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Online text: web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node137.html

Online text: web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node135.html

Online text: web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node134.html

Online text: web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node19.html

Cited passages: MIT Unified Engineering, §2.5.2: heat inward and work outward in the first-law balance.

Online text: cs.uwaterloo.ca/~watrous/TQI/TQI.2.pdf

Chapter 2 of the same authored book. Book publication is April 2018; no online chapter revision date is inferred.

Cited passages: Watrous, The Theory of Quantum Information, §2.1.2, printed pp. 62–64, Definition 2.5 and pure states; §2.3, Definition 2.34, measurement probabilities.

Online text: cs.uwaterloo.ca/~watrous/TQI/TQI.1.pdf

Chapter 1 of the authored book published by Cambridge University Press in April 2018. The online chapter does not establish a manuscript revision date.

Cited passages: Watrous, The Theory of Quantum Information, §1.1.2, printed pp. 17–18, equations (1.94)–(1.97), Hermitian, positive semidefinite and density operators; finite-dimensional setting.

Online text: www.bio.fsu.edu/easton/topic26.html

Topics carry the date and copyright label July 2000. Florida State University is the host of the author’s teaching module.

Cited passages: Topic 26, Conditioning-testing, stimulus duration and Figure 26: constant first response and recovery of the second response.

Online text: www.bio.fsu.edu/easton/topic15.html

Topics carry the date and copyright label July 2000. Florida State University is the host of the author’s teaching module.

Cited passages: Topic 15, Figure 15: bath conduction, stimulus artifact and delayed neural response.

Online text: www.bio.fsu.edu/easton/topic14.html

Topics carry the date and copyright label July 2000. Florida State University is the host of the author’s teaching module.

Cited passages: The Nerve Impulse Seen from Outside, Topic 14, Figure 14 and suction-electrode construction.

Online text: www.feynmanlectures.caltech.edu/I_28.html

Chapter 28 of Volume I. The book byline credits Richard Feynman, Robert B. Leighton and Matthew Sands; Caltech hosts the online edition.

Cited passages: Feynman, Leighton and Sands, Electromagnetic Radiation, §§28-2–28-3: transverse acceleration and dipole geometry.

Diastereomers are different stereoisomers that are not enantiomers. The meso compound form of tartaric acid is a diastereomer of either member of its enantiomer pair.

Chapter of Statistical Physics by David Tong, by David Tong. No chapter publication year is inferred from its HTML generation date.

Online text: davidtong.org/teaching/statistical-physics/statmechhtml/S1

Cited passages: Statistical Physics, §1.2.1, equation (1.2) and the independent-system entropy addition argument; §1.4: chemical potential differential; §1.4.3 Extensive and Intensive Quantities, equations (1.46)–(1.48) and the scaling discussion.

Chapter PDF: davidtong.org/pdfs/teaching/statistical-physics/statphys1.pdf

The whole-work byline is David Tong, University of Cambridge Part II Mathematical Tripos. No separately contributed chapter author, initial publication date or revision date is established.

Additional cited passages: §§1.2.1–1.2.2, printed pp. 5–10; §1.3, printed pp. 17–18, equation (1.19).

Online text: mc-stan.org/docs/stan-users-guide/finite-mixtures.html

Cited passages: Finite Mixtures, section Recovering posterior mixture proportions: Bayes formula for label probabilities conditional on the observation, component parameters, and mixing weights; Finite Mixtures, sections Latent discrete parameterization and Summing out the responsibility parameter: nonnegative normalized weights, categorical labels, and the sum of weighted normal densities.

For a closed two-level system with spontaneous emission population decay $P_e(t)=P_e(0)e^{-\Gamma t}$ and no additional dephasing, the natural linewidth has full width at half maximum $\Gamma$ in angular frequency, or $\Gamma /(2\pi )$ in hertz. The excited-state lifetime is $1/\Gamma$.

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.

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.

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.