How exactly does it follow mathematically from the 3rd law of thermodynamics that it's impossible to reach absolute zero?

Question

According to Wikipedia,

The third law of thermodynamics states that the entropy of a closed system at thermodynamic equilibrium approaches a constant value when its temperature approaches absolute zero. This constant value cannot depend on any other parameters characterizing the system, such as pressure or applied magnetic field. At absolute zero (zero kelvins) the system must be in a state with the minimum possible energy.

It seems like the 3rd law is really just saying that, as the temperature of a closed system goes to absolute zero, the entropy and energy must both approach minimum values. That makes intuitive sense based on the definition of entropy in terms of probability and microstates vs macrostates, since the lower the temperature of a system is, the lower the average kinetic energy is, and hence there are fewer possible states the system could be in. But why does that mean it's impossible for any system to reach absolute zero? Obviously it can't happen to a closed system, since lowering the temperature at all requires putting in work to reduce the entropy locally and of course the lost energy has to go somewhere else. But I don't see why that would imply it's then fundamentally impossible for even an open system to reach absolute zero. I suspect it's because it would require arbitrarily large amounts of energy to keep decreasing the entropy towards that minimum value, and Wikipedia backs this up:

The laws of thermodynamics indicate that absolute zero cannot be reached using only thermodynamic means, because the temperature of the substance being cooled approaches the temperature of the cooling agent asymptotically.

And yet, I don't see why this is the case. What's the actual math showing that a system can only approach absolute zero at a (presumably vertical) asymptote?

Answer 1

I suspect it's because it would require arbitrarily large amounts of energy to keep decreasing the entropy towards that minimum value

This is a true statement but only a consequence of the actual thing being referred to.

The laws of thermodynamics indicate that absolute zero cannot be reached using only thermodynamic means, because the temperature of the substance being cooled approaches the temperature of the cooling agent asymptotically.

This is an elaboration of the thing being referred to.

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What you should realise is that the 3rd law is a mishmash of different statements. The version that says $$\tag1\lim_{T\to0}S=0$$ is the one written on Wiki and books, but is actually not the one being referred to here. It could even be false—there are some quantum models that do not obey that.

Instead, what they are saying is that $$\tag2\lim_{T\to0}\frac{\mathrm dS}{\mathrm dT}=0$$ which is the statement that has better evidence for. This is the one that tells us that it is impossible to reach absolute zero, because if the asymptotic slope of entropy with temperature removal turns zero, then isotherms coincide with isentropes, and that means, including isobars and isochores, all processes will fail to remove entropy. So, the last steps you want to remove entropy from a system, will always fail.

Answer 2

The precise statement is that due to the vanishing of the temperature derivative of entropy moving toward zero temperature, it is impossible to reach absolute zero in a finite number of steps.

The reason, nicely illustrated in the figure in this Wikipedia page, is that a typical way of decreasing the temperature of a system is through a sequence of adiabatic and isothermal transformations. The vanishing of the slope of $S(T)$ close to $T=0$K makes it possible to reach absolute zero only as an asymptotic process of infinite steps.

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Additional comment

Let me add a few words about an important point. In principle, one could ask why we must stick to a sequence of linear isotherms and adiabats in the $S$,$T$ plane instead of using a transformation corresponding to a curved line bringing in one or few steps at absolute zero. The answer is that we do not have such a kind of transformation. For example, if we had a thermostat at absolute zero, we could put our system in contact with such a thermostat and wait. However, this project does not work: not only because we do not know any real system at absolute zero but also because, even if it existed, an infinite time would be required to equilibrate due to the universal vanishing of specific heat at zero temperature. Without the possibility of reducing the temperature by thermal contact, we are left with the possibility of adiabatic cooling, i.e., horizontal lines in the $S$,$T$ plane. At this point, even if one would use something different from an isotherm to reduce the entropy, an infinite number of transformations would be necessary to get absolute zero.

Answer 3

Since all, yes, all, processes are irreversible, that is entropy producing, and at $T \to 0$ the entropy of all substances is $S(T)=S_0+f(T)$ where $f(T)=\mathcal O(T)$, any amount of irreversibility would increase $S$ away from $S_0$.

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Callen: THERMODYNAMICS AND AN INTRODUCTION TO THERMOSTATISTICS,
page 281, Section 11-3

THE "UNATIAINABILITY" OF ZERO TEMPERATURE

It is frequently stated that, as a consequence of the Nernst postulate, the absolute zero of temperature can never be reached by any physically realizable process. Temperatures of $10^{-3}K$ are reasonably standard in cryogenic laboratories; $10^{-7}K$ has been achieved; and there is no reason to believe that temperatures of $10^{-10}K$ or less are fundamentally inaccessible. The question of whether the state of precisely zero temperature can be realized by any process yet undiscovered may well be an unphysical question, raising profound problems of absolute thermal isolation and of infinitely precise temperature measurability. The theorem that does follow from the Nernst postulate is more modest. It states that no reversible adiabatic process starting at nonzero temperature can possibly bring a system to zero temperature. This is, in fact, no more than a simple restatement of the Nernst postulate that the $T = 0$ isotherm is coincident with the $S = 0$ adiabat. As such, the $T = 0$ isotherm cannot be intersected by any other adiabat.