Annus Mirabilis · Interactive critical edition in preparation
Gas ionization bounds and counting model
Ionize a gas with ultraviolet light one quantum at a time.
Gas ionization by light
Gas ionization bounds and counting model
Static worked example
CurrentThese numbers match the current settings.
Model note
- Primary outputs incidentPower, frequency, ionizationEnergyPerMolecule, absorptionEfficiency, duration: Host calculation (lq09.acceptedInputs). Owner lq09.acceptedInputs.
- Primary outputs absorbedLightEnergy, quantumRate, absorbedQuantumRate, ionizationRate, ionizationCount, ionizedGramMolecules: Host calculation (photoelectric.ionizationCount). Owner photoelectric.ionizationCount.
- Primary outputs quantumEnergy, quantumEnergyEv, thresholdFrequency, thresholdWavelengthNm, excessEnergyEv, singleQuantumAllowed: Host calculation (photoelectric.ionizationBounds). Owner photoelectric.ionizationBounds.
- Accepted input revision 1.
- Snapshot version 1.
- Not modeled: Secondary ionization and cascade ionization by energetic photoelectrons in dense gases; Multi-photon ionization processes occurring at extreme optical field intensities; Molecular dissociation channels competing with direct ionization without charge separation; Collisional de-excitation and recombination kinetics over extended reaction times; Spatial beam divergence, gas column pressure gradients, and non-uniform absorption profiles; Detailed autoionization resonances and vibrational-electronic coupling manifolds.
Predict before the numbers
A quantum's energy hν is below the energy J needed to ionize one molecule. How many molecules does the light ionize, one quantum at a time?
Predict before the numbers
Under the paper's assumption that all absorbed light ionizes, what happens to the number of ionized molecules if you double the radiant power?
The result appears when you choose, say you have one in mind, or skip.
2901.59 THz: one quantum carries 12.00 eV
1.00 μW
Experiment settings ionization energy, absorption, exposure, what absorbed light does
10.00 eV
Worked example: a quantum of 2901.6 THz light carries 12 eV, 2 eV more than the 10 eV ionization energy, so molecules are ionized at 2.6 × 10¹¹ per second.
Can one quantum ionize a molecule?
hν = 12.00 eV; ionization energy per molecule 10.00 eV ; threshold ν₀ = 2418.0 THz, λ₀ = 124.0 nm
Quanta in, molecules ionized, each second
Einstein’s hypothesis: every absorbed quantum ionizes one molecule.
Values at these settings
| Quantity | Value |
|---|---|
| Frequency ν | 2901.59 THz |
| Ionization energy per molecule, J | 10.00 eV |
| Energy of one quantum, hν | 12 eV |
| Energy left over, hν − J | 2 eV |
| Light energy absorbed, L | 5.000 × 10−7 J |
| Quanta absorbed each second | 2.601 × 1011 |
| Molecules ionized each second | 2.601 × 1011 |
| Gram-molecules ionized, j | 4.318 × 10−13 mol |
Einstein’s 1905 checks in §9
Lenard, 1900: the longest wavelength that ionizes air. About 190 nm, so Rβν = ca. 6,4 · 1012 Erg per gram-equivalent, as printed. That is 6.65 volts per unit charge, a figure derived here, not printed. With modern constants, 190 nm is 6.53 eV per molecule.
Stark, 1902: the smallest measured ionization voltage for air, at platinum anodes, ca. 10 Volt, so λ₀ ≈ 126 nm; J = 9.6 × 1012 erg per gram-equivalent.
What this model leaves out
- Secondary ionization and cascade ionization by energetic photoelectrons in dense gases
- Multi-photon ionization processes occurring at extreme optical field intensities
- Molecular dissociation channels competing with direct ionization without charge separation
- Collisional de-excitation and recombination kinetics over extended reaction times
- Spatial beam divergence, gas column pressure gradients, and non-uniform absorption profiles
- Detailed autoionization resonances and vibrational-electronic coupling manifolds
The rule this laboratory evaluates
From src/physics/reference/photoelectric.ts, the audited TypeScript reference evaluator.
// Paper 1, §9: one quantum, one ionization.
// Threshold frequency: nu_0 = J / h
// If nu < nu_0: no single-quantum ionization; the count is not applicable.
// If nu >= nu_0:
// every absorbed quantum ionizes: j = L / (R*beta*nu), or N_ion = L / (h*nu)
// a declared share a ionizes: N_ion = a * L / (h*nu)
// the share is unknown: underdetermined, at most N_abs = L / (h*nu)If ultraviolet light ionizes a gas one quantum at a time, each quantum must carry at least the work needed to ionize one molecule, and the number of molecules ionized should equal the number of quanta absorbed. Einstein proposed the second statement as a test worth making.
§9 assumes that in the ionization of a gas by ultraviolet light each absorbed light quantum ionizes one gas molecule. Two consequences follow. First, the ionization work per gram-equivalent, J, cannot exceed the energy of the absorbed quanta: Rβν ≥ J, or per molecule hν ≥ J. Second, absorbed light energy L ionizes j = L/(Rβν) gram-molecules, for any gas that shows no appreciable absorption without ionization at that frequency. The instrument works both at its defaults: J = 10 eV per molecule, light of 2901.59 THz whose quanta carry 12.0 eV, 2.0 eV more than needed, so the threshold is 2418 THz, a wavelength of 124 nm. With 1 μW of light, half of it absorbed, for 1 s, L = 5 × 10−7 J, and 2.60 × 1011 quanta are absorbed; if each ionizes one molecule, 2.60 × 1011 molecules are ionized, 4.32 × 10−13 gram-molecules. If only a declared share ionizes, the count is that share of the quanta, 7.80 × 1010 for a share of 0.3. If the share is unknown, the lab reports the count as not fixed, with the absorbed quanta as its upper limit, and never more ions than quanta. Below the threshold, at 2000 THz (8.27 eV), no single quantum can ionize, and the count is not applicable rather than zero. A named gas must come with a cited source for its ionization energy.
Start with one quantum. Its energy is hν; at 2901.59 THz that is 6.626 × 10−34 J·s × 2.90159 × 1015 s−1 = 1.923 × 10−18 J, which is 12.0 eV. To ionize a molecule the quantum must supply the ionization work, here J = 10 eV, so it has 2.0 eV to spare. The lowest frequency that can do it is where hν equals J: ν = J/h = 10 eV/(4.136 × 10−15 eV·s) = 2.418 × 1015 s−1, or 2418 THz, and the wavelength there is c/ν = 124 nm. At 2000 THz a quantum carries only 8.27 eV, 1.73 eV short, so under §9's assumption nothing is ionized, however much light arrives, because no quantum can pool its energy with another. Now count. 1 μW for 1 s is 10−6 J of light; half is absorbed, so L = 5 × 10−7 J. Dividing by the energy of one quantum, 5 × 10−7/1.923 × 10−18 = 2.60 × 1011 quanta absorbed. If every absorbed quantum ionizes one molecule, 2.60 × 1011 molecules are ionized. A gram-molecule holds N = 6.022 × 1023 molecules, so that is 2.60 × 1011/6.022 × 1023 = 4.32 × 10−13 gram-molecules, which is Einstein's j = L/(Rβν) written per molecule: L divided by N hν. If only a share a of the absorbed quanta ionize, multiply by a: 0.3 × 2.60 × 1011 = 7.80 × 1010. If the share is unknown, the equation no longer fixes a number; all that survives is that ions cannot outnumber absorbed quanta, since each ion needs a quantum of its own. That is why the lab states an upper limit and not a value. §9 restricts j = L/(Rβν) to gases with no appreciable absorption unaccompanied by ionization, because light absorbed in some other way would add to L without adding ions.
Einstein compared the first consequence with two measurements. Lenard's largest wavelength effective in ionizing air, about 1.9 × 10−5 cm, gives Rβν = 6.4 × 1012 erg per gram-equivalent, about 6.5 eV per molecule, as an upper limit for J; Stark's smallest ionization voltage for air, about 10 volts at platinum anodes, gives the upper limit 9.6 × 1012, which Einstein called nearly equal, while noting in a footnote that inside the gas the ionization voltage for negative ions is five times larger. The count j = L/(Rβν) was offered as the test he thought most important, not as a result. Modern first ionization energies of oxygen and nitrogen molecules, about 12.1 and 15.6 eV, exceed both limits, so what Lenard and Stark measured cannot have been the single-quantum ionization of those molecules that §9 assumed. The lab's default of 10 eV follows Stark's figure.
The explanation
Full explanation
Each quantum must carry at least the work needed to ionize one molecule, which sets a frequency threshold, and the number of molecules ionized should equal the number of quanta absorbed. Einstein proposed that count as a test worth making.
Show every step of the investigation
Choose a gas with a cited ionization energy, set the frequency and the absorbed energy, and read whether the quanta clear the threshold and how many molecules they could ionize. A gas without a citation is refused rather than given a number.
An explanatory model, not an observation of nature. This embed starts from the laboratory’s worked defaults, not a saved run. Presentation options change the surrounding guide, never the numerical inputs.