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Electroneutrality means that positive and negative charges balance on the length scale under consideration. For a bulk electrolyte with ionic concentrations $c_i$ and charge numbers $z_i$, the approximation is ${\sum }_i{z}_i{c}_i=0$.

A solution containing sodium and sulfate ions therefore needs twice as many singly charged sodium ions as doubly charged sulfate ions if these are its only ions. Local charge separation still occurs near an electrode in an electric double layer. Thus bulk electroneutrality does not mean that every microscopic region is neutral, and it should not be imposed pointwise when modeling a charged interface.

An ecologist studies ecology.

A solar eclipse occurs when the Moon passes in front of the Sun for an observer on Earth. The same event appears different from different locations: an observer in the umbra sees totality, while an observer in the penumbra sees only part of the solar disk covered.

If the Moon's apparent angular diameter is too small to cover the Sun, the central alignment produces an annular eclipse instead. The comparison is between apparent sizes, which depend on both physical size and Euclidean distance. During totality the normally overwhelming solar disk is hidden, exposing the much fainter solar corona and allowing observations of nearby background stars.^[ref]

An azeotrope has equal coexisting liquid and vapor compositions: $y=x$. Ordinary fractional distillation cannot cross this composition boundary at fixed pressure; more packing does not remove it. A constant head temperature therefore does not establish a pure chemical substance.^[ref] The stipulated $\alpha =2$ model has $y>x$ throughout $0<x<1$, so it has no interior azeotrope and cannot represent this limit.

McMaster University's 1997/98 laboratory manual compares simple distillation and fractional distillation using charges of $25\mathrm{mL}$ each of cyclohexane and p-xylene. Part B uses a $100\mathrm{mL}$ round-bottom flask and a fractionating column. The protocol specifies a boiling chip, a heating mantle on a ring support, and the top of the thermometer bulb midway in the opening toward the laboratory condenser. Heating is adjusted for roughly one drop per second. Students record head temperature every $2\mathrm{mL}$ collected, up to $45\mathrm{mL}$, and plot both runs against collected volume on the same axes.^[ref]

Charge, atmospheric operation, and collection rate provide comparison conditions; the fractionating column is changed. A sharper separation of temperature regions would support improved fractionation. It would not directly measure product mole fractions or prove purity. The manual distinguishes collected volume from composition and states that this pair forms no azeotrope. No student dataset is supplied here, and the constant-relative volatility example is not assigned to these liquids.^[ref]

Consider a nonreacting two-component mixture processed into just two products, with no losses or unaccounted inventory. Let $F,D,B$ be feed, distillate, and bottoms amount of substance values in moles, and let $z,x_D,x_B$ be their component-A mole fractions. These are whole-stream compositions; $x_D$ is not automatically the instantaneous equilibrium vapor composition $y$ above the pot. Total and component balances giveThe component-B balance follows by subtracting the second equation from the first. Eliminating $B$ gives, for $x_D\ne x_B$,Nonnegative product amounts require $z$ between $x_B$ and $x_D$. For an original stipulated feed $F=1\mathrm{mol}$, $z=0.35$, $x_D=0.80$, and $x_B=0.10$,Check the component inventories:The distillate contains a fraction $(D{x}_D)/(Fz)=40/49\approx 0.816327$ of the original component A, despite its higher mole fraction $x_D=0.80$. This distinction between composition and recovery is useful when deciding how much enriched product to collect. These balances determine neither temperatures, heat input, nor the number of equilibrium stages; they do not prove that the specified products are attainable in a particular fractionating column.

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.

Fractional distillation brings rising vapor into contact with descending liquid in a fractionating column. Packing supplies contact area; it does not act as a sieve for individual molecules. Repeated exchange can enrich the rising vapor in the more volatile component. Some condensed liquid returns downward as reflux, rather than all being withdrawn as distillate.^[ref]

An equilibrium stage is an idealization of this contact. A real fractionating column has finite contact time and imperfect wetting and mixing; its length is not automatically a count of equilibrium stages. Product withdrawal and reflux affect the separation together. Thus the repeated ideal-stage enrichment example illustrates the composition advantage of repeated equilibration, but supplies neither a column design nor a specified reflux requirement.^[ref]

In simple distillation, pot vapor passes directly to the laboratory condenser. An initial small condensate sample can approximate vapor-liquid equilibrium with the current pot liquid. Removing enriched distillate depletes the pot's more volatile component. A collected fraction averages the changing condensate composition, rather than preserving the first-drop mole fraction.^[ref]

A laboratory condenser removes heat from vapor so that it becomes liquid. In distillation, the downstream laboratory condenser makes collectable distillate. Complete condensation, without loss or reaction, preserves the entering vapor's overall component mole fractions; cooling alone does not give another enrichment step. Partial condensation can instead leave two streams of different composition, requiring both vapor-liquid equilibrium and a distillation material balance.

Vacuum distillation carries out distillation at reduced pressure.

A disjoint-set data structure maintains a partition of a set. Its find operation returns a representative, equal for two elements exactly when they belong to the same block; its union operation merges two blocks. Kruskal's algorithm uses it to track connected components of a graph.

Start with a singleton set for every stored set element. Each disjoint-set find operation returns a disjoint-set representative; a disjoint-set union operation merges the containing blocks of a set partition. The interface does not split a block of a set partition or delete a member.^[ref]

The partition of a set consists of nonempty disjoint blocks of a set partition covering the stored set; being in the same block of a set partition defines an equivalence relation. A disjoint-set representative names a whole block of a set partition, without requiring a list of its members.

By Jan Gaspar. Boost library manual; its 2003–2013 interval denotes copyright, not a publication or revision date.

By David Howells and Paul E. McKenney. The Linux Kernel documentation hosts this authored guide; no publication date is asserted.

Cited passages: What is a circular buffer?; Measuring power-of-2 buffers; Using memory barriers with circular buffers, including The producer and The Consumer. Its reserved-slot and synchronization conventions are specific to the presented implementation.

By Tony Hoare, credited as C. A. R. Hoare. Communications of the ACM 12(10), 576–580, 583 (October 1969).

DOI: doi.org/10.1145/363235.363259

Paper scan: sites.cs.ucsb.edu/~kemm/courses/cs266/acmhoare69.pdf

Cited passages: §3.4, iteration; §4, reservations.

Proceedings of the eighth international educational computing conference, Piran, 15–17 May 2003; first printing, Ljubljana, 2003. ARNES hosts the PDF. The colophon identifies editors separately from contribution authors; no whole-volume personal authorship is assigned.

Inspected metadata: cover and colophon, PDF pages 1–2; ISBN 961-234-455-8.

By Vladimir Batagelj and Matjaž Zaveršnik. arXiv:cs/0310049v1, submitted 25 October 2003. This entry identifies the inspected preprint; the arXiv record separately lists a 2011 journal reference. arXiv hosts the metadata and experimental HTML conversion.

Inspected full text: arxiv.org/html/cs/0310049

Version record: arxiv.org/abs/cs/0310049v1

Inspected locators: §2, core definition; §§3–4.1, algorithm outline, Algorithm 1 and time-complexity analysis.

By Robert Tarjan, credited as Robert Endre Tarjan. Journal of the ACM 22(2), 215–225, April 1975. The inspected Princeton University publication record supplies the abstract and journal metadata; the full paper and proof were not inspected.

Institutional publication record: collaborate.princeton.edu/en/publications/efficiency-of-a-good-but-not-linear-set-union-algorithm/

DOI: doi.org/10.1145/321879.321884

Cited passage: abstract. Earlier technical reports are distinct versions.

Authored notes by Robert Tarjan, credited as R. Tarjan, Preceptor, on p. 1. Princeton University hosts the course document. Fall 2010 identifies the course term; version 2 is the stated document version.

Course notes: www.cs.princeton.edu/courses/archive/fall10/cos226/precepts/15UnionFind-Tarjan.pdf

Cited passages: p. 1, title and byline; p. 4, union by rank and rank bounds after path compression.