Mixture

A mixture contains two or more chemical substances combined in proportions that need not be fixed. A chemical compound instead has a definite composition: water has two hydrogen chemical atoms per oxygen chemical atom, whereas a sucrose and water mixture can contain different sugar proportions. A homogeneous mixture has uniform composition on the scale of representative samples; a heterogeneous mixture has distinguishable regions of different composition. A chemical solution is homogeneous, whereas sand dispersed in water is a particle suspension.^[ref]

The distinction concerns the sample, not merely its ingredient list. A vessel with dissolved sodium chloride and excess undissolved sodium chloride contains a chemical solution and an ionic crystal thermodynamic phase. The liquid portion can be homogeneous while the whole vessel is heterogeneous. Similarly, molecular oxygen dissolved in water is part of a chemical solution; visible gas bubbles introduce a separate gas thermodynamic phase.

A historical fuel gas mixture made by destructive distillation of coal. Its composition depends on the feed and retort conditions; it is not synonymous with natural gas or pure methane.

The complex condensable organic fraction produced during destructive distillation of coal.

The aqueous condensate and wash liquid from historical gasworks, containing dissolved Ammonia and other constituents.

The atmospheric gas mixture, principally nitrogen and oxygen near the Earth's surface, with variable water vapour and smaller amounts of other components. Gas relative density comparisons conventionally specify dry air as the reference.

The number of atoms of one constituent divided by the total number of atoms of the specified mixture.

A colloid contains dispersed particles in a continuous medium. A fine particle suspension can form a colloid; the particles in the Perrin sedimentation experiment illustrate this case.

A dispersion of gas bubbles in a liquid or solid matrix. Thin intervening films and interfacial forces govern stability; bubble diameter need not itself lie below a micrometre.

A fluid colloidal system. A solid-in-liquid example is fine silicon dioxide dispersed in water. The term can also include other fluid colloidal systems, so the dispersed and continuous constituents should be specified.

A force caused by confinement or restructuring of solvent near interfaces. It can depend on molecular layering and specific surface chemistry and need not have the range or sign of an electrostatic double-layer force.

A short-range solvent-mediated interaction associated with water organization and hydration near surfaces. It is distinct from an ideal point-ion double layer and can contribute repulsion at separations where continuum far-field models fail.

A dispersion of droplets of one liquid in another immiscible liquid. Its interfacial energy and kinetic persistence depend on droplet size, interfacial tension and stabilizing species.

Merging of droplets after their separating film fails. This differs from clustering of still-distinct droplets. Interfacial protection can delay merging without making the two bulk liquid phases molecularly miscible.

A surfactant-stabilized dispersion can remain macroscopically uniform while its two liquids remain distinct phases. Lower interfacial tension reduces the work of making droplets, and repulsive interfacial films can slow droplet coalescence. Kinetic emulsion stability does not by itself establish equilibrium molecular miscibility or a thermodynamically stable microemulsion.

An emulsion stabilized by particles adsorbed at the liquid–liquid interface. A partially wetting sphere has a detachment-energy scale $\pi r^2\gamma (1-|\cos \theta |{)}^2$, which can greatly exceed $k_{B}T$ for nanoparticles; complete wetting and insufficient coverage prevent a universal stability guarantee.

Aggregation caused when an adsorbing polymer chain binds more than one particle. Incomplete surface coverage can favour bridges, whereas a densely covered, well-solvated surface can instead be sterically stabilized.

Aggregation of colloidal particles when encounters lead to persistent sticking. Lowering an electrostatic barrier can change a suspension from reaction-limited to approximately diffusion-limited aggregation.

The distribution of particle dimensions in an ensemble, with number-, volume- or intensity-weighting specified. Broad size distributions can broaden a melting transition, frustrate colloidal crystallization and bias scattering-based size estimates. A mean size alone is then insufficient.

A spread of particle or molecular dimensions rather than a single size. A quantitative measure must state the weighting and statistic; particle-size variation and polymer chain-length variation are related examples but use different common indices.

The weight-average molar mass divided by the number-average molar mass. Its conventional written symbol is Đ. With number weighting it equals $1+\mathrm{Var}(M)/⟨M{⟩}^2$, so it is at least one and equals one only for a uniform molar mass. It measures one aspect of distribution breadth, not the full shape of a molar mass distribution.

An ordered arrangement of colloidal particles. Nearly monodisperse hard-sphere-like particles can order through packing entropy at sufficient concentration, while weak reversible attractions or templates can guide ordering. An irreversibly aggregated solid is not necessarily a colloidal crystal.

Resistance of a dispersion to aggregation or irreversible separation. Electrostatic and steric interactions and solvent compatibility can all matter.

For a sphere of radius $a$ and buoyant density difference $\Delta \rho$, define ${\mathrm{Pe}}_g=v_{s}a/D=m_{b}ga/(k_{B}T)$ using radius as the transport length. With Stokes law and Stokes–Einstein diffusion, ${\mathrm{Pe}}_g=4\pi \Delta \rho g{a}^4/(3k_{B}T)$. Replacing radius by diameter changes the convention by a factor of two. This local criterion does not guarantee a uniform suspension over a macroscopic vessel.

Setting the gravitational Péclet number to one gives the displayed particle-scale estimate. A macroscopic sample instead requires its height to be small compared with the sedimentation scale height. These different observational criteria give different size thresholds; “no sedimentation” is not an exact universal cutoff.

For noninteracting particles with positive buoyant mass $m_b$ in a gravitational field, a dilute thermodynamic equilibrium profile is proportional to $e^{-z/{\ell }_g}$ with ${\ell }_g=k_{B}T/(m_{b}g)$. Uniformity over a vessel height $H$ requires $H\ll {\ell }_g$. Requiring only ${\ell }_g$ to exceed a particle radius is a weaker microscopic gravity-versus-diffusion condition.

For gently curved interacting surfaces whose range and closest gap are small compared with their radii, integrate the local planar interaction over nearly parallel surface elements. With $W(s)$ the planar interaction free energy per area and $\Pi (s)=-W^{\prime }(s)$, the local gap for two spheres is $s\simeq h+r^2/(2R_{\mathrm{eff}})$, where $R_{\mathrm{eff}}^{-1}=a_1^{-1}+a_2^{-1}$. Thus $F=2\pi {\int }_0^{\infty }r\Pi (s)dr=2\pi R_{\mathrm{eff}}{\int }_h^{\infty }\Pi (s)ds=2\pi R_{\mathrm{eff}}W(h)$ when $W(\infty )=0$. Equal spheres have $R_{\mathrm{eff}}=a/2$; a sphere and plane have $R_{\mathrm{eff}}=a$. Use consistent surface boundary conditions when supplying the planar interaction.

For two equal spheres of radius $a$ separated by a surface gap $h\ll a$, a positive Hamaker constant $A_H$ gives the nonretarded Van der Waals force approximation $U=-A_{H}a/(12h)$ and $F=-dU/dh=-A_{H}a/(12h^2)$. The Derjaguin approximation derives the geometry factor from the planar interaction energy per area $W=-A_H/(12\pi h^2)$ and effective radius $a/2$. A sphere and plane instead have effective radius $a$, doubling the force at the same gap. Molecular repulsion, roughness and retardation limit these continuum formulas.

Particle aggregation is association of initially separate particles into larger clusters. For colloids, both the rate of Brownian motion encounters and the probability that an encounter produces attachment matter. Strong attraction with negligible repulsive barriers can give diffusion-limited cluster aggregation; a barrier that makes attachment unlikely can give reaction-limited cluster aggregation. Reversible association and bond rearrangement require additional kinetic assumptions.

Association of dispersed particles into larger clusters, often after repulsive stabilization is weakened. Salt screening, reduced surface charge, polymer bridging or depletion force can promote it. Clusters may sediment rapidly; preventing aggregation and preventing gravitational separation are distinct stability requirements.

In the standard single-seed model, individual particles execute random walks until first contact with a growing cluster, where they stick irreversibly. Exposed tips intercept more particles than screened inner regions, producing branched clusters. This differs from diffusion-limited cluster aggregation, in which whole existing clusters move and join one another. The two models need not share the same mass fractal dimension.

In reaction-limited cluster aggregation, encounters occur frequently compared with successful attachment: an activation or interaction barrier gives a small sticking probability. Repeated unsuccessful approaches allow clusters to sample more relative configurations before bonding, commonly producing denser aggregates than diffusion-limited cluster aggregation. Typical three-dimensional isotropic irreversible cluster models give mass fractal dimension around $2.1$. This structural contrast does not require previously formed bonds to break and restructure; it can arise from unsuccessful encounters alone. ^[ref]

In diffusion-limited cluster aggregation, mobile clusters undergoing Brownian motion attach with probability close to one when they encounter another cluster. Irreversible first-contact sticking tends to trap particles in exposed outer branches, producing open aggregates. In the common three-dimensional isotropic cluster model, the mass fractal dimension is roughly $1.8$. This is different from depositing individual random walkers onto one fixed cluster. Rotational motion, anisotropic attraction and bond restructuring can change the morphology.

For two dilute spherical species with absorbing contact and negligible interaction barriers, relative diffusion has coefficient $D=D_1+D_2$ and contact radius $R=a_1+a_2$. The steady concentration outside one target is $c(r)=c_{\infty }(1-R/r)$, solving the radial Laplace equation with zero contact concentration. Its inward diffusive flux is $4\pi DR{c}_{\infty }$, proving the displayed bimolecular rate coefficient. For identical particles, converting this encounter coefficient into a rate of loss of cluster number additionally needs the factor that prevents counting the same pair twice.

A colloidal glass is a disordered dense particle assembly with structural relaxation too slow on the observation timescale. Crowding, repulsion or short-range attraction can impede rearrangement. Kinetic arrest must be distinguished from an equilibrium phase transition to a crystal.

A colloidal gel is a mechanically persistent, space-spanning particle network arrested on the observation timescale. Attractions can produce such a network during aggregation or arrested phase separation. A gel is a dynamical, often nonequilibrium state; its appearance does not establish an additional equilibrium thermodynamic phase.

Polymer bridging occurs when one polymer chain attaches to more than one particle, producing an effective attraction or network connection. Surface affinity and incomplete coverage are essential to this mechanism. It should be distinguished from both steric stabilization by densely coated particles and the depletion force produced by nonadsorbing chains.

Steric stabilization is resistance to particle aggregation supplied by attached or adsorbed molecular layers. Bringing two coated particles close compresses or overlaps the layers, increasing their free energy through conformational and osmotic effects. It differs from the depletion force of nonadsorbing dissolved chains.

Repulsion associated with overlap of surface-bound layers, often through chain confinement and osmotic free-energy costs. Layer thickness, solvent quality and grafting/adsorption state matter; it is not simply a bare hard-sphere collision.

Stabilization by a surface-bound polymer layer that supplies both conformational/osmotic repulsion and electrostatic repulsion. Chain coverage, solvent quality and screening determine the resulting interaction barrier.

A depletion force is an effective interaction caused by exclusion of smaller particles or macromolecules from regions around larger objects. Overlap of the excluded regions increases the accessible volume for the smaller species. At fixed reservoir osmotic pressure ${\Pi }_p$, the associated pair free energy is $-{\Pi }_p$ times the overlap volume in the ideal dilute-depletant approximation. It is an entropic attraction for two large hard particles. Surface adsorption can introduce different interactions, so adding an arbitrary polymer does not by itself guarantee depletion attraction.

The Asakura-Oosawa model treats dilute nonadsorbing polymer coils as mutually interpenetrable ideal particles excluded from a layer around each colloid. For colloid radius $a$ and polymer exclusion thickness $R_g$, two exclusion spheres of radius $R_e=a+R_g$ overlap when $2a\le R<2R_e$. Adding their two spherical cap volumes gives $V_{\mathrm{ov}}=\pi (4R_e+R)(2R_e-R{)}^2/12$. Multiplication by $-{\Pi }_p$ gives the displayed pair free energy, with zero depletion interaction beyond $2R_e$. The range is controlled by polymer size and the depth by reservoir polymer concentration. At larger polymer concentration, polymer interactions and many-body overlap effects require care.

DLVO theory combines screened electrostatic repulsion from overlapping electric double layers with attractive Van der Waals forces to model colloid stability. A repulsive barrier can inhibit aggregation even when a deep short-distance attractive minimum exists. Increasing electrolyte concentration reduces the screening length and can lower that barrier. Molecular surface forces and polymer-mediated interactions require additions to this description.

A deep short-distance minimum can bind contacting particles, while a more distant shallow minimum can permit loose reversible aggregates. A repulsive barrier between them controls kinetic access to contact. Salt can lower that barrier through screening. Molecular-scale steric or hydration forces lie outside the simplest continuum dispersion-plus-double-layer model.

For a symmetric monovalent reservoir with number density $n$ of each species, linearizing Poisson-Boltzmann theory gives the displayed inverse screening length. A planar potential decays exponentially; a spherical far field also has a geometrical inverse-distance factor. Increasing salt reduces the range of repulsion. Fixed surface potential, fixed charge and charge regulation need not give identical interaction amplitudes.

Like-charged colloids can resist aggregation because bringing their electric double layers together costs free energy. In DLVO theory, screened electrostatic repulsion competes with attractive Van der Waals forces. A repulsive barrier much larger than $k_{B}T$ suppresses thermally activated aggregation, although the dispersion can remain metastable. Lower ionic strength generally lengthens the Debye–Hückel screening length, while multivalent counterions or added salt shorten it and can lower the barrier. The surface charge, charge regulation and solvent-dependent attraction also matter; screening length alone does not guarantee stability.

The critical coagulation concentration is the approximate electrolyte concentration separating slow, barrier-limited aggregation from rapid aggregation of a charge-stabilized colloid. In the simplest DLVO theory, increasing salt shortens the Debye–Hückel screening length and lowers the repulsive barrier until rapid aggregation is possible. It depends on particle charge, solvent, temperature and particularly counterion valence. It is a kinetic crossover, not a universal salt concentration or necessarily an equilibrium phase boundary. Aggregation just below this crossover can be reaction limited; exactly at or above the rapid-aggregation threshold, proximity to the threshold alone does not imply a small sticking probability.

Strong sensitivity of a lyophobic colloid's critical coagulation concentration to counterion valence. A high-surface-potential DLVO approximation gives $c_{\mathrm{ccc}}\propto z^{-6}$; specific adsorption and charge regulation can invalidate this simple scaling.

A screened Coulomb pair potential models the effective electrostatic interaction of particles in an electrolyte. For identical weakly charged spheres of radius $a$, charge $Q$, and Debye–Hückel screening length ${\kappa }^{-1}$, a common pairwise approximation is $U(R)=Q^2{e}^{-\kappa (R-2a)}/[4\pi \epsilon (1+\kappa a{)}^{2}R]$ for $R\ge 2a$. The near-contact interaction can require more detailed boundary conditions or nonlinear screening. Charge, screening length and temperature all matter to a resulting phase diagram.

A chemical solution is a homogeneous mixture in which designated solutes are distributed in a solvent. Representative equal-volume samples have the same composition after mixing. Individual molecules and ions remain distinguishable. The solute/solvent labels describe their roles in this mixture.^[ref]

A dissolved solute is dispersed as molecules, chemical atoms, or ions, rather than as the larger solid particles of a particle suspension. For example, sucrose retains its molecular formula during dissolution in water; a chemical reaction is not required. Dissolved sodium chloride supplies sodium cations and chloride anions, surrounded by water molecules through solvation. Dissolved does not mean chemically destroyed or converted into water.^[ref]

This makes a useful connection between microscopic identity and laboratory sampling. If a chemical solution is well mixed, the amount of substance in a measured portion follows from its concentration and volume. A scoop from a poorly mixed particle suspension need not represent the whole vessel. Visual clarity alone cannot establish composition or exclude a fine dispersion; the quantitative claim needs representative sampling.

A solution whose concentration has been established for use as a measurement reference, for example in titration. It can be prepared from a primary standard or established by titrant standardization. The word “standard” does not imply one universal concentration.

The distribution of an element among chemical forms distinguished by oxidation state, complexation or association with phases. Speciation affects solubility and partitioning.

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.

A chemical solution with a polymer as a solute. The polymer conformation and interactions depend on the solvent and concentration.

Association of multiple polymer chains into a larger solution object.

The extent to which a solvent favors polymer–solvent contacts relative to polymer–polymer contacts.

A solution above the polymer overlap concentration but below the concentrated regime, with interpenetrating chains. Local polymer structure is characterized by a correlation length or concentration-blob size. Scaling formulas describe a broad semidilute regime most reliably when the concentration greatly exceeds overlap while remaining small compared with one in volume fraction.

In a planar good-solvent solution, a correlation blob of $g$ monomers has size $\xi \sim \ell g^{3/4}$. Its area fraction is $\varphi \sim g{\ell }^2/{\xi }^2\sim g^{-1/2}$, giving $g\sim {\varphi }^{-2}$ and $\xi \sim \ell {\varphi }^{-3/2}$. Each blob contributes order $k_{B}T$ to the osmotic free energy, so the surface osmotic pressure is $\Pi \sim k_{B}T/{\xi }^2\sim (k_{B}T/{\ell }^2){\varphi }^3$. This regime requires many blobs per chain and area fraction below close packing. In the dilute regime entire chains are the independently translating objects and $\Pi \sim k_{B}T\varphi /(N{\ell }^2)$. The two expressions match at the two-dimensional polymer overlap fraction ${\varphi }^{\ast }\sim N^{-1/2}$.

A region of size $\xi$ containing a typical correlated chain segment whose internal monomer concentration matches the bulk concentration. If it contains $g$ statistical units with isolated size law $\xi \sim b{g}^{\nu }$ and unit volume $v_0$, concentration matching gives $\varphi \sim g{v}_0/{\xi }^3$. Eliminating $g$ gives $\xi \sim b(\varphi b^3/v_0{)}^{-\nu /(3\nu -1)}$. Numerical prefactors and distinctions among scattering, osmotic and geometric correlation lengths require additional definitions.

In a three-dimensional semidilute good solvent, take $g$ statistical segments per blob with size $\xi \sim a{g}^{3/5}$ and concentration matching $\varphi \sim g{a}^3/{\xi }^3$. Eliminating $g$ gives the displayed relations. A developed blob regime requires $1\ll g\ll N$ and a volume fraction well below close packing. Within a blob the chain retains isolated swollen statistics; on larger scales interpenetrating chains screen excluded-volume effects. The exponents use $\nu =3/5$ and the site-volume assumption. For general $\nu$, $\xi \sim a{\varphi }^{-\nu /(3\nu -1)}$.

For an ideal chain inside a blob, $\nu =1/2$ and $\xi \sim b\sqrt{g}$. Together with $\varphi \sim g{v}_0/{\xi }^3$, this gives the displayed concentration-blob size. Under the site-volume approximation $v_0\sim b^3$, $\xi \sim b/\varphi$ and $g\sim {\varphi }^{-2}$. The estimate describes local overlap rather than the full chain size and requires $1\ll g\ll N$ for a well-developed semidilute regime.

A solvent under conditions where the leading effective pair excluded volume of a long dilute polymer vanishes. In the simplest Flory–Huggins solution theory this means $\chi =1/2$. The chain is approximately an ideal chain, while three-segment interactions can still affect the osmotic pressure of a concentrated solution. A particular polymer-solvent pair often meets this condition only near its theta temperature.

The overlap concentration is the concentration at which distinct dilute polymer coils first substantially interpenetrate. It corresponds to roughly one chain per coil volume. If a chain contains $N$ repeat units, has molar mass $M$ and occupies a coil volume $V_{\mathrm{coil}}$, then its chain number density is approximately $1/V_{\mathrm{coil}}$, its repeat-unit number density is $N/V_{\mathrm{coil}}$, and its mass concentration is $M/(N_A{V}_{\mathrm{coil}})$. A scaling convention uses $V_{\mathrm{coil}}\sim R^3$; a commonly chosen geometric convention uses $V_{\mathrm{coil}}=4\pi R_g^3/3$. These give different order-one prefactors because a coil has no sharp surface. For an ideal chain, $R\sim b{N}^{1/2}$ and a site-volume estimate $v_0\sim b^3$ gives overlap volume fraction ${\varphi }^{\ast }\sim N^{-1/2}$. A good solvent chain with the Flory approximation $R\sim b{N}^{3/5}$ instead gives ${\varphi }^{\ast }\sim N^{-4/5}$.

The polymer overlap volume fraction counts the segment volume of one chain divided by its coil volume. If the segment volume is $v_0$ and $R\sim b{N}^{\nu }$, then ${\varphi }^{\ast }\sim (v_0/b^3)N^{1-3\nu }$. The site-volume convention $v_0\sim b^3$ gives $N^{-4/5}$ for the three-dimensional Flory approximation and $N^{-1/2}$ for an ideal chain. A geometric coil volume such as $4\pi R_g^3/3$ changes a prefactor; segment length and segment volume are not generally interchangeable without a model convention.

For polymers restricted to an interface, overlap means roughly one coil per coil area, $n_c{R}^2\sim 1$. A chain contains $N$ monomers with typical occupied area ${\ell }^2$, so its overlap area fraction is ${\varphi }^{\ast }\sim N{\ell }^2/R^2$. With the planar good-solvent exponent $\nu =3/4$ from the two-dimensional Flory approximation, this gives ${\varphi }^{\ast }\sim N^{-1/2}$. The same exponent happens to occur for a three-dimensional ideal chain's volume fraction, but its geometric origin is different.

A poor solvent favors effective attraction between polymer segments relative to solvent contacts. A dilute flexible chain tends to contract, and sufficiently unfavorable conditions produce a compact polymer globule. Solvent quality depends on the polymer, solvent and temperature. The simple Flory–Huggins solution theory identifies the poor-solvent side of the theta condition with $\chi >1/2$; this does not guarantee that every finite-density mixture is inside its spinodal.

A polymer globule is a compact conformation favored by effective segment attractions in a poor solvent. If its internal segment density is independent of chain length, its volume is proportional to $N$ and a size measure scales as $N^{1/3}$. The proportionality between physical volume and the cube of a radius of gyration depends on shape and density profile. Compactness alone supplies scaling, not a uniquely determined numerical monomer volume.

The theta temperature is the temperature at which the leading effective two-segment interaction of a dilute long polymer vanishes, so its large-scale size is approximately that of an ideal chain. In the simplest Flory–Huggins solution theory this is the condition $\chi =1/2$. It is the infinite-chain limit of the upper critical temperature in the usual energetic model, not the critical temperature of every finite chain.

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.

An incompressible, equal-site-volume mean-field approximation for two polymers of degrees of polymerization $N_A,N_B$. Here $f$ is mixing free energy per site and $\varphi$ is the volume fraction of species $A$. For a composition-independent Flory interaction parameter, local stability requires $1/(N_A\varphi )+1/[N_B(1-\varphi )]-2\chi >0$. The thermodynamic critical point is ${\varphi }_c=\sqrt{N_B}/(\sqrt{N_A}+\sqrt{N_B})$ and ${\chi }_c=(N_A^{-1/2}+N_B^{-1/2}{)}^2/2$.

Connecting segments into chains reduces their independent translational mixing entropy per segment by factors $1/N_A$ and $1/N_B$. Consequently the tolerable positive contact penalty tends to zero as both chains become long. Equal chain lengths have ${\chi }_c=2/N$; favorable unlike interactions can still produce a miscible polymer pair.

For a single-site solvent and polymer length $N$, expand $(1-\varphi )\ln (1-\varphi )=-\varphi +{\varphi }^2/2+{\varphi }^3/6+O({\varphi }^4)$ in Flory–Huggins solution theory. The displayed result follows. Its linear term changes reference chemical potentials; its quadratic term sets the effective two-segment excluded volume. The latter vanishes at the theta temperature, while the positive cubic term remains. Taking $\varphi f^{\prime }-f$ yields the osmotic pressure in Flory-Huggins solution theory, whose cubic coefficient is $1/3$ rather than the free energy's $1/6$.

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.

For a strongly segregated, symmetric long-chain polymer blend, take the square-gradient functional $F/(k_{B}T{\rho }_0)=\int [\chi \varphi (1-\varphi )+a^2({\varphi }^{\prime }{)}^2/(24\varphi (1-\varphi ))]dz$. Its first integral equates the two terms, giving ${\varphi }^{\prime }=\sqrt{24\chi }\varphi (1-\varphi )/a$. Thus $\varphi =[1+\tanh (z/\ell )]/2$ with $\ell =a/\sqrt{6\chi }$. The inverse-central-slope width $w=1/{\varphi }^{\prime }(0)=2\ell$ differs by a factor two from the tanh-argument scale. The continuum approximation is marginal when these lengths approach the statistical segment size. An observed width can additionally include roughness and instrument broadening.

Let $f(\varphi )$ be the mixing free energy per lattice site in Flory–Huggins solution theory, $v_0$ the site volume, and $\varphi$ the polymer volume fraction. The solvent mixing chemical potential is $f-\varphi f^{\prime }$, so its balance against the pure-solvent pressure contribution gives $\Pi =(\varphi f^{\prime }-f)/v_0$. For composition-independent $\chi$,The first term is the ideal chain translational contribution; the second encodes the effective two-segment excluded volume. At the theta temperature condition it vanishes but the positive third-order term remains. Treating $f$ as a free-energy density without dividing by $v_0$ loses the physical dimensions of osmotic pressure.

The Flory interaction parameter measures the excess free-energy cost of unlike contacts in Flory–Huggins solution theory. For a simple lattice of coordination number $z_c$ and contact energies ${ϵ}_{ps}$, ${ϵ}_{pp}$ and ${ϵ}_{ss}$,Real effective parameters may also contain entropic contributions and composition dependence. With site volume $v_0$ and composition-independent $\chi$, the effective two-segment excluded volume is $v_{\mathrm{eff}}=v_0(1-2\chi )$. Positive, zero and negative $v_{\mathrm{eff}}$ correspond respectively to a good solvent, the theta temperature condition and a poor solvent. In particular, $\chi =1/2$ means cancellation of leading repulsive and attractive contributions; it does not mean that every molecular interaction vanishes.

For composition-independent $\chi$, the spinodal of Flory–Huggins solution theory is $2\chi =1/(N\varphi )+1/(1-\varphi )$. At its thermodynamic critical point, a second condition $-1/(N{\varphi }^2)+1/(1-\varphi {)}^2=0$ gives $1-{\varphi }_c=\sqrt{N}{\varphi }_c$, proving the displayed values. Hence ${\varphi }_c\sim N^{-1/2}$ and ${\chi }_c-1/2\sim N^{-1/2}$. Under $\chi =C/T$, the theta temperature is $\Theta =2C$ and $T_c=\Theta /(1+N^{-1/2}{)}^2$. Thus $T_c$ approaches a constant while $\Theta -T_c$ scales as $N^{-1/2}$. Converting ${\chi }_c$ to a temperature always requires a specified or inferred $\chi (T)$ law.

A good solvent favors contact with polymer segments strongly enough that the effective excluded volume is positive. A long dilute flexible chain swells relative to an ideal chain; the three-dimensional size exponent is approximately $0.588$, often replaced by $3/5$ in a scaling estimate. This designation depends on the polymer, solvent and temperature, rather than on the solvent alone.

The denominator of molar concentration is the final volume of the chemical solution, including the dissolved solute. It is not the volume of solvent initially poured in. A volumetric flask fixes the final total volume; weighing the solute supplies its mass, and a specified molar mass converts that mass to amount of substance.^[ref]

Consider an original illustrative account, not reported measurements. Suppose $2.922\mathrm{g}$ of pure dry sodium chloride, with stipulated molar mass $58.44\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, is completely dissolved and quantitatively transferred into a chemical solution whose final volume is $250.0\mathrm{mL}$. Assume no losses, no additional chemical reactions, and complete dissociation of an ionic compound. ThenHere analytical concentration counts the sodium chloride inventory, not intact dissolved $\mathrm{NaCl}$ molecules. Under these assumptions, the amount of substance of sodium cations and of chloride anions is separately $0.05000\mathrm{mol}$; each has molar concentration $0.2000\mathrm{mol}{\mathrm{L}}^{-1}$. Their combined ionic amount of substance is $0.1000\mathrm{mol}$, but this does not double either the salt inventory or the mass. Each sodium cation still contains a sodium atomic nucleus, and each chloride anion a chlorine atomic nucleus. Counting chemical elements survives dissociation of an ionic compound.

The mass ledger is different from the volume ledger. If the final chemical solution had an illustrative net mass of $252.922\mathrm{g}$, the water contribution would be $252.922-2.922=250.000\mathrm{g}$, assuming nothing entered or escaped except those two components. The ratio $2.922/252.922\approx 0.011553$ expresses the solute share of total mass; the separate mass density would be $252.922/250.0=1.011688\mathrm{g}\mathrm{m}{\mathrm{L}}^{-1}$ before rounding. These stipulated quantities are mutually consistent bookkeeping inputs, not a measured mass density for saltwater. Neither establishes that $250.0\mathrm{mL}$ of water was added.

More generally, with full transfer and conserved solute, $n=m/M$ and $c=m/(MV)$. No additive volume or ideal chemical solution assumption is needed when $V$ is the measured final volume. If only a fraction $f$ of the weighed solute reaches the flask, the actual result is $c=fm/(MV)$. This short calculation explains why an undissolved residue or an incomplete rinse creates a systematic deficit even when the final calibration mark is read correctly.

A calibration mark can be tested against an independent mass measurement. The Buffalo State-hosted Laboratory Experiment 1. Calibration of Volumetric Glassware uses a $25\mathrm{mL}$ pipette, filler, conical flask, weighing bottle, balance, and thermometer. It specifies clean, freely draining glassware, bubble-free filling, an eye-level liquid meniscus reading, gravity drainage, and retaining the residual liquid allowed for by the pipette's calibration. Degassed water is delivered to a tared bottle; ten weighings and initial/final temperature readings give repeated delivered volumes using the water mass density. Their mean and standard deviation assess delivery and repeatability. The balance and thermometer must themselves be checked before claiming accuracy. The same handout describes weighing an empty volumetric flask and then weighing it filled to its mark.^[ref]

The independent inference is $V=m_{\mathrm{water}}/{\rho }_{\mathrm{water}}(T)$. Agreement between repeats constrains scatter; agreement with the nominal delivery additionally needs trustworthy mass density, temperature, and balance readings. This experiment tests delivered or contained volume under its conditions. It does not prove that a subsequent solute transfer is complete or a prepared chemical solution is homogeneous.

The University of Wisconsin–Madison teaching module supplies the complementary preparation demonstration. Dissolved solute and vessel rinsings reach the volumetric flask; the contents reach room temperature before a dropper adjusts the eye-level liquid meniscus to the calibration mark. The module excludes a stirring bar during filling because its displaced volume is absent from the calibration, and requires mixing the neck liquid with the bulk chemical solution. Its self-check identifies incomplete dissolution and transfer as a cause of lower-than-intended molar concentration.^{[ref][ref][ref]}

The observable endpoints are a liquid boundary at a mark and a delivered mass, not a direct count of solute particles. An overfilled flask has excessive final volume; removing some well-mixed chemical solution also removes solute, so it cannot recover the intended inventory merely by restoring the mark. Measurement uncertainty, contamination, incomplete transfer, and sampling before full mixing limit the inferred molar concentration. The apparatus demonstrates why the words “dilute to” refer to final chemical solution volume.

A liquid meniscus is the curved boundary where a liquid meets a vessel wall. In the transparent water examples above, the bottom of this curve is aligned with the calibration mark at eye level. Reading from above or below changes the apparent alignment without changing the contained volume.

The concentration-based ionic strength is $I_c=\frac{1}{2}{\sum }_i{c}_i{z}_i^2$, where $c_i$ is each dissolved ion's concentration and $z_i$ its signed electric charge number. It has the same units as $c_i$; some formulas instead use the dimensionless ratio $I_c/c^{\circ }$. All dissolved ions contribute, including those that balance the electric charge of a buffer solution's conjugate base. Ionic strength helps describe the interactions represented by activity coefficients. See Hippler and Metcalfe, activity coefficients and dimensionless ionic strength.

A solute is a component of a chemical solution distinguished from its solvent, typically present in a smaller proportion. There can be several solutes. A concentration must name both what is counted and the denominator: molar concentration uses amount of substance per final chemical solution volume; a solute-to-solvent mass ratio has another denominator.^[ref]

“Added sodium chloride” and “dissolved sodium cations” describe different levels of the same inventory. Dissociation of an ionic compound changes the grouping of the particles; it preserves their chemical elements. A dissolved ion remains an ion, and dissolved sucrose remains molecular unless a separate chemical reaction occurs. Thus the name of the weighed solute need not be the name of every dissolved chemical species.

A solvent is the component designated as the medium of a chemical solution, commonly the component present in greatest proportion. Water is a common solvent, not a necessary one. A mixture of mostly ethanol with a little water can designate ethanol as solvent and water as solute. Chemical solutions can also be gas or solid mixtures; molecular nitrogen is the principal medium of ordinary well-mixed air. When proportions are comparable, state the convention explicitly.^[ref]

The solvent is part of the final sample. Choosing it influences solubility and solvation; measuring its added volume alone does not determine the final chemical solution volume. Solvent and solute masses add in a closed account, whereas the final volume should be measured rather than inferred by adding pre-mixing volumes.

A solvent containing an appropriate proton-donating group, often an O–H or N–H group. It can hydrogen bond, shuttle protons and quench strongly basic organometallic reagents.

A polar solvent lacking an acidic hydrogen-bond-donor group. Relative to strongly protic solvation, it can leave anionic nucleophiles more reactive, while counterion and solubility effects still matter.

Hydrodynamic size describes the effective dimensions of a dissolved molecule moving with its surrounding solvent. It depends on molecular geometry, internal shape, and solvent conditions, rather than just molar mass. SEC pore access probes size-dependent accessibility of the packing.

The hydrodynamic radius is the radius of a hypothetical no-slip sphere with the same translational diffusion coefficient as the dissolved species. In the dilute continuum regime the Stokes–Einstein relation giveswhere $k_B$ is the Boltzmann constant, $T$ the absolute temperature, $\eta$ the dynamic viscosity of the solvent, and $D$ the translational diffusion coefficient. This effective radius need not equal the radius of gyration or a geometric molecular radius. It is a useful descriptor for SEC molar mass calibration, rather than a complete specification of how every shape enters every pore.

Approximating a dilute polymer coil as a hydrodynamic object of size $R$ gives $D\sim k_{B}T/(\eta R)$. If a stipulated model uses the theta-chain end-to-end distance $R=2a\sqrt{N}$ as a literal solid-sphere radius, Stokes law gives $D=k_{B}T/(12\pi \eta a\sqrt{N})$. The scaling matches theta-solvent hydrodynamic coupling, but a permeable coil's hydrodynamic radius is not generally equal to its end-to-end distance.

Dilution lowers a solute's concentration by adding solvent. For conserved amount of substance, molar concentration satisfiesBoth $V_1$ and $V_2$ are chemical solution volumes; $V_2$ is the final total, not the added solvent.^[ref]

The proof is simply the definition $n=cV$ applied before and after. Its useful assumptions are full transfer of the selected portion, no loss or gain of the counted solute, and complete mixing. An ideal chemical solution approximation is not required for this inventory equation. If chemical equilibrium redistributes dissolved chemical species, use a conserved analytical concentration or chemical element inventory rather than assuming each individual species is conserved. This conservation statement does not by itself preserve the pH: a chemical equilibrium may change as the chemical solution is diluted.

Serial dilution repeatedly transfers a measured portion of one well-mixed chemical solution and dilutes that portion to a new final volume. With transferred volume $v_j$ and final volume $V_j$ at step $j$, conserved solute givesThis product follows by repeated substitution. Each step conserves the solute in the portion transferred into its receiving flask, not the entire inventory in the preceding flask. Uniform mixing makes the selected fraction representative.

Continue the original solution mass and volume bookkeeping example. A $10.00\mathrm{mL}$ portion of the $0.2000\mathrm{mol}{\mathrm{L}}^{-1}$ stock contains $0.002000\mathrm{mol}$ of analytical sodium chloride, corresponding to $0.11688\mathrm{g}$ before rounding. After dilution to $100.0\mathrm{mL}$, its analytical concentration is $0.02000\mathrm{mol}{\mathrm{L}}^{-1}$. A new $5.000\mathrm{mL}$ portion of that mixed chemical solution contains $0.0001000\mathrm{mol}$; dilution to $50.00\mathrm{mL}$ gives $0.002000\mathrm{mol}{\mathrm{L}}^{-1}$. The two factors $10.00/100.0$ and $5.000/50.00$ multiply to $1/100$.

The remaining stock holds $0.05000-0.002000=0.04800\mathrm{mol}$; the remaining first dilution holds $0.002000-0.0001000=0.001900\mathrm{mol}$. Including the final flask restores the starting inventory:This is a direct check that making a dilute final sample does not destroy the solute. Most of it remains in earlier vessels.

Serial dilution makes a small final concentration accessible using measurable portions instead of a single extremely small transfer. Its limitation is that systematic errors propagate: if each transferred volume is $1\%$ below its intended value while final volumes are correct, two steps give $0.99^2=0.9801$ times the intended concentration, a $1.99\%$ deficit. Repeated readings alone do not remove that bias. The gain in manageable volume therefore depends on calibrated transfer and representative mixing.

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.

In the ideal solution and ideal gas vapor approximation, each component's partial pressure equals its liquid mole fraction times its pure-component vapor pressure at the same temperature:These pressure formulas neglect the change in the pure-liquid reference between its saturation pressure and $P$; appreciable real gas behavior also requires a more general treatment. The ideal solution definition in terms of chemical potentials does not require either approximation.^[ref]

A short derivation explains the reference choice. At vapor-liquid equilibrium, each component has the same chemical potential in the liquid and vapor. For an ideal gas mixture, chemical potential of an ideal gas giveswhere $p^{\circ }$ is a reference pressure. Approximate ${\mu }_i^{\ast }(T,P)$ by ${\mu }_i^{\ast }(T,p_i^{\ast })$ and subtract the pure-component equation from the mixture equation:Taking the exponential function gives Raoult's law. Summing partial pressures and normalizing gives $P$ and $y_i$; this last step uses the ideal gas vapor assumption separately from the liquid model.

The law is named after François-Marie Raoult, who measured vapor pressures and freezing behavior of chemical solutions.^[ref]

For the stipulated A/B ideal solution, put $x=x_A$. At the fixed temperature $300\mathrm{K}$, Raoult's law reduces toThe relative volatility is $\alpha =p_A^{\ast }/p_B^{\ast }=4$. For $0<x<1$,Hence every interior vapor-liquid equilibrium pair in this chosen model has A-enriched vapor; neither pure endpoint is enriched. Conversely,For example, $y_A=1/2$ corresponds to $x_A=1/5$, $p_A=p_B=16\mathrm{kPa}$ and $P=32\mathrm{kPa}$. Equal vapor mole fractions therefore need not mean equal liquid mole fractions.