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Preferential occupation of distinct crystal sublattices by different chemical elements. Favourable unlike-neighbour bonding can stabilize a periodic alloy arrangement despite its lower configurational entropy than a random alloy.

Randomly distributing $n$ species with equal fractions over $N$ sites gives $\Omega =N!/[(N/n)!{]}^n$. Stirling approximation gives $S=k_{B}N\ln n$, or $R\ln n$ per mole of sites. Configurational entropy alone does not decide whether a multicomponent alloy is a stable single thermodynamic phase.

The crystalline expression diverges at $\varphi ={\varphi }_{\mathrm{cp}}\simeq 0.907$, where the cage free area vanishes. This is physically different from the spurious $\varphi =0.5$ pole produced by extrapolating the low-density fluid approximation. A real compression path cannot be inferred by following that fluid pole and then abruptly switching curves. At low density it follows a fluid equation of state, while at sufficiently high density an ordered arrangement can leave more accessible rattling configurations than a disordered crowded arrangement. This is entropy-driven hard-sphere crystallization, here in two dimensions: allowed hard-core configurations have the same interaction energy, so configurational entropy drives the phase competition. An equilibrium first-order coexistence interval would have common pressure and chemical potential, giving a constant-pressure segment as mean density changes. The two asymptotic expressions alone neither locate that interval nor determine the detailed two-dimensional melting sequence. They require a consistent free-energy comparison and trustworthy approximations in the transition region.

For $N$ distinguishable lattice sites occupied by $N_i$ mutually indistinguishable atoms of each species, the number of configurations isUsing Stirling's approximation,For one mole, $N=N_A$ and $k_B{N}_A=R$, soEquiatomic configurational entropy follows when the composition has $X_i=1/n$, givingThe reference is the unmixed configurational arrangement, and the model assumes random site occupation without short-range order. It counts configurational entropy, not all vibrational, electronic or magnetic entropy. A large number of elements does not by itself guarantee a stable single solid solution: competing thermodynamic phases and the enthalpy term also enter the Gibbs free energy.

Both starting arrangements converge toward the same chemically ordered alloy, exhibiting chemical ordering in alloys. This supports a lower free energy for the ordered unlike-metal neighbour pattern than for the segregated core-shell or disordered precursor at the final temperature. In a simple nearest-neighbour model, ordering is favoured whenwith consistent bond-energy conventions. Bulk mixing and ordering then compensate the loss of configurational entropy on cooling. Annealing alone does not supply numerical bond or surface energies, and elastic mismatch or adsorbed ligands can also matter.

The final morphology does not show a thick single-metal skin: the surface-energy difference is not large enough to overwhelm the bulk preference for ordering throughout the particle. Some atomic-scale surface enrichment is compatible with the drawing, so exact equality of the two surface free energies is not established.

If B has a much lower surface free energy, Surface segregation of B and an A-rich interior become favourable. A nearly continuous B shell is likely if enough B is present and the surface-energy gain exceeds the cost of disrupting favourable A-B bonds; otherwise only partial enrichment occurs.

Take $f$ as the tensile force exerted on the band, so reversible stretching supplies work $fdx$. At fixed other mechanical variables,Equality of mixed derivatives of the Helmholtz free energy yields the thermoelastic Maxwell relation for a stretched bandFor entropy-dominated rubber elasticity, increasing extension reduces configurational entropy, consistently giving a tensile force that increases with temperature at fixed extension. A force defined as the band's restoring force on its support has the opposite sign.

Configurational entropy counts the arrangements compatible with a macroscopic configuration: $S=k_B\ln \Omega$. Let $x=L/(Na)$, $N_{\downarrow }=N(1+x)/2$ and $N_{\uparrow }=N(1-x)/2$. The multiplicity is $N!/(N_{\downarrow }!N_{\uparrow }!)$. Stirling's approximation therefore givesWith downward $L$ positive, the gravitational potential energy is $-mgL$. Minimizing the Helmholtz free energy $F=-mgL-TS$ gives $\mathrm{artanh}x=mga/(k_{B}T)$, henceRaising temperature reduces the extension: the entropic preference for many up/down arrangements opposes alignment by the load. For small loads, $L\simeq N{a}^{2}mg/(k_{B}T)$.

For the positive-feedback entropic chain with replacement force $f(L)=f_{0}L/a$, $f_0>0$, the conservative potential is $U(L)=-{\int }_0^{L}f(s)ds=-f_0{L}^2/(2a)$. Using $-f(L)L$ would overcount it by a factor two. The stationary equation becomesExpansion about $x=0$ givesFor $T>T_c$ the stable extension is zero. Below $T_c$, nonzero minima appear, with $x\simeq \pm \sqrt{3(T_c-T)/T}$ close to the transition and $|x|\to 1$ as $T\to 0$. Restricting to downward extensions selects the positive branch. The onset is continuous in this model, with a nonanalytic mean-field change at $T_c$; a finite jump in $L$ does not follow from the printed force law, despite the question's discontinuity wording. A finite chain has rounded fluctuations rather than a literal thermodynamic singularity.

This is the same self-consistency equation as the zero-field mean-field treatment of the Ising model: the segment orientation corresponds to a spin, extension to magnetization, and the extension-dependent force to the self-consistent molecular field. The competition between alignment energy and binary configurational entropy explains the shared temperature dependence.

Gibbs free energy and its minimum. For fixed atom number $N$, there are $N+n$ sites andconfigurations with $n$ indistinguishable vacancies. Their configurational entropy is $k_B\ln W$. Relative to the vacancy-free Gibbs free energy $G_0$, creating $n$ vacancies costs energy $nϵ$ and pressure-volume work $npv$. Put $A=ϵ+pv>0$. ThenAt leading thermodynamic order, Stirling's approximation givesThe minimum obtained from the stationary point below isIf the perfect crystal's internal energy and other entropy are chosen as zero, its pressure term is $G_0=pNv$. More generally the information given fixes the vacancy contribution rather than the absolute bulk free energy. The pressure-volume term must be retained because the number of sites, and hence the volume, changes.

Equilibrium vacancy population. Differentiating the extensive free energy at fixed $N,T,p$ givesThe stationary point is consequently the unique interior minimum. Write $q=e^{-A/(k_{B}T)}$; its condition is $n/(N+n)=q$, giving the requested thermodynamic result for equilibrium vacancies with variable lattice size,Substitution back into $G$ cancels the terms containing $n_{\ast }$ and yields the boxed minimum. In the dilute-defect limit $A\gg k_{B}T$, $n_{\ast }\simeq N{e}^{-A/(k_{B}T)}$. The Bose-like denominator is a consequence of the changing number of lattice sites and combinatorial entropy; it does not assert that vacancies here are an ideal gas of bosonic particles.

There is a finite-size correction to a variable-site vacancy model if “average” is interpreted exactly. The microscopic count written above gives the relative isothermal-isobaric sumsoIts relative equilibrium free energy is $-k_{B}T\ln \mathcal{Y}=(N+1)k_{B}T\ln (1-q)$. Thus the source's $N$ numerator is correct to leading extensive order, but is not the exact finite-$N$ mean for that stated configuration count. A free-energy minimum is likewise not generally identical to the mean of a finite distribution. These distinctions vanish relatively as $N\to \infty$.

Thermal volume. Each atom or vacancy occupies volume $v$, so at the same thermodynamic accuracyFor the exact counting model, replace $n_{\ast }$ by $(N+1)q/(1-q)$ in $v(N+⟨n⟩)$. The vacancy-free limit is $Nv$ as $T\to 0$. This model isolates expansion from vacancies, taking $v$ and $ϵ$ constant; it omits ordinary phonon thermal expansion. Its growth at high temperature should not be extrapolated to arbitrarily large vacancy fractions in a stable crystal, and $A>0$ is essential for a normalizable variable-site ensemble.

A chain of $N$ up/down segments of length $a$ has extension $L=Nax$ and binary configurational entropy. An external force $f(L)=f_{0}L/a$ has potential $-f_0{L}^2/(2a)$, obtained by integrating the force. Its mean-field free-energy minima obey the displayed equation. Expansion gives $F-F(0)=N{k}_B(T-T_c)x^2/2+N{k}_{B}T{x}^4/12+\cdots$, so the ordered extension emerges continuously, $|x|\sim \sqrt{3(T_c-T)/T}$, rather than jumping. The equation matches zero-field mean-field theory of the Ising model; finite systems round the ideal singular behavior.

A missing atom at a lattice site that would be occupied in the reference crystal. A vacancy defect is a point defect and can contribute configurational entropy because many choices of empty sites can have the same macroscopic energy. Formation energy and pressure-volume work oppose this entropy gain. Vacancies can be present in thermal equilibrium; they need not all be preparation mistakes. Their allowed number and creation mechanism depend on whether the lattice size, atom number and surface reservoirs are fixed.