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The partial pressure of a component of a gas mixture is the pressure it would exert alone at the same temperature and total volume. For an ideal mixture, $p_i=n_{i}RT/V$ and $p={\sum }_i{p}_i$. The chemical potential of an ideal gas uses $p_i$ for that component rather than the total pressure $p$.

The seven named mixtures of the 1864 British Pharmacopoeia are Mistura Ammoniaci, Mistura Amygdalae, Mistura Creasoti, Mistura Cretae, Mistura Ferri Composita, Mistura Guaiaci and Mistura Scammonii. Their compositions are as follows; avoirdupois ounces are used for solids and fluid ounces for liquids.

The category is an official preparation name, not every possible bedside combination of medicines. Camphor water and acacia mucilage are not additional entries under Mistura: the 1864 names are Aqua Camphorae and Mucilago Acaciae. ^[ref]

Chemical elements are the constituents from which ordinary chemical compounds are built; many familiar materials are mixtures of those chemical compounds and sometimes free chemical elements. In the historical operational definition, a chemical element could not be resolved into simpler substances by chemical means. In the modern definition, its chemical atoms all have the same atomic number. Ordinary chemical reactions rearrange those chemical atoms rather than creating a new chemical element.

Air is a mixture, chiefly of nitrogen and oxygen, whose components can be separated or selectively consumed. For example, burning phosphorus in a suitable enclosed sample removes most of the oxygen and leaves chiefly nitrogen; the residual gas has different properties from the original air. Air also contains carbon dioxide, variable water vapour and smaller amounts of other gases. Its variable composition is incompatible with treating it as one elementary substance.

Water is a chemical compound: electrolysis produces molecular hydrogen and molecular oxygen, approximately in the volume ratio $2:1$ at the same temperature and pressure. Their recombination produces water again:Thus water has fixed composition but is chemically decomposable, whereas air has no fixed single chemical formula. Neither belongs with the chemical elements merely because it is common and widespread on the Earth's surface.

For a conditional spin measurement of identical fermions addressing the $z$ component in one spatial mode, the Born rule givesConditioned on this result, the mode spin state is $|{\uparrow }_a{\downarrow }_b⟩$; the other outcome produces $|{\downarrow }_a{\uparrow }_b⟩$. The spin product is not a spin-exchange eigenstate. The measurement has selected an addressed mode and recorded an outcome, so this conditional spin description distinguishes the measured subsystem.

The full electron wavefunction still obeys fermionic exchange symmetry. For example, the conditional spin-up outcome in spatial mode $a$ is represented in particle coordinates byIt is antisymmetric under full particle exchange. Thus the loss of a definite symmetry of the conditional spin product does not mean that measurement turns identical electrons into distinguishable particles. If the outcome is discarded, the mode spin state is instead the equal mixture of the two opposite-spin products. The measurement must specify which spatial mode is addressed for the conditional statement to be operationally meaningful.

For an equilibrium contact angle measured through the liquid, Young's equation balances the tangential interfacial forces:The pure-ethanol measurement therefore fixes the solid–vapour surface energy asThe inferred surface energy is $40.7\mathrm{mJ}{\mathrm{m}}^{-2}$, equivalently $40.7\mathrm{mN}{\mathrm{m}}^{-1}$. The equality of these units follows because creating interfacial area costs energy.

For the equal-fraction mixture, the stated linear relations give ${\gamma }_{SL}=20.5\mathrm{mN}{\mathrm{m}}^{-1}$ and ${\gamma }_{LV}=28.5\mathrm{mN}{\mathrm{m}}^{-1}$. Holding the solid–vapour surface energy fixed,The small decrease in ${\gamma }_{SL}$ increases the numerator of Young's equation, but the increase in ${\gamma }_{LV}$ has the larger relative effect and reduces $\cos \theta$. Consequently the contact angle increases and the mixture wets the surface less well. This inference uses equilibrium wetting on an unchanged smooth solid; a changing surface composition or contact-angle hysteresis would require additional measurements.

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.

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]

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 a two-component mixture, let $x$ be component A's mole fraction in the liquid and $y$ its mole fraction in the coexisting vapor. At interior compositions, define the relative volatility of A with respect to B by the ratio of composition odds:Thus $\alpha >1$ means A is enriched in the vapor, while $\alpha =1$ means $y=x$. Solving this definition givesHere a constant relative volatility is a stipulated model of vapor-liquid equilibrium, not a measured property of a named chemical solution. Real relative volatility can vary with composition, temperature, and pressure. The relation specifies no absolute temperature scale: even fixing $\alpha =2$ does not determine a boiling-temperature curve or a heat requirement.

For the state-space model $S_t=G{S}_{t-1}+w_t$, $X_t=F{S}_t+v_t$, assume independent Gaussian process noise and measurement noise of variances $W,V$, independent of the prior state and previous observations. A normal distribution for the filtered state, with mean $m$ and variance $P$, predicts mean $Gm$ and variance $R=G^{2}P+W$. Here $R$ denotes predicted state variance, rather than the observation-noise covariance notation used in the vector equations above. Gaussian conditional expectation then gives posterior mean $Gm+K(X-FGm)$ and posterior variance $RV/(F^{2}R+V)$, where the Kalman gain is $K=FR/(F^{2}R+V)$.

Mere weak white noise assumptions do not imply normal posterior distributions: independent process noise with a Rademacher distribution makes the next state a mixture of two shifted normal distributions. The Kalman filter can still describe a best affine state estimate under appropriate second-order assumptions, but that is distinct from an exact Gaussian conditional law.

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 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.

The equilibrium pressure of a vapor in contact with its liquid or solid at a specified temperature. For pure liquid–gas phase coexistence with negligible interface effects, it is the pressure on their phase coexistence curve. In a gas mixture, distinguish the component's partial pressure from the total pressure; the pure-substance Integrated Clausius-Clapeyron relation does not describe the total pressure of arbitrary humid air.