= Solution
The <Third law of thermodynamics>, in its perfect-crystal form, states that the <entropy> of a pure, perfectly ordered <crystal> with a unique <ground state> tends to zero as $T\to0$. Its more general entropy-difference formulation says that the <entropy> difference between equilibrium states related by a finite change of a regular external control parameter tends to zero at <absolute zero>:
$$
\lim_{T\to0}\left[S(T,X_2)-S(T,X_1)\right]=0.
$$
The common limiting <entropy> is independent of $X$. Ground-state degeneracy or frozen disorder can produce <residual entropy>, so zero <entropy> is not an unconditional assertion about every sample.
In the usual regular thermodynamic setting, <absolute zero> cannot be attained from positive <temperature> by a finite sequence of cooling operations with finite resources. An entropy-temperature plot shows why a finite reversible adiabatic step cannot reach the common zero-temperature endpoint from a state with positive <entropy> above that endpoint: an <isentropic process> follows a horizontal line, while all the regular <entropy> curves meet only at $T=0$.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2021/paper-2-a2-zero-temperature.png]
{title=Entropy curves and a cooling staircase approaching absolute zero}
The figure illustrates this with two possible settings having $S-S_0=a_1T$ and $S-S_0=a_2T$, where $0<a_1<a_2$. First change the setting isothermally from the upper to the lower curve, rejecting <heat>; then change it reversibly and adiabatically back to the upper curve. <Entropy> conservation on the latter step gives
$$
a_1T_n=a_2T_{n+1},\qquad T_{n+1}=\frac{a_1}{a_2}T_n,
\qquad T_n=T_0\left(\frac{a_1}{a_2}\right)^n>0
$$
for every finite $n$. The staircase approaches the origin but reaches it only after infinitely many such stages. Irreversibility generates <entropy> and cannot improve on the reversible endpoint.
This is a thermodynamic illustration of <unattainability of absolute zero>, not a claim that every cooling apparatus has this particular linear <entropy> law. Its conclusion presumes regular equilibrium states and bounded control parameters; dynamical statements about finite cooling time additionally require physical assumptions about the available interactions and cooling rates.
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