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The entropy of mixing is the entropy change between initially separate components and their final mixture, with initial and final temperatures, pressures and component amount of substance values specified. For an ideal solution made from pure liquids at the same temperature and pressure, let $n={\sum }_i{n}_i$. ThenThe right-hand expression is the ideal solution result; the general definition does not require ideality. Each interior mole fraction has a negative logarithm, so mixing distinct components gives positive entropy of mixing. At a pure endpoint, use the limit $x\ln x\to 0$ as $x\to 0^{+}$.

For one mole of the chosen A/B ideal solution with $x_A=0.4$,For the same total amount of substance, equal mole fractions maximize this binary entropy of mixing: the composition function $s(x)=-x\ln x-(1-x)\ln (1-x)$ hasso its unique maximum is $s(1/2)=\ln 2$. This is a comparison between different prepared compositions, not a prediction that a closed sample changes its component inventory to become equimolar.

Flory–Huggins solution theory is an incompressible lattice description of mixing a polymer with a solvent. A chain occupies $N$ connected sites, while a solvent molecule occupies one; the displayed mixing free energy is per site. The first two terms are translational entropy of mixing contributions and the last is a mean-field excess contact interaction. A positive interaction parameter $\chi$ penalizes polymer–solvent contacts. If the site volume is $v_0$, the physical free-energy density is $f/v_0$ and the polymer volume fraction gives monomer number density $\varphi /v_0$. Composition-dependent interaction parameters require extra derivative terms in stability calculations.

The Gibbs free energy of mixing compares the final mixture's Gibbs free energy with that of its initially separate components at the same temperature and pressure. For an ideal solution, subtract the pure-component reference from $G={\sum }_i{n}_i{\mu }_i$:For one mole of the chosen A/B ideal solution at $300\mathrm{K}$ and $x_A=0.4$,The individual chemical potential changes are $RT\ln 0.4\approx -2.28554\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ for A and $RT\ln 0.6\approx -1.27417\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ for B. Weighting these by $0.4\mathrm{mol}$ and $0.6\mathrm{mol}$ reproduces the total Gibbs free energy of mixing.

If ideality holds in a neighborhood of the specified temperature and pressure, differentiation at fixed component amounts givesThus the ideal mixing Gibbs free energy decrease comes entirely from the entropy increase; the mixing enthalpy and volume changes vanish. Ideality at a single isolated state alone does not determine temperature or pressure derivatives.^[ref]

The same formula rules out spontaneous separation into two liquid thermodynamic phases within this model. Per mole, the mixing Gibbs free energy $g_{\mathrm{mix}}=RT[x\ln x+(1-x)\ln (1-x)]$ hasAdding the pure-component reference contributions changes only a term linear in $x$. The total molar Gibbs free energy is therefore a strictly convex function: a separated pair of different compositions has a larger composition-weighted Gibbs free energy than the uniform sample with the same inventory. Such phase separation requires physics beyond this ideal solution model.

Counting permutations of identical classical particles as different states produces a spurious entropy of mixing when otherwise identical gases are combined. The dilute ideal gas partition function $Z_N=Z_1^N/N!$ removes that label multiplicity and gives an extensive entropy in the thermodynamic limit. For different species, genuinely distinct composition information still produces a entropy of mixing. quantum statistics supplies the underlying identical-particle counting.

An ideal solution is a chemical solution whose component chemical potentials depend on composition throughHere $n_i$ is an amount of substance, $x_i$ its mole fraction, $R$ the molar gas constant, and ${\mu }_i^{\ast }$ the pure-component liquid reference at the same temperature $T$ and pressure $P$. This model applies to every component across the composition range; it is stronger than an approximation valid only near a pure solvent.^[ref]

With this pure-liquid standard state convention, the thermodynamic activity satisfies ${\mu }_i={\mu }_i^{\ast }+RT\ln a_i$, so an ideal solution has $a_i=x_i$, or activity coefficient ${\gamma }_i=a_i/x_i=1$.^[ref] This statement concerns the liquid. Assuming that a coexisting vapor is an ideal gas mixture is an additional approximation. Its mole fractions $y_i$ generally differ from the liquid mole fractions $x_i$.

For an original two-component example, stipulate $T=300\mathrm{K}$ and pure-component vapor pressures $p_A^{\ast }=80\mathrm{kPa}$ and $p_B^{\ast }=20\mathrm{kPa}$. A and B are invented distinguishable components; these numbers are chosen model inputs, not material data. Combine an ideal solution with an ideal gas vapor, neglect the pure-liquid reference's pressure correction, and exclude other gases. At liquid mole fractions $x_A=0.4$ and $x_B=0.6$, Raoult's law and addition of partial pressures giveThus the coexisting vapor contains about $72.73\%$ A by amount of substance, although the liquid contains only $40\%$. The total pressure is neither pure-component vapor pressure, and $x_A$ cannot be substituted for $y_A$. This enrichment is the basis of distillation.

The usefulness of the ideal solution is that one composition law connects chemical potentials, vapor-liquid equilibrium, entropy of mixing and Gibbs free energy of mixing. Its liquid components can interact strongly: the defining condition is their composition dependence, rather than the absence of all interactions as in an ideal gas model.

The substituted molecule has the larger entropy. Increased mass and moment of inertia reduce translational and rotational spacings and make more states thermally accessible. The largest contribution here is the loss of the twofold rotational indistinguishability of the oxygen ends, contributing $R\log 2$. It is a rotational symmetry number correction for one isotope-defined species, not an entropy of mixing between different molecules.

For equal-size molecules, let $f$ be the mixing Helmholtz free energy per lattice site. Flory–Huggins solution theory reduces to the regular solution model:The first two terms are minus the combinatorial entropy of mixing; the last is the excess cost of unlike contacts. A positive Flory interaction parameter penalizes such contacts. Additive terms linear in composition change the chosen pure-component reference but do not change stability.

A small composition fluctuation at fixed mean composition costs energy proportional to $f^{\prime \prime }$. Thus local stability requires $f^{\prime \prime }>0$, and the spinodal isInside this curve, $f^{\prime \prime }<0$ and arbitrarily small composition fluctuations grow. The minimum of ${\chi }_s$ is at $\varphi =1/2$, giving the critical value ${\chi }_c=2$. At the thermodynamic critical point, $f^{(2)}=f^{(3)}=0$ while $f^{(4)}>0$.

For $\chi =0$, $f$ is strictly convex with its minimum at $1/2$. At $\chi =2$, the central quadratic term vanishes but a positive quartic term remains. At $\chi =2.5$, there are two minima and a central maximum; the spinodal compositions are $(1\pm \sqrt{1-2/\chi })/2\simeq 0.2764,0.7236$. The common-tangent construction gives coexisting compositions ${\varphi }_b\simeq 0.1448$ and $1-{\varphi }_b$, satisfying $\ln [{\varphi }_b/(1-{\varphi }_b)]+\chi (1-2{\varphi }_b)=0$. Between a binodal endpoint and the spinodal, the homogeneous state is locally stable but metastable; global stability requires the convex envelope of $f$.

For two species of chains with the same polymerization degree $N$, the number of independently translating objects per occupied site is reduced by $N$. The symmetric Flory-Huggins polymer blend hasThe symmetry still places the thermodynamic critical point at ${\varphi }_c=1/2$, but nowThe entropy of mixing per monomer is much smaller than for simple liquids, so even a weak unfavorable interaction can cause phase separation for long chains. If the energetic interaction parameter is $\chi =A/T$ with $A>0$, this model gives $T_c=AN/2$; the conclusion relies on that specified temperature law.

Write $z_c$ for the lattice coordination number and ${ϵ}_{AA},{ϵ}_{BB},{ϵ}_{AB}$ for contact energies. Counting the excess of unlike contacts relative to the unmixed reference gives the Flory interaction parameterUnlike contacts that are less favorable than the mean like contact give $\chi >0$. For temperature-independent contact energies, $\chi$ is proportional to $1/T$, so heating weakens this energetic preference relative to the entropy of mixing. Effective parameters in real solutions can also include entropic terms, often represented schematically by $A/T+B$; the pure contact-energy model does not explain every possible temperature dependence.

The Flory interaction parameter $\chi$ measures the excess cost of unlike segment contacts. In a simple lattice with coordination number $z$, contact energies ${ϵ}_{AA},{ϵ}_{BB},{ϵ}_{AB}$ and temperature $T$,A positive Flory interaction parameter penalizes unlike contacts; a negative value favors them. Mixing still depends on the balance with entropy of mixing, rather than the sign of $\chi$ alone.

For two polymers of degrees of polymerization $N_A,N_B$, let $\varphi$ be the volume fraction of species $A$. The binary Flory-Huggins polymer blend has mixing free energy per lattice siteThe logarithmic terms favor mixing. Their factors $1/N_A$ and $1/N_B$ arise because an entire chain translates as one object; connecting many segments greatly reduces the mixing entropy per segment. Hence long polymers can demix for a much smaller positive Flory interaction parameter than small molecules.

For composition-independent $\chi$, the local stability criterion isPositive curvature means stability to small composition fluctuations, zero curvature locates the spinodal, and negative curvature means spontaneous growth of fluctuations. Actual two-phase coexistence is determined by the common-tangent construction, or equality of both species' chemical potentials; its boundary is the binodal. A locally stable but globally unfavourable homogeneous mixture can lie between the binodal and spinodal.

Setting the second and third derivatives of $f$ to zero givesFor the symmetric Flory-Huggins polymer blend, ${\varphi }_c=1/2$ and ${\chi }_c=2/N$. For a polymer solution with $N_B=1$ and long chains of species $A$, ${\chi }_c\to 1/2$, consistent with the theta solvent limit. An effective Flory interaction parameter can depend on temperature and composition; in the simple energetic case $\chi \propto 1/T$, heating encourages mixing, but that temperature trend is not universal.

The original illustration uses stipulated $N_A=N_B=100$. At $\chi =0.01$ the mixing free energy is convex, whereas at $\chi =0.03$ a central unstable region appears. The horizontal common-tangent construction in the latter case joins the symmetric coexisting compositions; its contacts differ from the spinodal boundaries. These are model curves, not supplied experimental data.

Meaning of the mixing free energy. The supplied expression is the Flory–Huggins solution theory mixing free energy per lattice site, or per solvent-sized reference cell. Although the question calls it “per particle,” it cannot mean per polymer molecule: one chain occupies $N$ sites. Let the site volume be $v_0$, the total number of sites be $M=V/v_0$, and the chain and solvent counts be $n_p,n_s$, sowhereThe first term is the polymer's translational entropy of mixing. Its factor $1/N$ expresses that $N$ connected segments translate as one molecule. The second is the solvent's translational mixing entropy. Both are negative in the mixing free energy for $0<\varphi <1$, favoring mixing.

The final term is a mean-field excess contact contribution. The Flory interaction parameter $\chi$ measures the relative preference for unlike polymer-solvent contacts compared with like contacts. In the simplest lattice model with coordination number $z$ and pair energies ${ϵ}_{ps},{ϵ}_{pp},{ϵ}_{ss}$,Positive $\chi$ penalizes unlike contacts and promotes separation; negative $\chi$ favors them. Effective real-solution values can also include non-enthalpic effects, so their temperature dependence need not be exactly a constant divided by $T$. The subsequent derivatives assume composition-independent $\chi$ as in the supplied model.

Osmotic pressure and the special value one half. At fixed chain count, $\partial \varphi /\partial V=-\varphi /V$. Differentiating $F_{\mathrm{mix}}=(V/v_0)k_{B}Tg(\varphi )$ with respect to volume, relative to the pure-solvent reference, givesThe resulting osmotic pressure in Flory-Huggins solution theory isThe leading term is the ideal osmotic pressure of chains, since their number density is $\varphi /(N{v}_0)$. At $\chi =1/2$ the second-virial term vanishes: effective two-segment attraction and excluded-volume repulsion balance. This is the theta solvent condition for a sufficiently long dilute chain; the positive third-order contribution remains, so a finite-concentration solution is not a completely noninteracting gas. The site-volume factor is necessary to give physical pressure units.

For $\chi <1/2$, the effective pair excluded volume is positive. In a good solvent, the isolated chain expands relative to a Gaussian coil, with $R\sim b{N}^{\nu }$, $\nu \simeq 0.588$ or $3/5$ in the Flory approximation. For $\chi >1/2$, attraction wins in a poor solvent and a sufficiently long chain collapses toward a dense polymer globule, with asymptotic size $R\propto N^{1/3}$. Close to the theta condition, finite-chain crossover effects matter. At $\chi =1/2$, $R\sim b{N}^{1/2}$ to leading scaling order. This theta criterion is not the exact finite-$N$ phase-separation critical point; the lattice model has ${\varphi }_c=1/(1+\sqrt{N})$ and ${\chi }_c=(1+1/\sqrt{N}{)}^2/2$.

Overlap concentration. The polymer overlap concentration is reached when there is roughly one isolated coil in its own pervaded volume. If the coil size is $R$ and a repeat unit occupies volume $v_0$,With $R\sim b{N}^{\nu }$, the general scaling law isThe usual single-segment-volume estimate $v_0\sim b^3$ gives ${\varphi }^{\ast }\sim N^{1-3\nu }$: $N^{-1/2}$ in a theta solvent, approximately $N^{-4/5}$ in a good solvent, and order one for compact globules. A coil has no sharp surface, so choosing a radius of gyration or a geometric coil volume changes numerical prefactors.

For the stated polystyrene at $N=1500$ and $\chi =1/2$, the conventional scaling estimate isThis is a volume-fraction estimate, not a molar concentration. It uses the implicit site-volume approximation; the question does not provide an independent repeat-unit volume to determine a more precise prefactor.

Semidilute length scale. Beyond overlap, a semidilute polymer solution is characterized locally by a concentration blob size $\xi$. Suppose one blob contains $g$ statistical units. At the theta condition the subchain is Gaussian, so $\xi \sim b\sqrt{g}$, while matching its internal concentration to the bulk gives $\varphi \sim g{v}_0/{\xi }^3$. Eliminating $g$ gives the concentration-blob size in a theta solvent:Using the supplied statistical step length $b=0.67\mathrm{nm}$ and $\varphi =0.05$,The same convention gives $g\sim {\varphi }^{-2}=400<N=1500$, and the whole ideal chain has $R\sim b\sqrt{N}\simeq 26\mathrm{nm}$. Thus the blob is smaller than the chain. Because $\varphi /{\varphi }^{\ast }\simeq 1.94$ is only modestly above overlap, the value is a scaling estimate in a crossover region, not a precision asymptotic measurement. A statistical step length is not automatically a physical monomer diameter: if $v_0$ differs from $b^3$, the more general estimates are ${\varphi }^{\ast }\sim (v_0/b^3)N^{-1/2}$ and $\xi \sim v_0/(b^2\varphi )$, with the missing prefactor left explicit.

The Flory–Huggins solution theory treats a polymer solution as an incompressible lattice mixture, with one solvent molecule occupying one site and one chain occupying $N$ connected sites. Thus the supplied $f$ is naturally the mixing free energy per lattice site; if a site has volume $v_0$, the free-energy density is $f/v_0$.

The term $k_{B}T(\varphi /N)\ln \varphi$ is the polymer translational entropy of mixing contribution. There are only $1/N$ as many independently translating chains as monomers, which explains the factor $1/N$. The term $k_{B}T(1-\varphi )\ln (1-\varphi )$ is the solvent entropy of mixing contribution. Both lower the mixing free energy for $0<\varphi <1$.

The term $k_{B}T\chi \varphi (1-\varphi )$ is the mean-field excess interaction contribution from replacing like contacts with polymer–solvent contacts. In the simplest energetic lattice model,where $z$ counts neighboring contacts and the ${ϵ}_{ij}$ are pair contact energies. Positive $\chi$ penalizes unlike contacts and can favor demixing. Because chain connectivity suppresses the polymer's translational mixing entropy, relatively modest unfavorable interactions can cause separation of long chains. More general effective $\chi$ can also include entropic effects.

Particles within each species are indistinguishable, while the two species are physically distinguishable and their numbers are fixed. Independence makes the canonical partition function factor by species:Since $Z_1\propto T^{3/2}$ at fixed volume, $U=-{\partial }_{\beta }\ln Z=3N{k}_{B}T/2$. The canonical entropy is thereforeFor large species populations, Stirling's approximation gives the equivalent Sackur-Tetrode equation formFor structureless particles with the same translational partition function, $Z_1=V/{\lambda }^3$, where $\lambda$ is their common thermal de Broglie wavelength. More generally the supplied common constant factor in $Z_1$ should be retained in the entropy.

Writing $x_i=N_i/N$ separates the total-number contribution from the composition term:The last term is the entropy of mixing relative to separated ideal components at the same temperature and pressure. This is the entropy of a two-species classical ideal gas. Artificially labeling otherwise identical chemical atoms would not justify dividing by separate factorials or adding a mixing entropy. The pure-species limit is obtained by $x\ln x\to 0$.

A regular solution keeps the ideal combinatorial entropy of mixing while allowing a nonzero energetic preference for unlike contacts. For a symmetric binary mixture with equal molecular sizes, the displayed free energy per site combines mixing entropy with a contact penalty $\chi \varphi (1-\varphi )$. Its spinodal satisfies $1/\varphi +1/(1-\varphi )-2\chi =0$. At the symmetric thermodynamic critical point, the third composition derivative also vanishes, giving ${\varphi }_c=1/2$ and ${\chi }_c=2$. If $\chi =A/T$, the critical temperature is $A/2$. This is the equal-size limit of Flory–Huggins solution theory.

For two equal-site-volume polymers of chain length $N$, the mixing free energy per lattice site is $f/(k_{B}T)=[\varphi \ln \varphi +(1-\varphi )\ln (1-\varphi )]/N+\chi \varphi (1-\varphi )$. Each chain is one translating object, reducing the entropy of mixing per monomer by $N$ relative to the regular solution model. Its curvature is $f^{\prime \prime }/(k_{B}T)=1/[N\varphi (1-\varphi )]-2\chi$. The symmetric thermodynamic critical point has ${\varphi }_c=1/2$ and ${\chi }_c=2/N$. Long chains can therefore demix under a very small unfavorable Flory interaction parameter.