By David Tong.

Additional online chapter: www.damtp.cam.ac.uk/user/tong/dynamics/two.pdf

Additional cited passages: 2. The Lagrangian Formalism, §2.1, printed pp. 10–12, equations (2.2)–(2.8); §2.3, printed pp. 17–19, ideal holonomic constraints; §2.5.2, printed p. 28, equations (2.68)–(2.72); §2.6 and §2.6.1, printed pp. 38–42, especially equations (2.130)–(2.132). These are authored lecture notes by David Tong, hosted by Cambridge DAMTP; no chapter publication date or separately contributed byline is established.

Publisher: University of Tennessee physics teaching site.

Online text: labs.phys.utk.edu/mbreinig/phys421/modules/m3/Refracting%20systems%20details.html

Cited passages: Refracting systems (details), definition of principal planes, unit lateral magnification, and final imaging equation; Refracting systems (details), principal-plane definition and final paragraph on object/image distances; University of Tennessee physics teaching resource, Refracting systems (details), final paragraph: object and image distances must refer to the principal planes.

By Karlsruhe Institute of Technology, Faculty of Physics teaching laboratory.

Publisher: Karlsruhe Institute of Technology, Faculty of Physics.

This source is credited to an organization.

Online text: labs.physik.kit.edu/prakt-nebenfach-manuals/Experiments/Brennweise_d%C3%BCnne_Linsen/

Cited passages: Karlsruhe Institute of Technology, Brennweite dünner Linsen, Durchführung and Aufgaben 1a and 2: target, illumination check, rail readout, repeated Bessel measurements and colour controls.

By PHYWE Systeme GmbH & Co. KG.

Publisher: PHYWE Systeme GmbH & Co. KG.

Publication information: www.phywe.de/physik/waermelehre-thermodynamik/kinetische-gastheorie-und-gasgesetze/joule-thomson-apparatur_800_1731/

This source is credited to an organization.

Online text: phywe-itemservice.s3.eu-central-1.amazonaws.com/sites/DMS-Phywe/PROD/de-DE/item/phy_operatingmanuals/04/04361-00/0436100E.PDF

Operating-instructions version dated 2024 in the printed page footers.

Cited passages: PHYWE, 2024 instructions, p. 2, section 4, Hints concerning the experiment: pressure limit, equilibration, drift, averaging and zero compensation; PHYWE, Joule-Thomson apparatus 04361-00, 2024 instructions, pp. 1–2, sections 2–4: heat exchanger, frit, manometer, probes and difference readout.

James M. Fiore's circuit-analysis laboratory manual connects network equations with actual voltage and current measurements. Testing Kirchhoff's voltage law and Kirchhoff's current law requires consistent choices of reference directions and node potentials, while the measuring instrument itself can load the circuit. Comparing predicted and measured values therefore tests both the circuit model and the measurement arrangement. This gives a physical reason to distinguish ideal components from a laboratory realization.^[ref]

An engineering educator whose joint teaching text connects the first law of thermodynamics with practical energy bookkeeping. The change in stored internal energy differs from the heat and work transferred during a process. Separating a state function from transfers across a system boundary helps compare different ways of taking the same material between two states.^[ref]

An engineering educator whose joint course notes use thermodynamics to analyze energy conversion. Their treatment of a moving piston distinguishes the gas pressure from the external pressure that determines work on the surroundings. The two can be equated only under suitable mechanical conditions, preventing a misleading use of equilibrium formulas during rapid or freely expanding processes.^[ref]

An engineering educator whose joint thermodynamics text emphasizes choosing a thermodynamic system before applying an energy balance. A heater can transfer energy as electrical work across one boundary and as heat across another. The distinction depends on the selected system, making explicit boundaries essential when interpreting an apparatus.^[ref]

Adsorption is accumulation of a substance at an interface. Binding a solute to a packing surface can delay its passage through a chromatography column. Ideal size-exclusion chromatography separates through SEC pore access without such binding; extra retention from adsorption requires a different interpretation.

For a gas contacting a solid, adsorption concentrates molecules at the boundary; absorption takes material into the bulk. A porous solid can have a large internal surface area: binding on pore walls is still adsorption, even though those walls lie inside the particle. Both processes can occur together. Desorption returns adsorbed material to the surrounding phase. The useful question is therefore not just how much material disappeared from the gas, but where it went and whether it returns when the pressure falls.

An adsorption site model turns this spatial distinction into counting. With twelve identical adsorption sites and three bound molecules, the surface coverage is $\theta =3/12=1/4$. Filling six gives $\theta =1/2$; filling twelve gives a complete monolayer. These are configurations illustrating the count, rather than measured arrangements or claims that all real surfaces have twelve sites.

A crystal structure specifies the chemical species and spatial arrangement of the chemical atoms or ions in a crystal. For an ideal periodic structure, separate two ingredients: a Bravais lattice specifies the translations that repeat the arrangement, and an atomic basis of a crystal specifies the chemical motif repeated at every lattice point. The outward shape, or crystal habit, is a different description: the internal arrangement is what this article describes.

For a concrete two-dimensional model, choose primitive lattice translation vectors $a_1=(2,0)$ and $a_2=(0,1.5)$ in arbitrary length units. At every lattice point $R=m{a}_1+n{a}_2$, with integers $m,n$, place species A at $R$ and a different species B at $R+(0.65,0.55)$. Translating by $a_1$ or $a_2$ preserves both the positions and their chemical labels. Translating by $(0.65,0.55)$ sends one A position to a B position, so it is not a translation symmetry of the labeled structure. A picture of dots with the labels erased can therefore conceal essential chemical information.

A primitive unit cell contains one lattice point, but may contain several chemical atoms. A conventional unit cell can contain several lattice points to display the symmetry more clearly. Thus the conventional cube of a face-centered cubic lattice contains four lattice points, whereas a body-centered cubic lattice has two. Neither lattice name alone specifies which chemical species occupies the repeated motif. The NaCl conventional cell and coordination example makes the distinction explicit.

This separation is useful experimentally: interplanar spacings constrain the repeating geometry, while relative wave intensities also depend on the atomic basis of a crystal through the crystal basis structure factor. The Bragg crystal reflection experiment of 1913 describes the X-ray spectrometer, angular scans and detector checks in detail. Its conversion from angle to length is developed under Bragg's law; a geometrically possible reflection can still have zero wave intensity.

William Lawrence Bragg's 1913 structure determination supplied a complementary test. He directed X-rays through sodium chloride sections cut parallel to a cube face and recorded spots on photographic plates. Comparing section thicknesses tested how transmission altered the pattern: the relative spot strengths changed. Comparing related salts led him to two interpenetrating face-centered cubic lattices, one for each kind of constituent. He explicitly distinguished evidence for that arrangement from the still-needed identification of a single chemical atom with each scattering center. The photographs were evidence for a structural model, not direct pictures of labeled ions.^[ref]

A neuroscientist studies the nervous system, connecting its structure, activity, development or evolution to function. Experiments can address processes from ion channels to animal behaviour; an explanation at one scale need not replace measurements at another.

A single-molecule experiment resolves a response attributable to an individual molecule, rather than only an average over many molecules. Optical detection need not resolve its constituent chemical atoms in an image: repeated photon absorption by the same molecule can produce a measurable response. One detected photon is not automatically one different molecule.

The measured channel matters. Direct photon absorption detection measures a change in transmitted light. Fluorescence detection collects photons emitted after excitation; a fluorescence excitation spectrum varies the illuminating frequency while observing that emission. Both can probe excitation, but their backgrounds and readouts differ. A labeled particle cartoon explains the interpretation; it supplies no experimental evidence by itself.

Such measurements expose differences that a population average can conceal. Suppose two groups of molecules switch between different internal quantum states. Their average response could vary smoothly even though each molecule changes in discrete steps. Resolving an individual response tests that interpretation directly, subject to the measurement's time resolution, sensitivity, and particle assignment. The following experiment makes that assignment through spectral selection and controlled perturbations.

Four water molecules contain eight hydrogen and four oxygen chemical atoms, twelve constituent chemical atoms altogether. Counting the constituents does not mean they have been separated by breaking chemical bonds. The same sample has different counts because the specified entities differ.

For $N$ water molecules, their amount of substance is $n=N/N_{\mathrm{A}}$, while the constituent hydrogen and oxygen inventories correspond to $2n$ and $n$ respectively. The exact Avogadro constant is $N_{\mathrm{A}}=6.02214076\times 10^{23}\mathrm{mo}{\mathrm{l}}^{-1}$. The mole applies to specified entities, including chemical atoms, molecules, and ions.^[ref] Thus four water molecules correspond to approximately $6.6422\times 10^{-24}\mathrm{mol}$ of water, but twice that amount of substance of constituent hydrogen chemical atoms. A macroscopic conversion through measured mass additionally needs the relevant molar mass, as in water mass to molecular count.