In thermodynamics, the third law of thermodynamics states that no finite sequence of cyclical processes processes can succeed in cooling a body to absolute zero. [1] In other words, it is found by experiment, one needs to do an ever increasing, and ultimately infinite, amount of work to remove energy from a body as heat as its temperature approaches absolute zero. In short, the third law, also known as the Nernst heat theorem, states that as absolute zero is approached, the entropy change ΔS for a chemical or physical transformation approaches zero:

$\lim_{T \to 0} \Delta S = 0$

This phenomenon was determined by German physicist Walther Nernst in 1906.

History
One of the first proto-statements of the outline of the third law of thermodynamics comes from the 1898 experimental findings of Scottish physical chemist James Dewar, who had been thinking about the nature of cold since the age of ten (1852) after having fallen through the ice of a pond and thereafter suffering a period of prolonged illness. Specifically, in 1898, following many years of low temperature research, Dewar solidified hydrogen by first evacuating hydrogen gas molecules from a dewar of hydrogen; then used charcoal to absorb the molecules, which resulted to cool the liquid to the solidification point; lastly he applied suction to solid hydrogen to reach a then new low-temperature record of thirteen degrees above absolute zero, at which point he commented that “there or thereabouts our progress is barred.” In commentary on the efforts involved in moving downward in these difficultly decreasing ranges of temperature, Dewar wrote: [5]

“To win one degree low down the scale is quite a different matter from doing so at higher temperatures; in fact, to annihilate these few remaining degrees would be a far greater achievement than any so far accomplished in low-temperature research.”

In 1905-06, German physicist Walther Nernst proposed a new heat theorem, which provided a means of determining free energies, and therefore equilibrium points, of chemical reactions from heat measurements. [2] In the years to follow, German physicist Max Planck was the first to emphasize that the new heat theorem could be considered as a new entropy theorem; in that, it supplied a means of fixing the otherwise undetermined additive constant:

S = k log W + constant

in the expression for the entropy of a condensed system. [3] Planck published this interpretation of the new heat theorem in a new and revised edition of his textbook on thermodynamics, indicating in the preface, dated 1910, that he had begun to consider the physical significance of Nernst’s theorem in areas outside the scope of thermodynamics.

In December of 1910, German physical chemist Otto Sackur submitted a paper to the Annalen in which he argued for a connection between the “new law of thermodynamics”, i.e. Nernst’s heat theorem, now known as the third law of thermodynamics, and the role played by the quantum constant, i.e. Planck’s constant h, in statistical calculations of entropy. [3] Upon reading this paper, Planck seized on this logic in a move to eliminate the additive constant to the statistical entropy equation. In 1914, Planck outlined what was then called Nernst’s heat theorem, in his “hypothesis of quanta”, as such: [4]

“For the hypothesis of quanta as well as the heat theorem of Nernst may be reduced to the simple proposition that the thermodynamic probability of a physical state is a definite integral number, or, what amounts to the same thing, that the entropy of a state has a quite definite, positive value, which, as a minimum becomes zero […]. For the present, I would call this proposition as the very quintessence of the hypothesis of quanta.”

References
1. Atkins, Peter. (2007). Four Laws - that Drive the Universe. Oxford: Oxford University Press.
2. Nernst, Walther. (1921). “Studies in Chemical Thermodynamics”, Nobel Lecture, Dec. 12.
3. Planck, Max. (1915). The Theory of Heat Radiation, (pg. xxxiii). Springer (reprint).
4. (a) Planck, Max. (1915). The Theory of Heat Radiation, Engl. Transl. Morton Masius, Preface to second edition, pp. VI, VIII. Philadelphia.
(b) Garola, Claudio, Rossi, Arcangelo, and Sozzo, Sandro. (2006). Foundations of Quantum Mechanics: Historical Analysis and Open Questions – Cesena 2004 (pg. 101). World Scientific.
5. Shachtman, Tom. (1999). Absolute Zero and the Quest for Absolute Cold (pg.167-78). Mariner Books.