John Tyndall's radiant heat, water vapour and the "blanket": a primer
Project folder: Researchers/Tyndall-John. It holds five works, all by John Tyndall (1820-1893), professor at the Royal Institution (RI) in London: On Calorescence (Philosophical Transactions 1866), On Radiation (the Rede Lecture, Cambridge, 1865), Heat, a Mode of Motion (third edition, 1868), Six Lectures on Light (America, 1872-73) and New Fragments (1892, collected essays). All five PDFs have a usable text layer (OCR); every number a verdict turns on was checked on rendered page images by the deep readers.
Source note (repeated from claims.md and arbiter.md). This is not a source gap, but there are limits that carry into everything below: - Everything in the folder is by Tyndall. On his own priority (water-vapour absorption, calorescence, the glacier theory, his reconstruction of Mayer's calculation) he is an interested party. Every independent account (Magnus, Lecher and Pernter, Rontgen, Very, Akin, Stokes, Foote, Franz, Jackson's 2019 history) comes from the Investigator's web notes and was seen only as quoted there; no outside document was downloaded. - Missing and unread by anyone: journal page 1 and Plate I of the 1866 paper (Muller's solar curves), Tyndall's galvanometer calibration table, Magnus 1861/62, Lecher and Pernter 1880/81, Strachey's July 1866 paper, Foote's 1856 text, Tyndall's Contributions to Molecular Physics (1872/73) and Mayer's own paper. - (Revised after the quality check, 2026-10-09.) Heat chapters XI (paras 435-466), XII, XIII and XIV (paras 647-684) have now been deep-read with page images; the CO2 percentages 48.0-74.3 below are image-confirmed (Heat p.431). The other chapters of Heat remain digest only. No verdict turns on them alone. - Where a statement rests on recalled knowledge only, it is tagged [memory].
Page numbers: "p.N" is the PDF page. In Heat the printed page is 27 lower in the early chapters (PDF p.97 = printed p.70) and 29 lower by Ch. XII (per the Ch. XII deep read).
Source tags: [read] = from the project documents (directly or via the deep reads); [web] = Internet source with URL (most via the Investigator's notes); [memory] = recalled, with confidence; [computed] = calculated for this primer and reproduced by example.py.
1. The idea in one paragraph
Tyndall set out to find out whether the invisible "radiant heat" given off by warm bodies passes freely through gases. Using a long brass tube closed with rock-salt windows (rock salt lets heat rays through where glass does not), a heat source at one end and a sensitive heat detector at the other, he found in 1859 that the main gases of air (oxygen, nitrogen) and hydrogen let almost all of it through, while gases made of several kinds of atom (carbon dioxide, ammonia, ethylene, and above all water vapour) stop a large part of it, and that a gas that absorbs a ray also gives that ray out when hot. He drew the conclusion that the thin trace of water vapour in the air acts as a blanket: sunlight passes through it, but the heat the ground radiates back is partly held, so the vapour keeps the surface warm and controls night frost, desert cold and mountain cold. He also showed that most of the energy of a bright lamp is invisible (about eight parts invisible to one part visible for an electric arc), and that if the visible light is filtered out, the invisible rays alone can be focused to set paper on fire and make platinum glow ("calorescence"). Much of this is now textbook physics and the founding experiment of the science of the greenhouse effect; some of his interpretations (the "ether", rays being "transmuted" into light, glacier ice not being viscous) have been replaced, and several of his single numbers belong to his apparatus rather than to nature.
2. What you need to know first
Heat capacity, the ideal gas and the first law of thermodynamics. The heat capacity per unit mass (the "specific heat", Tyndall's "capacity for heat") is the heat needed to warm 1 unit of mass by 1 degree. For a gas it depends on what you hold fixed: at constant volume all the heat goes into the molecules' motion (c_v); at constant pressure the gas also expands and pushes on its surroundings, doing work, so more heat is needed (c_p). The ratio gamma = c_p / c_v is 1.40 for air. The ideal gas law P V = m R_s T (pressure times volume equals mass times the specific gas constant times absolute temperature) tells you how much the gas expands. The first law says heat added = change of internal energy + work done. Textbooks: Halliday, Resnick and Walker, Fundamentals of Physics (the chapters on temperature, heat and the kinetic theory of gases); D. V. Schroeder, An Introduction to Thermal Physics (2000), chapter 1.
Blackbody radiation (Planck's law). Every body above absolute zero radiates electromagnetic waves. A perfect absorber (a "blackbody") at absolute temperature T radiates a spectrum given by Planck's law; the total power per unit area is sigma T^4 (Stefan-Boltzmann law), and the peak wavelength is about 2898 micrometres-kelvin divided by T (Wien's law). A body at 300 K peaks near 10 micrometres (deep infrared); a 3000 K lamp filament peaks near 1 micrometre (near infrared, just beyond red); the sun (about 5800 K) peaks near 0.5 micrometre (green). The visible band is only about 0.40-0.76 micrometres. Schroeder chapter 7 covers the photon gas and Planck's law.
The electromagnetic spectrum. Visible light, infrared ("radiant heat", Tyndall's "obscure" or "dark" or "extra-red" rays) and ultraviolet are all the same kind of electromagnetic wave with different wavelengths lambda; frequency nu = c / lambda, where c is the speed of light. Each quantum (photon) carries energy E = h nu = h c / lambda (h is Planck's constant), so a blue photon carries more energy than a red one. D. J. Griffiths, Introduction to Electrodynamics, chapter 9 (electromagnetic waves).
Absorption along a path (Beer-Lambert law) and saturation. When a beam crosses an absorbing gas, each thin slice removes a fixed fraction, so intensity falls exponentially: I = I_0 exp(-k L), where L is path length and k is an absorption coefficient proportional to the number of absorbing molecules per unit volume. For a single wavelength that is the whole story. For a broad beam crossing a gas that absorbs only in narrow bands, the band centres soon absorb everything and further gas only nibbles at the band edges: total absorption then grows much more slowly than the amount of gas ("saturation"). Any atmospheric radiation book covers this, for example G. W. Petty, A First Course in Atmospheric Radiation (2nd ed., 2006).
Kirchhoff's law of radiation (1859-60). At each wavelength, a body's emissivity (how well it radiates compared with a blackbody) equals its absorptivity (the fraction it absorbs). Good absorbers are good emitters at the same wavelengths. Petty, and any thermal physics text.
Why some molecules absorb infrared and others do not. Infrared is absorbed when the wave's frequency matches a vibration (or rotation) of the molecule and that motion changes the molecule's electric dipole moment (the separation of its positive and negative charge), so the wave's electric field has something to push on. N2, O2 and H2 are two identical atoms: stretching them leaves no dipole, so they are almost transparent. CO2 (bending mode at 15 micrometres), H2O (6.3 micrometres, and a rotation band beyond about 17 micrometres), NH3, CH4, N2O and O3 have vibrations that change the dipole, so they absorb. This is the "dipole selection rule"; you do not need quantum mechanics beyond the idea that a molecule absorbs at its own natural frequencies. P. Atkins, Physical Chemistry, the chapter on vibrational spectroscopy.
The greenhouse energy balance. The Earth absorbs about 240 W/m^2 of sunlight averaged over its surface. A blackbody radiating 240 W/m^2 sits at (240 / sigma)^(1/4) = 255 K, but the mean surface is about 288 K. The 33 K difference is the greenhouse effect: gases and clouds that absorb the surface's infrared radiate part of it back down. R. T. Pierrehumbert, Principles of Planetary Climate (2010), or Petty.
Tyndall's instrument. A thermopile (many small thermocouples in series) gives a voltage proportional to the heat falling on it; a galvanometer needle turns in proportion to the current. Tyndall used two thermopile faces set against each other ("compensated"), so the needle reads only the difference between the beam through the tube and a balancing beam, which let him see small changes on a large signal. The needle's angle is proportional to heat only up to about 30 degrees; above that he converted angles to heat with a calibration table [read: Heat p.327-328].
3. The textbook view
Symbols used throughout: - P : pressure; V : volume; T : absolute temperature (kelvin K, or degrees Rankine, R = absolute Fahrenheit; 1 K = 1.8 R). - m : mass; rho : density (mass per volume). - c_p, c_v : specific heats at constant pressure and constant volume; gamma = c_p / c_v. - R_s : specific gas constant of air, 287.05 J/(kg K); for air, P = rho R_s T. - J : the mechanical equivalent of heat, the amount of work equal to one unit of heat (4.186 joules per calorie; about 778 foot-pounds per Btu, where 1 Btu heats 1 lb of water 1 degree F). - lambda : wavelength; h : Planck's constant; c : speed of light; k_B : Boltzmann's constant; sigma = 5.67 x 10^-8 W m^-2 K^-4. - I, I_0 : beam intensity after and before a path L; k : absorption coefficient per unit length.
3.1 Mayer's relation. At constant pressure, warming a mass m of ideal gas by dT takes heat m c_p dT; at constant volume, m c_v dT. The difference has gone into the work P dV of pushing back the surroundings, and for an ideal gas P dV = m R_s dT. If heat is measured in calories and work in joules, the bookkeeping needs a conversion factor J:
J (c_p - c_v) = R_s, so J = R_s / (c_p - c_v)
This is Mayer's relation (1842). It gives the mechanical equivalent of heat from three numbers anyone could look up in the 1840s: the gas law, the specific heat of air and gamma. It assumes the gas does no internal work when it expands (true for an ideal gas, and checked by Joule's free-expansion experiment).
3.2 Planck's law. The power radiated per unit area per unit wavelength by a blackbody at temperature T is
M(lambda, T) = (2 pi h c^2 / lambda^5) / (exp(h c / (lambda k_B T)) - 1)
Integrating over all wavelengths gives sigma T^4. Integrating over a band (for example 0.40-0.76 micrometres) gives the power in that band. The fraction that is visible rises steeply with temperature; for any real lamp the infrared dominates.
3.3 Absorption by gases. For one wavelength, I = I_0 exp(-k L) with k = n s (n molecules per unit volume, s the absorption cross-section). Absorption depends on the column density n L (molecules per unit area of beam), not on how the molecules are packed, provided the lines do not change shape. In reality pressure broadens lines (collisions blur the frequency), and phase changes (vapour to liquid) shift and broaden bands strongly, especially for hydrogen-bonded liquids like water. A broadband source seen through a band absorber shows saturation: doubling the gas less than doubles the absorption.
3.4 Selection rules. Infrared absorption needs a change of dipole moment (section 2). Raman scattering, a different process, needs a change of polarisability; N2 and O2 are Raman-active but infrared-inactive. N2 and O2 do absorb very weakly through collisions and (O2) magnetic transitions, orders of magnitude below CO2 [memory: medium-high].
3.5 Thermal emission versus fluorescence. A body heated by absorbed radiation shares the energy among all its atoms and then emits a Planck-like spectrum for its own temperature. Fluorescence is different: a single absorbed photon is re-emitted, usually at a longer wavelength (Stokes's rule, 1852: the emitted light has lower frequency than the exciting light). One infrared photon cannot become one visible photon, because it does not carry enough energy; but many infrared photons can heat a foil until its thermal glow reaches the visible.
3.6 The greenhouse effect today. The lower atmosphere is optically thick in the water-vapour and CO2 bands and fairly clear in the 8-12 micrometre "window". In a current model attribution the present greenhouse effect is about 50 per cent water vapour, 25 per cent clouds and 20 per cent CO2, with the shares depending on how overlapping bands are allocated; and water vapour follows temperature (it condenses out when air cools) whereas CO2 does not, so CO2 sets the level and vapour amplifies it [web: https://www.giss.nasa.gov/research/briefs/2010_schmidt_05/, Schmidt, Ruedy, Miller and Lacis 2010].
3.7 Ice. Polycrystalline glacier ice creeps continuously under constant stress; the strain rate is roughly proportional to the cube of the stress (Glen's flow law, 1955) [web: https://data-ww3.ifremer.fr/BIB/Glen_PRSLA1955.pdf]. Regelation (ice melting under pressure and refreezing when the pressure is released) is real but the melting-point shift is tiny, about -0.0075 K per atmosphere [computed, from the Clausius-Clapeyron equation, per the mathematician's review].
4. Where Tyndall departs
Tyndall is unusual in this collection: he is not an outsider. Most of what he claimed became the textbook. The "departures" below are mostly where his 1860s interpretation, or a single number, went further than his measurements, and three priority questions where others (Foote, Franz, Akin) have a claim. Each row starts with the real physics his work rests on.
| Topic | Tyndall's position | Textbook position today | Where they part |
|---|---|---|---|
| Gas absorption (claim 1) | O2, N2, H2, dry air "practically transparent"; compound gases absorb tens to thousands of times more (CO2 972, NH3 5460, ethylene ("olefiant gas") 6030 times air at 1 inch of mercury) [read: On Radiation p.22-23; Heat p.349-350]; a gas radiates what it absorbs | Same pattern, explained by the dipole rule; Kirchhoff's law | Not on the pattern. His "air = 1" is the smallest deflection he could read (about 1 degree), not zero, and he says organic impurity can raise it fiftyfold [read: Heat p.380-381]; so the big ratios are lower bounds for his tube, source and calibration, not constants of nature. His chlorine 60 and bromine 160 should be near zero by the dipole rule (probably impurity or the floor effect; an inference). "Radiates what it absorbs" rests on 4 static and 13 dynamic measured pairs: the rank order agrees in every case, but absorption / radiation runs from 0.40 to 0.87, so the data show rank agreement and rough proportionality, not equality; exact equality comes from Kirchhoff's theory [read: Heat Ch. XI, deep read]. Ozone also absorbs: 20 to 136 times its carrier oxygen, depending on how it was made, with its concentration "perfectly unmeasurable", so these rank preparations, not a coefficient [read: page image Heat p.366-367] |
| Water vapour as blanket (claim 2) | The vapour in air absorbs 70-90 times as much as the dry air carrying it; per molecule about 16,000 (80 x 200) [read: Heat p.380, 387-388, 401]; "at least 10 per cent" of the earth's heat stopped within ten feet [read: Heat para 480]; vapour is the dam that keeps the surface warm [read: New Fragments p.106] | Vapour is the largest single greenhouse absorber; controls night cooling, desert and mountain temperature ranges | Not on the core. The ten-foot figure is an extrapolation from a 4-ft tube and laboratory sources to the 288 K Earth, and his own 4-ft figures range from 6 to 20 per cent depending on the source (see section 5, Part C). He never apportioned the effect between vapour and CO2; reading him as "CO2 is minor" turns a silence into a ranking: against a CO-flame source his own CO2 absorbs 48 to 74 per cent [read: page image Heat p.431]. He could not know that vapour is a feedback and CO2 a control |
| How much of a lamp's radiation is invisible (claim 3) | Arc light: invisible 7.7 times visible by area under the spectrum curve, 8 by filtering (89 transmitted, 11 absorbed; an earlier "1 in 10" would give 9); sun "about 2" (quoted from Muller) [read: 1866 p.7, 11; Heat p.458-459] | Planck's law: 7.7 means a blackbody near 3000 K; the sun gives about 1.07 | Not on the arc. Muller's solar 2 is high by about a factor two (cause unknown; his curves are not in the folder). It was measured at the ground, after the atmosphere's vapour had removed some infrared (Tyndall himself says so, Heat para 573), so the true ratio above the air would be higher still, which makes the gap with the modern 1.0 larger, not smaller |
| Calorescence (claim 4) | Focused dark rays make platinum glow: a "transmutation" of rays to higher refrangibility, "the reverse of Stokes's rule"; through some blue glasses the glow looked pink, possibly a "gap" in the spectrum [read: 1866 p.17-18, 23-24] | Ordinary thermal incandescence: the foil is heated and glows by Planck's law (section 3.5) | The interpretation. Stokes himself wrote to Tyndall on 13 January 1865 that this "is not a little different from fluorescence" [web: https://epsilon.ac.uk/view/tyndall/letters/Tyndall2215]. Tyndall's own letter to Croll the next day already describes heating ("hit the mass with waves of slow recurrence ... obtain in return ... waves of quick recurrence") [web: https://epsilon.ac.uk/view/tyndall/letters/Tyndall2216] |
| Absorption is molecular (claim 5) | Absorption belongs to the molecule, selected by resonance between wave period and atomic periods; same absorption when the same molecules are squeezed into a shorter tube (18.4 vs 18.8 per cent, 25.7 vs 25.6 ...) or condensed to liquid (ether: vapour 66.7, liquid 67.2 per cent) [read: New Fragments p.103-105] | Resonance and molecular origin: yes. Column-density rule: holds for weak, broad bands; fails where pressure broadening or the liquid phase reshapes bands: in liquid water the rotation lines vanish, hydrogen bonding broadens the bands, and the stretch bands move from 3657 and 3756 to about 3450 and 3615 cm^-1 (wavenumber, waves per centimetre) [web: https://en.wikipedia.org/wiki/Water_absorption]; liquid water also absorbs across 8-12 micrometres where the vapour is nearly clear [memory: high, not fetched] | His tested vapours (ether, amyl compounds and other organic and halide liquids, ten non-water substances in all) are the case where the rule works; water vapour was never in the test ("Aqueous vapour, unmixed with air, condenses so readily that it cannot be directly examined" [read: page image Heat p.418]), and his conclusion for water is an extrapolation of rank (para 515). Generalising the rule to all gases and to water goes beyond the data. His "ether entangled in the molecule" has no modern counterpart |
| Heat and light (claim 6) | Same ether wave, differing in period; heat is a mode of motion; Mayer's method gives J = 771.4 against Joule's 772 [read: Heat p.97-99] | Electromagnetic waves (Maxwell 1865, Hertz 1887-88) [memory: high]; no ether; J = 778 ft-lb per Btu | The ether. And the 771.4 agreement is partly luck (Part A) |
| Glaciers (claim 7) | Ice is not viscous under tension; glaciers move by fracture and refreezing (regelation) [read: Heat p.211-212] | Ice creeps (Glen 1955); regelation helps at the bed | Tyndall himself conceded ice "behaves like a viscous body" under pressure and viscosity "accounts, certainly, for half the facts" [read: Heat p.211-212]; Forbes called the two views "complementary" [web: https://www.sciencehistory.org/stories/magazine/under-pressure/] |
The dispute about water vapour. Tyndall's vapour result was challenged for twenty years, and the challenge was real. Gustav Magnus (1861/62) and Buff found no vapour action and blamed films of moisture on the rock-salt plates; Lecher and Pernter (1880/81) confirmed Tyndall on dry gases but found moist air "a little more transparent than a vacuum" [web, per Investigator, quoting Tyndall's 1882 Bakerian Lecture: https://archive.org/details/philtrans03250633]. Tyndall answered with controls: plates removed (an open tube still gave 45 degrees deflection against 0), air from Hyde Park, Epsom Downs and the Isle of Wight (about 70 times each), and a pressure series in which absorption was proportional to pressure within about 2 per cent (16, 32, 49, 64, 82, 98 against 16, 32, 48, 64, 80, 96) [read: Heat p.386-388]. Rontgen (1884), using a different method (the pressure rise of the irradiated gas) chosen to avoid the film problem, found "moist air absorbs rays of lower refrangibility" and wrote that many physicists still regarded the question as open [web: https://archive.org/details/sim_annalen-der-physik_1884_23_9]. Frank Very's 1900 review for the US Weather Bureau sided with Tyndall but asked for "further elucidation" of his humid-air table [web: https://archive.org/details/atmosphericradia00veryrich]. Evidence on the other side also includes William Ferrel, the American meteorologist, who as quoted by Very accepted Magnus's film explanation as "really to have been the case, according to experiments which have since been made"; Very then rebutted it [web: https://archive.org/details/atmosphericradia00veryrich, per Investigator]. Magnus himself did not give up: Tyndall's closing note in Heat says Magnus had published a further paper "controverting" the vapour results, "by which I am not convinced", and promises to examine it; the book gives neither its date nor its content, and this one sentence by Tyndall is the only record of it in the folder [read: Heat p.533]. New Fragments adds that Magnus's radiant-heat work on gases occupied his last years "in discussion with Tyndall" [read: New Fragments p.256]. The dispute was settled by new methods and by spectroscopy, not by silence. One caution: much of what the folder and the Investigator found is Tyndall's side ("Magnus denies the action; but he has not a foot to stand upon", to Hooker [web: Epsilon letter 1941]); Magnus's and Lecher and Pernter's raw data were not seen by anyone in this review.
Tyndall's own statements with no experiment on record, kept. Two sets of statements bear on claim 2 and rest on Tyndall's word alone; none is rejected, and none has an independent test (arbiter verdict for each: Never properly tested). - The planets. Answering an unnamed book (probably Whewell's Of the Plurality of Worlds, 1853 [memory: medium]) that held the distant planets uninhabitable by the inverse-square law, he wrote that "a layer of air two inches in thickness, and saturated with the vapour of sulphuric ether ... I find ... would cut off fully 35 per cent. of the planetary radiation" while letting sunlight through, so that "a comfortable temperature might be obtained on the surface of the most distant planet" [read: Heat para 546, p.436; also On Radiation p.44, "proved by experiment"]. No table, source or temperature is given. A 1-inch shell of olefiant gas would absorb "at least 26 per cent" (para 545; pressure and source of the thin-layer figure not stated). This is the greenhouse argument extended to other planets. - The Sahara: "Ice has been formed in this region at night" [read: Heat p.395]; no source is given.
The field evidence in Heat Ch. XIV is mostly other people's and has no humidity readings: Wells's dew theory (1818; screens and a cloud cut the grass chill from about 12 F to 2 F), Glaisher's height profile, Pouillet's and Melloni's figures [read: Heat Ch. XIV, deep read]. Tyndall's own contributions there are interpretive: vapour as radiator; a dry-air supplement to Wells on Bengal ice-making, resting on one quoted remark that the best ice nights have "very little dew ... after midnight"; and his gloss ("Doubtless") that Pouillet's 25 and 40 per cent solar absorption by the atmosphere is mainly vapour's. Those are solar figures and must not be pooled with the terrestrial "ten feet". In summer 1866 on the Titlis he "repeated with a hand spectroscope the observations of M. Janssen on the absorption-bands of aqueous vapour" [read: New Fragments p.472], but gives no result, so it is a recorded attempt, not evidence either way; Janssen's own work is a lead for the next Investigator run.
Priority. Eunice Foote (1856, sunlit cylinders of gas; CO2 and moist air warmed more) and R. Franz (1855, absorption by gases; 3.54 per cent apparent absorption even for dry air in his tube) published before Tyndall's 1859 work; the historian Roland Jackson (2019) treats it as simultaneous discovery and notes that Tyndall did not cite Foote [web: https://royalsocietypublishing.org/doi/10.1098/rsnr.2018.0066]. Tyndall's 1859 Note itself has no numbers and no climate statement; the climate argument appears from 1861 [web: https://archive.org/details/jstor-111604; Jackson 2019]. On calorescence, Karoly Akin ("calcescence", British Association 1863, with G. Griffith at Oxford, a 2-ft mirror in sunshine) had the same experiment; Sabine and Armstrong sided with Tyndall, and Tait (1868) credited Akin with "Calcescence or Calorescence" [web, per Investigator, Epsilon letters 2075-2208]. Akin's physics record stops after 1865; he went into Hungarian academic and political life (Academy member 1868) and died poor. The arbiter sets out the competing explanations (the dispute drove him out; a change of career; no laboratory post; deliberate suppression, for which there is no evidence but which cannot be ruled out because every account of the dispute in the folder is Tyndall's) and names the records that would decide: Akin's 1863 BA paper, his letters in the Tyndall Correspondence volumes 8-9, and the Hungarian Academy archive. For balance in weighing Tyndall as an interested party: on singing flames he wrote that "all the merit of priority rests with Count Schaffgotsch" [read: Heat p.293], a case where he conceded priority; it sits beside the Akin record, not in place of it.
Influence outside mainstream physics. Edwin Babbitt's Principles of Light and Color (1878) took up Tyndall's "calorescence" in an alternative theory of colour [web, per Investigator: sites.md, Google research note 2026-10-08], an example of how a superseded interpretation can live on outside physics.
A dissenting modern view, recorded. B. D. Macdonald (viXra:1504.0165, 2015, not peer reviewed) argues that Tyndall's thermopile misread the experiment and that N2 and O2 absorb because they are Raman-active [web: https://vixra.org/abs/1504.0165]. The reviewers' answer: Raman activity and infrared activity follow different selection rules (section 3.4); the residual absorption by N2 and O2 is real but tiny, and Tyndall's "1" already allows for a small residual.
5. Worked example
Three calculations, each from numbers Tyndall printed. Part A is the main one: it is a complete energy balance, done exactly as Tyndall did it in front of his audience, and it shows how a "perfect" agreement can hide two errors. Parts B and C are shorter and connect his radiation numbers to Planck's law and Beer-Lambert. All are reproduced by example.py:
.venv/bin/python review/Tyndall-John/example.py
Part A. The mechanical equivalent of heat from a cubic foot of air (Heat, paras 76-81)
Tyndall's setting [read: Heat p.97-99, checked against the extracted text]: a cylinder with a base of 1 square foot holds 1 cubic foot of air at 0 C (32 F) under a freely moving lid. Heat the air until its volume doubles. Because volume is proportional to absolute temperature at constant pressure, the absolute temperature must double: a rise of 273 C, "or 490 F" (strictly 491.4 F). The lid rises 1 foot.
Step 1. Work done. The lid pushes back the atmosphere, "a pressure of 15 lbs. on every square inch". Over 144 square inches that is a force of 144 x 15 = 2,160 lb, moved 1 foot:
W = 2,160 ft-lb
Step 2. Heat put in at constant pressure. "The weight of our cubic foot of air is 1.29 oz"; its capacity for heat is 0.24 (water = 1). So the air needs as much heat as 1.29 x 0.24 = 0.3096 oz of water, which Tyndall writes 0.31 oz. Heating 0.31 oz of water through 490 F is the same heat as heating 0.31 x 490 = 151.9 oz = 9.49 lb of water through 1 F. Tyndall: "9 1/2 lbs."
Q_p = 9.5 lb-F (pounds of water heated 1 degree F, i.e. Btu)
Step 3. Heat that would have been needed at constant volume. Using gamma = c_p / c_v = 1.42 ("1.42 : 1 = 9.5 : 6.7"):
Q_v = 9.5 / 1.42 = 6.69, written 6.7 lb-F
Step 4. The difference became work. The gas at constant pressure took 9.5 - 6.7 = 2.8 lb-F more heat, and the only extra thing it did was lift 2,160 lb by 1 ft. If heat and work are the same thing in different units (energy conservation) and the expanding air does no internal work (ideal gas):
2.8 lb-F of heat = 2,160 ft-lb => J = 2,160 / 2.8 = 771.4 ft-lb per lb-F
Tyndall: "a quantity of heat sufficient to raise 1 lb. of water 1 Fahr. in temperature, is competent to raise a weight of 771.4 lbs. a foot high" [read: Heat p.99]. He then sets this beside Joule's 772 and says "Such a coincidence relieves the mind of every shade of uncertainty" [read: Heat para 92].
Step 5. What the chain really computes. Write the steps in symbols. Work W = P V_0 (the volume grows by V_0). Heat at constant pressure Q_p = rho V_0 c_p T_0, where T_0 is the starting absolute temperature (the rise equals T_0 because T doubles). The excess is Q_p (1 - 1/gamma) = rho V_0 (c_p - c_v) T_0. So
J = W / excess = P / (rho T_0 (c_p - c_v)) = R_s / (c_p - c_v)
The volume cancels, and P / (rho T_0) is just the gas constant of air. Tyndall's lecture demonstration is Mayer's relation of section 3.1 in disguise (example.py checks this symbolically). That tells you which inputs matter: pressure, density, temperature, c_p and, above all, gamma.
Step 6. Sensitivity to gamma. J is proportional to gamma / (gamma - 1). Taking logarithms and differentiating:
d ln J / d ln gamma = -1 / (gamma - 1) = -2.5 at gamma = 1.40
A 1 per cent error in gamma makes a 2.5 per cent error in J, because the answer rests on the difference of two nearly equal heats (9.5 and 6.7). With Tyndall's other data and no intermediate rounding [computed]:
| gamma | J (ft-lb per lb-F) |
|---|---|
| 1.400 (modern) | 797 |
| 1.410 | 783 |
| 1.420 | 770 |
| 1.421 (Tyndall's "accurate number") | 769 |
| 1.430 | 758 |
So with the modern gamma Tyndall's own data would have given 797, 3 per cent above Joule.
Step 7. Modern constants, and the two errors that cancelled. Put in modern values: 1 atmosphere = 14.696 lb per square inch = 2,116.2 lb per square foot; dry air at 0 C has density 0.08073 lb per cubic foot (= 1.29 oz, Tyndall had this right); c_p = 0.2400 Btu per lb per F; T_0 = 273.15 K = 491.67 R; gamma = 1.400 [memory: high for all five]:
J = 2,116.2 / (0.08073 x 0.2400 x 491.67 x (1 - 1/1.400)) = 2,116.2 / 2.722 = 777.5 ft-lb per Btu [computed]
That is the accepted 778 (the arbiter's 777.9 uses 491.4 R; the difference is in the fourth figure). Mayer's method is sound. Now put back Tyndall's choices one at a time [computed]:
| Change from the modern case | J | Change |
|---|---|---|
| none | 777.5 | |
| 15 psi instead of 14.696 | 793.6 | +2.1 per cent |
| gamma 1.421 instead of 1.400 | 749.8 | -3.6 per cent |
| both | 765.3 | -1.6 per cent |
| plus his roundings (490 F, 0.31, 9.5, 6.7, 2.8) | 771.4 |
Tyndall rounded the atmosphere up and used a gamma that was 1.5 per cent too high; the two errors pull in opposite directions, and two-figure rounding of the intermediate results (2.81 written 2.8) carried the rest. The agreement with Joule's 772 to 0.1 per cent is therefore partly luck; the method is good to a few per cent, and so the "coincidence" is not evidence of 0.1 per cent accuracy. This is not a charge of fudging: Tyndall printed every rounding openly.
Step 8. The same thing in SI. With R_s = 287.05 J/(kg K) = 0.28705 J/(g K), c_p = 0.2400 cal/(g K) and c_v = c_p / 1.400 = 0.1714 cal/(g K): J = 0.28705 / 0.06857 = 4.186 J per calorie, the accepted value [computed].
A note on Mayer. Tyndall says Mayer's first paper (1842) "contains merely an indication of the way in which he had found the equivalent; but does not contain the details of the calculation" [read: Heat p.99]. So 771.4 is Tyndall's reconstruction with 1868 data, not Mayer's number. Mayer's own published figure, 365 kilogram-metres per kilocalorie (about 665 ft-lb per lb-F), is attested for his 1845 booklet; whether it was already in the 1842 paper is unconfirmed [web: https://en.wikipedia.org/wiki/Julius_von_Mayer]. His lower value reflects the poorer specific heats available to him [memory: medium].
Part B. How much of a glowing source is invisible? (claims 3 and 4)
Tyndall found the infrared of his arc lamp to be 7.7 times the visible by measuring areas under his spectrum curve, and 8 times by filtering (a solution of iodine in carbon bisulphide absorbed 11 per cent of the beam and he took that 11 per cent to be the visible light: 89 / 11 = 8.1) [read: 1866 p.7, 11]. Can Planck's law tell us what kind of source that is?
Step 1. Integrate Planck's law (section 3.2) from 0.76 to 15 micrometres (infrared, the range rock salt passes) and from 0.40 to 0.76 micrometres (visible), and take the ratio [computed]:
| T (K) | infrared : visible |
|---|---|
| 2500 | 18.3 |
| 2800 | 10.5 |
| 3000 | 7.74 |
| 3500 | 4.26 |
| 4000 | 2.73 |
| 5772 (sun) | 1.07 |
Step 2. The ratio 7.7 corresponds to a blackbody at 3004 K; 8 to 2977 K. The hottest part of a carbon arc (the positive crater) is nearer 3800-4000 K [memory: medium-high], but the lamp also radiates from cooler glowing carbon, so an effective 3000 K is physically reasonable. Tyndall's two methods agree within 4 per cent and share only one assumption (where the eye puts the red limit). Caveats: both pass through a calibration table that was never published (it moves the iodine result from 94.3 per cent raw to 86.8 per cent transmitted), and taking all of the iodine's absorption as visible is Tyndall's inference ("may be inferred"); he concedes a little absorption beyond the red [read: 1866 p.11].
Step 3. For the sun, the same calculation gives 1.07, and the measured spectrum above the atmosphere is roughly 49 per cent infrared and 43 per cent visible [memory: high]. Muller's "about twice", which Tyndall quotes, is high by about a factor two; why is unknown, because Muller's curves are not in the folder. Note which way the atmosphere pushes: Muller measured at the ground, after water vapour had removed part of the sun's infrared, and Tyndall uses exactly that to explain why the sun's ratio is lower than the arc's [read: Heat para 573]. Above the air the ratio would be higher than 2, so the gap with the modern 1.0 is larger, not smaller. (If you define visible as 0.38-0.78 micrometres instead, the 3000 K ratio is 6.9 and the sun's about 0.94; the reviewers' slightly different numbers come only from this choice.)
Step 4. Making platinum glow with invisible rays. A thin blackened foil heated by a beam loses heat from both faces at 2 sigma T^4. To reach the Draper point (about 798 K, where a dull red glow becomes visible [web: https://en.wikipedia.org/wiki/Draper_point]) it needs 2 x 5.67 x 10^-8 x 798^4 = 4.6 W/cm^2; for white heat near 1500 K, 57 W/cm^2 [computed]. Focused to a few square millimetres, a watt or two does it, easily supplied by a large arc lamp whose output is mostly infrared. Energy is conserved and the second law is satisfied: the foil (about 1500 K at most) is cooler than the source (about 3000 K). Nothing is "transmuted": the absorbed infrared heats the foil, and the foil glows at its own temperature.
Step 5. A slip worth noticing. Tyndall wrote that red waves are "immensely more powerful" than blue. Per quantum it is the other way round: E = h c / lambda, so a 420 nm (blue) photon carries 700 / 420 = 1.67 times the energy of a 700 nm (red) photon [computed]. His experiment (visible light decomposes amyl nitrite vapour, the infrared does not) fits the photon picture: a chemical bond is broken by one photon with enough energy, not by a large number of weak ones. Why the arbiter labels the sentence "Internally inconsistent" rather than merely out of date: the mathematician reads "measured by the amount of energy which they carry" as energy per wave, and per wave the energy goes with frequency, so the sentence has the order backwards; the arbiter keeps the label on the ground that this contradicts the wave theory Tyndall himself was using, in which the faster vibration carries more energy per wave for the same amplitude. It is an error of statement, not of the experiment. Be aware of the limit of this reading: the clean statement "energy per quantum is proportional to frequency" is post-Tyndall (Planck 1900, Einstein 1905), and Tyndall may have meant the energy of the whole red part of a beam, which in a lamp is indeed larger; a reader may fairly judge the label strict. The arbiter's label stands as given.
Step 6. An exercise: white-hot platinum. Tyndall's filter gave the white-hot platinum spiral as "1 part luminous in 24", i.e. infrared : visible = 23 [read: page image Heat p.457-458; the same chapter's table prints 4.6 per cent absorbed, 1 in 21.7, unreconciled]. Using the Part B integral, a blackbody with ratio 23 is at about 2400 K (2395 K; ratio 20.7 gives about 2440 K) [computed]. But platinum melts at about 2041 K [memory: high], where a blackbody's ratio is about 61, i.e. visible only 1 part in 62. So the spiral looks "too visible" for its temperature. Two readings fit, and neither has been tested: (a) the iodine method under-counts visible light (the calibration tables are not in the folder); or (b) platinum is not a blackbody: its emissivity is higher in the visible than in the infrared [memory: medium], which raises its visible share at about 2000 K. Measuring the spectrum of a hot platinum wire with a cheap spectrometer would decide between them.
Part C. The ten-foot column and saturation (claim 2)
Step 1. Tyndall's number. Ordinary moist room air in an open tube absorbed 7.9 per cent of the heat from a boiling-water source [read: Heat p.386]. The path length is not stated; the tube was 4 ft 3 in (51 in), and the deep reader inferred about 21 in. In a 4 ft polished tube the same kind of source gave "over 10 per cent" [read: Heat p.386-387], and from that he wrote that "at least 10 per cent" of the earth's heat is stopped within ten feet [read: Heat para 480, p.387].
Step 1b. Tyndall's own proof that the source matters. In Heat Ch. XII he sent several sources through the same 4 ft polished tube, comparing undried laboratory air with dried air (which absorbed 0 in the two hydrogen-flame runs where it was measured) [read: page images Heat p.428-429]:
| Source | Absorbed by undried laboratory air |
|---|---|
| Platinum spiral, faintly glowing | 5.8 per cent |
| Platinum spiral plunged in a hydrogen flame | 8.6 per cent |
| Hydrogen flame | 17.2 per cent |
| Hydrogen flame, new burner, "some days subsequently" (a humid day) | 20.3 per cent |
Same tube, same kind of air, a factor of 3.5 between sources and days. Tyndall calls the spiral's figure "one-third of the quantity absorbed in the case of the flame of hydrogen". The hydrogen flame's radiation is emitted by hot water vapour, so it falls in the same bands in which cold water vapour absorbs (Kirchhoff's law again; Tyndall's word is "accord" between the periods). The humidity of the air is never stated, and whether the "dry air" also had its CO2 removed is not said. Two more facts from the same chapter: among solid sources the coolest (a Leslie cube at 212 F) is absorbed most [read: Heat p.423, 425], which favours "at least" for a 15 C ground; but the highest figures need a source that radiates in water's own bands, which wet ground does not. His step from "water is opaque to vapour's radiation" to "vapour is opaque to the radiation from water" (para 544) is an inference; no experiment with a liquid-water source is reported [read: Heat p.434].
Step 2. Beer-Lambert extrapolation. Transmission 0.921 over L gives k = -ln(0.921) / L. Then 10 ft (3.05 m) transmits exp(-k x 3.05) [computed]:
| If 7.9 per cent was over | k (per metre) | Absorbed in 10 ft |
|---|---|---|
| 21 in | 0.154 | 37.5 per cent |
| 51 in | 0.064 | 17.6 per cent |
Scaling his 4 ft figures the same way gives about 23 per cent in 10 ft from "over 10 per cent", 14 per cent even from the faint platinum spiral's 5.8, and 38 to 43 per cent from the hydrogen flame's 17.2 and 20.3 [computed]. Every one of these exceeds Tyndall's printed lower bound ("at least 10 per cent" in Heat), so on his own data the number is conservative, not contradicted. (The Rede lecture's "10 to 15 per cent" is a separate statement with no experiment cited.) The open question is whether the lower bound survives two things: saturation (once the band centres are black, more path adds little, so straight exponential scaling overestimates), and the change from his laboratory sources to the cooler 288 K Earth (Step 4).
Step 3. Saturation in Tyndall's own tables. At 1 inch of mercury CO2 absorbs 972 "air units" per equal pressure of air, which is 972 / 30 = 32.4 units of a whole atmosphere of air; at 30 inches (30 times as much gas) it absorbs 90. Thirty times the gas, 90 / 32.4 = 2.8 times the absorption [read: Heat p.347-350; computed]. Against a carbon-monoxide flame (whose radiation is emitted by hot CO2 and so falls exactly in CO2's bands), CO2 absorbs 48.0 per cent at 1 inch and 74.3 per cent at 10 inches, growing only as about the 0.19 power of the amount [read: page image Heat p.431; computed]. This is Kirchhoff's law and saturation in one table, and Tyndall measured it in the 1860s.
Step 4. The source matters. Tyndall's source was at about 373 K; the Earth radiates at about 288 K. The share of blackbody power in the main bands [computed]:
| Band | 373 K source | 288 K Earth |
|---|---|---|
| Water 6.3 micrometre band (5.5-7.5) | 15 per cent | 7.5 per cent |
| Window (8-12) | 29 per cent | 25 per cent |
| CO2 15 micrometre band (13-17) | 15 per cent | 19 per cent |
| Water rotation band (beyond 17) | 24 per cent | 38 per cent |
The laboratory number cannot be carried straight to the Earth. The reviewers' rough band counts give "a few per cent up to about 10 per cent" for 3 m of typical surface air; nobody ran the line-by-line calculation (defined in section 7) that would settle it (the arbiter's web check for a published chart value failed). So "at least 10 per cent within ten feet" stays a plausible but untested extrapolation, and Tyndall's word "certain" is too strong.
Step 5. The size of the blanket. (240 / 5.67 x 10^-8)^(1/4) = 255 K for an Earth with no absorbing gases, against about 288 K observed [computed]. Tyndall's qualitative claim, that without the vapour "garment" the surface would be much colder, is right in sign and large in size.
6. Evidence table
Verdicts are the arbiter's (arbiter.md), on the Step 2 scale.
| Claim | Tested? | Result | Replicated? | Verdict |
|---|---|---|---|---|
| 1. O2, N2, H2 transparent, compound gases absorb strongly; emission = absorption | Yes, Tyndall from 1859 (rock-salt tube, compensated thermopile) | Strong absorption by compound gases; air at his detection floor | Yes: Lecher and Pernter on dry gases (1880), Rontgen (1881/84, different method), photoacoustic work (Bell, Tyndall 1880-81), a century of spectroscopy; Franz (1855) and Foote (1856) before him | Established; reciprocity Established as rank agreement and rough proportionality (absorption / radiation 0.40-0.87 over 17 measured pairs), not exact equality; the numerical table only as a source-specific ranking (lower bounds) |
| 2A/C/D. Water vapour absorbs strongly and controls the earth's heat loss | Yes, Tyndall 1861-81 with many controls; meteorological series (Strachey, Martins and Bravais) as quoted by him | Positive; Magnus, Buff (no effect) and Lecher and Pernter (moist air "more transparent than a vacuum") disagreed | Rontgen 1884 by an independent method; Very 1900; modern spectroscopy | Established |
| 2B. At least 10 per cent of terrestrial heat stopped within ten feet | No (extrapolated from a 4 ft tube and a 100 C source) | Not measured on terrestrial radiation; his own 4 ft tube gives 5.8 to 20.3 per cent depending on the source; every extrapolation exceeds his 10 per cent lower bound, but saturation and the cooler Earth are unchecked | No | Plausible, untested as stated |
| 2. Ratios 70-90 and 16,000 per molecule | Measured against a purity-limited reference | Rise as the reference air is purified | No | Not testable as stated (as constants) |
| 2. Tyndall's unsourced statements: 2 in of ether-saturated air cuts 35 per cent of planetary radiation; 1 in of olefiant gas at least 26 per cent; night ice in the Sahara | Tyndall's word only ("I find"; no table, source or temperature) | No experimental record in the folder | No | Never properly tested |
| 2E. CO2 "minor" (as read into Tyndall) | Tyndall never apportioned the effect | His own tables show CO2 a strong absorber (48 to 74 per cent against a CO flame, image-confirmed) | n/a | Historically superseded but instructive |
| 3. Arc invisible : visible = 7.7 (area), 8 (filter); iodine filter passes heat, blocks light | Yes, Tyndall 1865-66, two methods | Agree within 4 per cent; fits a 3000 K blackbody | Glow and ignition at a dark focus also by Akin and Griffith (1863/65); no independent measurement of the ratio found | Established (arc, source-specific) |
| 3. Sun about 2 (Muller) | Not in the folder; measured at the ground | Modern value about 1.0-1.1 above the air, so the ground figure is even further off | No | Never properly tested |
| 4A. Dark rays make platinum and carbon glow | Yes | Glow with continuous spectrum, also in vacuum | Yes (Akin and Griffith, Croll, others) | Established |
| 4B. Glow is "transmutation", reverse of Stokes's rule | Interpretation | Replaced by thermal emission; Stokes objected in 1865 | n/a | Historically superseded but instructive |
| 4C. Pink glow through blue glass ("gap") | Tyndall once, conditional, asked for confirmation | Not followed up | No | Never properly tested (ignored, not dismissed) |
| 5. Absorption molecular, selected by resonance | Yes (Tyndall; photoacoustics; spectroscopy) | Confirmed; dipole rule replaces "periods of atoms" | Yes | Established |
| 5. Same absorption when compressed or condensed (weak form, non-water vapours) | Tyndall only, single pairs, no error bars, one source; water vapour not tested (extrapolated by rank) | Agreement within 0.5 points; rank order kept for ten liquids and vapours | No independent repeat | Plausible, untested independently (non-water vapours only) |
| 5. Same for all gases and phases (strong form) | Modern spectroscopy | Fails for pressure-broadened lines and liquid water | n/a | Tested, not supported (band shifts in liquid water [web: https://en.wikipedia.org/wiki/Water_absorption]; the 8-12 micrometre point [memory: high]) |
| 5. "Ether entangled in the molecule"; "red more powerful than blue" | n/a | Superseded; the colour sentence is wrong per quantum | n/a | Historically superseded but instructive; Internally inconsistent |
| 6. Heat and light the same wave; heat is motion; J = 772 (Mayer method 771.4) | Yes (Joule; every later measurement) | J = 778 ft-lb/Btu; 771.4 agrees partly by luck | Yes | Established; ether Historically superseded but instructive |
| 7. Regelation | Yes | Real | Yes | Established |
| 7. Ice not viscous; flow is fracture plus regelation | Glen 1955 creep tests | Ice creeps (strain rate about stress cubed) | Yes | Tested, not supported (strong form); Historically superseded but instructive in its time |
| 7. Ice Age needs abundant vapour and strong sun | No; "improved condenser" never specified | Superseded by orbital forcing with feedbacks [memory: high] | n/a | Not testable as stated |
7. What a learner should take away
What is solid. - Gases made of two identical atoms are almost transparent to infrared; molecules whose vibrations change their dipole absorb strongly at their own frequencies; a gas emits where it absorbs. Tyndall established this pattern with a hand-built instrument, and it has been confirmed by every later method. - Water vapour is the main absorber in the lower atmosphere, transparent to most sunlight and partly opaque to the earth's heat. That asymmetry is the greenhouse effect, and Tyndall's 1860s writing on it (night frost, deserts, mountains) holds up qualitatively. - A glowing lamp radiates most of its energy as invisible infrared, and the invisible part can be isolated and focused. His 7.7-8 ratio is what Planck's law gives for a source near 3000 K. - Heat is energy; Mayer's relation J = R_s / (c_p - c_v) gives the mechanical equivalent from the properties of air, and it works (777.5 against 778).
What is wrong, and why the mistake is instructive. - Ratios against a reference you cannot measure ("air = 1") are lower bounds, not constants. They grow every time you purify the reference, as Tyndall's own water-vapour ratio grew from 15 to 90. - Close agreement is not proof of accuracy. Tyndall's 771.4 matched Joule's 772 because two errors of 2 and 3.6 per cent cancelled, in a quantity that amplifies errors in gamma 2.5 times. Always ask how sensitive a result is to its inputs. - Calorescence is heating, not ray-by-ray conversion. The lesson is to separate an observation (the foil glows) from the story told about it; Stokes saw the difference in 1865, and Tyndall's private letter shows he partly did too. - A laboratory number does not transfer to a different source, path or temperature (the "ten feet" figure). Spectra matter: the same gas absorbs very differently from a 373 K cube, a CO flame and the 288 K Earth (Tyndall's own 4 ft tube of room air took 5.8 per cent from a faintly glowing platinum spiral and 17.2 to 20.3 per cent from a hydrogen flame), a point Tyndall himself made in 1859 ("the self-same gas exercises a different action upon different qualities of radiant heat") [web: https://archive.org/details/jstor-111604]. - "Ice is not viscous" is the case where the interested party won the argument in his lifetime and later measurement went the other way. Winning a public dispute is not the same as being right.
What is genuinely open (small, but real). - The right ten-foot figure for real surface air against a 288 K Earth, and which path length belonged to his 7.9 per cent. A line-by-line calculation would settle the first: instead of treating each absorption band as a block, it adds up the absorption of every individual spectral line of water vapour and CO2 (hundreds of thousands of lines, each with its measured strength and width at the given pressure and temperature) across the whole spectrum, then integrates against the source's Planck curve. The line data come from HITRAN (the high-resolution transmission molecular absorption database, maintained at the Harvard-Smithsonian Center for Astrophysics [memory: high]), the standard catalogue used for such calculations. Contributions to Molecular Physics (1873) may settle the second. The humidity of Tyndall's 4 ft runs (5.8 to 20.3 per cent), and whether his dried air also had its CO2 removed, are not recorded in the folder. - What Muller's solar ratio of "about 2" measured. - The date and content of Magnus's later paper "controverting" the vapour results, known only from Tyndall's one sentence (Heat p.533). - What Tyndall saw when he repeated Janssen's water-vapour band observations on the Titlis in 1866 (he gives no result). - Whether Tyndall's pink glow through blue glass is anything but colour perception in dim light: a cheap experiment (section 8, question 5) would end it. - Why Magnus and Lecher and Pernter got negative results: plausible artefacts are offered (heated glass plates radiating, moisture films), but partly by interested parties; their raw data were not examined. - The Tyndall-Akin priority and why Akin's physics record stops after 1865. - Why the vapour table in Heat differs from the Rede lecture's by a factor 1.3-2.6 (apparatus or source not stated); do not mix the two tables.
Safety. Tyndall put his own eye at the focus of the iodine-filtered beam [read: 1866 p.16; he adds "I do not recommend the repetition"]. Near-infrared light (0.76-1.4 micrometres) passes through the eye to the retina and can burn it without any sensation of brightness. Do not repeat it. Carbon bisulphide is also highly flammable; Tyndall records three fires with it [read: 1866 p.13].
8. Check yourself
- Tyndall's water-vapour ratio rose from 15 times to 90 times "air" as he purified his dry air. Why does that make the ratio a lower bound rather than a measurement of how much water vapour absorbs compared with air?
- In Part A, use Tyndall's data with gamma = 1.40 and 15 psi. What value of J do you get, and why did Tyndall not get it?
- A tungsten lamp filament runs at about 2800 K. Roughly how many times more infrared than visible power does it emit (use the table in Part B)? What does that say about the efficiency of incandescent lamps as light sources?
- CO2 at 1 inch absorbs 32.4 units and at 30 inches 90 units. If absorption followed Beer-Lambert for a single wavelength with weak absorption, what would you expect at 30 inches? What explains the difference?
- Design a cheap test of Tyndall's "pink glow" claim (a gap in the spectrum of glowing platinum seen through blue glass). What would you measure, and what result would settle it either way?
9. Further reading
Mainstream. - D. V. Schroeder, An Introduction to Thermal Physics (Addison-Wesley, 2000): ideal gases, heat capacities, blackbody radiation. - D. J. Griffiths, Introduction to Electrodynamics: chapter 9 on electromagnetic waves. - G. W. Petty, A First Course in Atmospheric Radiation (2nd ed., Sundog, 2006): Beer-Lambert, band absorption, Kirchhoff's law, the atmospheric window, at undergraduate level. - R. T. Pierrehumbert, Principles of Planetary Climate (Cambridge, 2010): the greenhouse effect from first principles. - G. A. Schmidt, R. A. Ruedy, R. L. Miller and A. A. Lacis, "Attribution of the present-day total greenhouse effect", J. Geophys. Res. 115, D20106 (2010); summary at https://www.giss.nasa.gov/research/briefs/2010_schmidt_05/. - R. Jackson, "Eunice Foote, John Tyndall and a question of priority" (title [memory: medium]), Notes and Records of the Royal Society (2019/2020): https://royalsocietypublishing.org/doi/10.1098/rsnr.2018.0066. - F. W. Very, Atmospheric Radiation (US Weather Bureau Bulletin G, 1900), the contemporary review of the Tyndall-Magnus dispute: https://archive.org/details/atmosphericradia00veryrich. - J. W. Glen, "The creep of polycrystalline ice", Proc. R. Soc. A 228 (1955): https://data-ww3.ifremer.fr/BIB/Glen_PRSLA1955.pdf. - The Tyndall Correspondence (Epsilon project), letters with Stokes, Croll, Magnus's critics, Akin: https://epsilon.ac.uk/.
Tyndall's own works (in the folder unless stated). - Heat, a Mode of Motion (3rd ed., 1868): Ch. III (Mayer's method, Part A above), Ch. X-XI (gases and water vapour), Ch. VI (glaciers). - On Radiation, the Rede Lecture (1865): the gas table and the blanket argument in brief. Validated text on Wikisource: https://en.wikisource.org/wiki/On_Radiation_(Rede_Lecture). - "On Calorescence", Phil. Trans. 156 (1866): the arc spectrum, the iodine filter, the pink glow. Complete copy (with the missing page 1 and Plate I): https://archive.org/details/jstor-108936 (not in the folder). - Six Lectures on Light (1873) and New Fragments (1892), especially "Atoms, Molecules, and Ether Waves" (1882) on conservation of molecular action. - 1859 Note, Proc. R. Soc. 10, 37-39: https://archive.org/details/jstor-111604 (not in the folder). - Contributions to Molecular Physics in the Domain of Radiant Heat (1872/73), the full research memoirs: https://archive.org/details/contributionstom00tynd (not in the folder). - Bakerian Lecture 1881, Phil. Trans. 173 (1882), his reply to Lecher and Pernter: https://archive.org/details/philtrans03250633 (not in the folder).
Answers
1. Tyndall did not measure the absorption of dry air; he set it at the smallest deflection he could read (about 1 degree) and said it was "probably less even". The ratio is (vapour absorption) / (air absorption). If the true air absorption is smaller than the assumed 1, the true ratio is larger than the printed one. Each time he purified the air, the reference fell and the ratio rose (15, 30, 40 ... 70, 90). A ratio whose denominator is an upper bound is a lower bound. What it does show reliably is that the vapour, though only about half a per cent of the molecules, absorbs far more than all the rest of the air.
2. Unrounded Q_p = 1.29 x 0.24 x 490 / 16 = 9.48 lb-F; excess = 9.48 x (1 - 1/1.40) = 2.71; J = 2,160 / 2.71 = 797 ft-lb. Tyndall used gamma = 1.421, the value from the experiments then accepted; with it the excess is 2.81 (rounded 2.8) and J = 771. His 15 psi (instead of 14.7) pushed J up by 2 per cent and his gamma pulled it down by 3.6 per cent. With 14.696 psi and gamma 1.40 you get 777.5, the modern value.
3. About 10.5 times as much infrared as visible (2800 K row). Under a tenth of the radiated power is visible, so an incandescent lamp is mostly a heater; that is Tyndall's point about the electric light, and the reason LEDs, which emit almost only in the visible, use a fraction of the power for the same light.
4. For weak absorption at one wavelength, absorption is proportional to the amount of gas: 30 x 32.4 = 972 units. Tyndall measured 90, only 2.8 times the 1-inch value. The gas absorbs in bands; at 1 inch the band centres already absorb most of what the source emits there, and adding gas only darkens the band edges (saturation). Pressure broadening widens the bands somewhat but cannot restore proportionality. This is why the effect of adding CO2 to the atmosphere grows roughly with the logarithm of the amount rather than in proportion [memory: high].
5. Heat a platinum foil electrically (a known current, so the temperature can be set) to a faint dull red in a dark room. Measure the transmission spectrum of the blue glass with a cheap spectrometer, and measure the foil's emission spectrum through the glass, not just its colour by eye. Planck's law times the glass's transmission predicts the spectrum you should see. If the measured spectrum matches the prediction (a continuous glow with only the glass's own absorption bands), there is no "gap" and the pink tint is the glass passing red and blue while cutting green, plus poor colour vision in dim light. If the foil's own spectrum, measured without the glass, shows a dip at the same place, Tyndall's gap would be real. Either result settles a question that has been open since 1866. (Do not look into any focused infrared beam while doing this.)
Corrections to earlier documents
- This primer, Part C Step 2 (earlier version) said the Beer-Lambert extrapolations to ten feet (17.6 and 37.5 per cent) did not match "Tyndall's printed claims". That was misleading: his claim in Heat is a lower bound ("at least 10 per cent"), and both extrapolations, like the scaling of his own 4 ft figures (about 23 and 38 per cent), exceed it. Corrected above, following qe.md section 4 and the arbiter's QE revision item 3. Verdict unchanged.
- This primer, source note and Part C Step 3 (earlier version) called the CO2 CO-flame percentages (48.0-74.3) "OCR, not image-checked". They are image-confirmed (Heat p.431, qe.md spot check 3). Caveat removed.
-
Earlier versions left out Tyndall's 35 per cent ether-shell and Sahara-ice statements, Ferrel's support for Magnus, and Babbitt; they are restored in section 4 (qe.md retention items).
-
Numerical note for the orchestrator (unchanged): arbiter.md item 2 gives the modern Mayer-method value as 777.9 using T_0 = 491.4 R; with T_0 = 273.15 K = 491.67 R it is 777.5 (example.py). Both round to the accepted 778, and no verdict changes. Arbiter item 4's "17.7 per cent if to 51 inches" is 17.6 per cent with 7.9 per cent exactly (17.7 with 7.95); immaterial.