic inverse of the heat engine is the refrigerator / heat pump. In this case, we are interested in removing heat from the cold reservoir, so we are interested in refrigeration (if we are interested in the hot reservoir, it is a heat pump). The 2nd law limit on the coefficient of performance for a refrigerator is

$$COP_{ideal} = \frac{T_{C}}{T_{H}-T_{C}}$$

which clearly goes to 0 with absolute temperature. In my opinion, this makes it unlikely we could construct a device that makes 0 Kelvin, as it should require infinite work for any finite extent.