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§2 · what spectral constants determine

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A derivation, to an approximation

A spectral fit is not yet a free-light hypothesis

What can the low-frequency coefficient determine about molecular scale?

The 4 printed paragraphs this passage explains

§2, paragraph 1: Einstein will show that Planck's determination of the elementary quanta, the constants that fix the size of atoms, does not depend entirely on Planck's theory of black radiation.
§2, paragraph 2: Planck's formula, which fits all experience so far, reduces for long wavelengths and high densities to the form found in section 1. Matching the two coefficients gives N = 6.17 × 10²³, so a hydrogen atom weighs 1.62 × 10⁻²⁴ gram: Planck's own value, in fair agreement with other methods.
§2, footnote 1: The footnote cites Planck's paper in Annalen der Physik, volume 4, page 561, 1901.
§2, paragraph 3: So the theoretical foundations of section 1 work better the higher the energy density and the longer the wavelength, and they fail completely for short wavelengths and low densities.

Write the historical spectral constants as $A$ and $B$ in this explanatory notation. At low $\nu/T$ , expanding the exponential denominator gives a term proportional to $\nu^2 T$ . Its coefficient can be compared to the classical allocation from §1.

\frac{A\,\nu^{3}}{\exp\left(\frac{B\,\nu}{T}\right) - 1} \approx \frac{A}{B}\,\nu^{2}\,T

At low frequency relative to temperature, the spectrum approaches A over B times frequency squared times temperature.

\frac{A}{B} = \frac{8\,\pi\,R}{N_A\,c^{3}}

N_A = \frac{B}{A}\,\frac{8\,\pi\,R}{c^{3}}

Equating the coefficients gives N equal to B over A times eight pi R over light speed cubed.

This comparison is about coefficients and their experimental provenance. In the modern SI, $R$ equals $N_A k_B$ . Inserting that identity and the modern spectral constants recovers the defined $N_A$ ; it is not independent evidence for a molecular count.

Planck’s resonators belong to the matter that emits or absorbs radiation. Einstein’s proposed energy elements concern the radiation itself. Moving between those statements requires an argument.

Open the foundation: Fractions and ratios

Show every step here: A spectral fit is not yet a free-light hypothesis

A tool for this step

Orders of magnitude

How big, how fast and how energetic are the things in the 1905 papers, measured against each other?

Two numbers of the same order differ by less than a factor of ten. Numbers several orders apart belong to different worlds, and comparing them usually tells you which effect can be ignored.

Sizes. A water molecule is about 0.3 nanometres across and a Brownian grain about 1 micrometre: the grain is roughly 3,000 times larger. The light-quanta paper prints the mass of a hydrogen atom as 1/N gram, 1.62 × 10⁻²⁴ g.

Speeds against light. The ratio v/c is about 3.3 × 10⁻⁶ for a rifle bullet at 1,000 m/s, 8.3 × 10⁻⁷ for a jet at 250 m/s, and 9.9 × 10⁻⁵ for the Earth in its orbit at 29.8 km/s. The corrections of relativity go as the square of v/c, so for the Earth they are about 10⁻⁸. Only Kaufmann's fast electrons, a large fraction of the speed of light, reached speeds where they are not small.

Energies of light. One quantum hν of red light (650 nm) carries about 1.91 electron volts, green (530 nm) 2.34 eV, and ultraviolet (250 nm) 4.96 eV. The light paper's photoelectric estimate, about 4.3 volts, is of the same order.

Molecular kicks. An estimate of how often water molecules strike a grain 1 μm across gives about 1.6 × 10¹⁹ blows a second, and about 6 × 10¹⁹ for a grain of 1 μm radius. The assumptions are rough, so the honest statement is 10¹⁹ to 10²⁰ blows a second: far too many for any single blow to be seen.

Worked example: Two estimates, done with powers of ten

The Earth: v/c = 29,800 ÷ 299,792,458 ≈ 9.9 × 10⁻⁵. Squared, that is about 9.9 × 10⁻⁹, near 10⁻⁸.
Water's molecules: 1,000 kg per cubic metre divided by 0.018015 kg per gram-molecule, times 6.02 × 10²³, is 3.34 × 10²⁸ molecules per cubic metre.
Molecules near 600 m/s crossing a surface at a quarter of n times their mean speed give about 5 × 10³⁰ strikes per square metre each second.
A sphere 1 μm across has a surface of 3.1 × 10⁻¹² m², so it takes about 1.6 × 10¹⁹ strikes a second. The inputs are rough, so the result is an order of magnitude, 10¹⁹ to 10²⁰.

JavaScript is off, so the controls below cannot change anything. The construction shows its first setting, and its closing paragraph, “What it shows, in words”, describes the rest.

Try it: placing numbers on a scale of powers of ten

Each line below runs from one power of ten to another, with a tick at every factor of ten. Pick two numbers to compare them, and zoom the lines to the stretch between them.

Comparewith

The width of a Brownian grain is 3,300 times the width of a water molecule: 3.5 powers of ten apart.

Sizes, the whole range

Each line runs from 10⁻¹⁰ at the left to 10⁻⁵ at the right, measuring length in metres. Each tick is ten times the one before.

The width of a water molecule: 3.0 × 10⁻¹⁰ m
One wavelength of 250 nm ultraviolet: 2.5 × 10⁻⁷ m
One wavelength of 650 nm red light: 6.5 × 10⁻⁷ m
The paper's spread in one second: 7.9 × 10⁻⁷ m
The width of a Brownian grain: 1.0 × 10⁻⁶ m
The paper's spread in one minute: 6.2 × 10⁻⁶ m

What it shows, in words

On a scale of powers of ten, equal steps are equal factors of ten. A water molecule and a Brownian grain sit three and a half powers of ten apart: the grain is about 3,300 times larger. The Earth's speed is about a ten-thousandth of the speed of light, and its square, the size of relativity's corrections, is a hundred million times smaller than one. Kaufmann's fast electrons are not placed here: the site holds no dataset for their speeds yet.

Where this lesson stops

This lesson places numbers on a scale of powers of ten. Converting the papers' printed units into SI units is a separate step, with its own conventions.

Letters as printed: the paper writes Planck's two constants as α and β, and this explanation names them A and B. Here β divides a frequency by a temperature inside an exponential, so it is a constant of the radiation law, Wien's constant; it is not the relativity paper's β. L, written c below, is the speed of light.

Planck's formula, which fitted all the observations then available, gives the spectral density as $\rho_\nu=\frac{A\nu^3}{\exp(B\nu/T)-1}$ (p. 136).
Name the ratio in the exponent $y=B\nu/T$ . It has no units: B is in seconds times kelvin, ν in inverse seconds, T in kelvin.
For small y, $\exp(y)=1+y+y^2/2+\ldots$ , so $\exp(y)-1=y(1+y/2+\ldots)$ , which approaches y as y becomes small. Small y means low frequency or high temperature, that is, large T/ν: long wavelengths and dense radiation.
Replace $\exp(B\nu/T)-1$ by $B\nu/T$ and divide: $A\nu^3$ divided by $B\nu/T$ is $\frac{A}{B}\nu^2T$ .

\frac{A\,\nu^{3}}{\exp\left(\frac{B\,\nu}{T}\right) - 1} \approx \frac{A}{B}\,\nu^{2}\,T

At low frequency relative to temperature, the spectrum approaches A over B times frequency squared times temperature.

This has the same form as the classical allocation of §1, $\frac{R}{N}\frac{8\pi\nu^2}{L^3}T$ : both are a constant times $\nu^2T$ . For the two to agree at every low frequency and every temperature, the constants in front must be equal.
Set them equal, then multiply both sides by N and by B/A to leave N alone.

\frac{A}{B} = \frac{8\,\pi\,R}{N_A\,c^{3}}

N_A = \frac{B}{A}\,\frac{8\,\pi\,R}{c^{3}}

Equating the coefficients gives N equal to B over A times eight pi R over light speed cubed.

Worked with the paper's inputs. The page prints β as $4{,}866\cdot10^{-11}$ and α as $6{,}10\cdot10^{-56}$ . The provenance receipt records the exponent of α as a misprint, and this edition computes with $A=6.10\times10^{-57}$ .
R and L are not printed as numbers in §2. With the standard values of 1905, $R=8.31\times10^{7}$ erg per kelvin per gram-molecule and $L=3\times10^{10}$ cm per second, supplied as inputs: B/A is $7.98\times10^{45}$ , $8\pi R/L^3$ is $7.74\times10^{-23}$ , and their product is $N=6.17\times10^{23}$ .
The page prints $N=6{,}17\cdot10^{23}$ , and adds that one atom of hydrogen weighs 1/N gram. Check it: $1/(6.17\times10^{23})=1.62\times10^{-24}$ gram, the number the page gives.
The misprinted exponent cannot be the one used: the same arithmetic with $6.10\times10^{-56}$ gives $6.17\times10^{22}$ , ten times smaller than the number printed.
The paper notes that this is the value Planck found, and that it agrees satisfactorily with values found by other routes. Its point is narrower than it may look: Planck's determination of N depends only to a degree on his theory of radiation, because only the long-wavelength form of the spectrum enters.
The conclusion the paper draws from §§1 and 2: its theoretical foundations serve better the greater the energy density and the wavelength of the radiation, and fail completely for short wavelengths and small densities. §4 works in exactly that region.
Keep the evidence straight. In today's SI, R is defined as $N_Ak_B$ , so putting modern constants into this formula returns the defined $N_A$ and tests nothing about molecules.
Keep the two claims apart. Planck's resonators belong to the matter that emits and absorbs; the energy elements the paper goes on to propose belong to the radiation itself. Getting from one to the other takes the argument of §§3 to 6.

A tool for this step

Fractions and ratios

What does dividing one quantity by another tell you?

The fraction 6/2 is 3, because 2 fits into 6 three times; it also says that 6 is three times 2. Every fraction a/b works the same way. The bottom number can never be zero, since no count of zeros adds up to 6.

When the two quantities have different units, their ratio is a new quantity with a unit of its own: 12 micrometres travelled in 4 seconds is 3 micrometres per second. When they have the same unit, the units cancel and the ratio is a plain number: 2 particles out of 8 is 0.25 of the group.

Worked example: Fourteen metres in ten seconds

A steady walk covers 14 metres in 10 seconds.
Divide the distance by the time: 14 divided by 10 is 1.4, and the unit divides with it, so the rate is 1.4 metres per second.
After 60 seconds the same walk has covered 84 metres, and 84 divided by 60 is 1.4 again. The walk is steady, so the ratio does not depend on how long it is watched.
Metres and seconds are different units, so neither cancels and what comes out is a new quantity that keeps a unit of its own.
The drawing below is the other case: 2 particles out of 8 has the same unit above and below, they cancel, and 0.25 is left as a plain number.

One in four and two in eight, drawn side by side

The filled dots are the ones that moved. Under each row, a bar with that share of its length shaded. The two groups have different sizes and the two shaded stretches are the same length.

The shaded stretch is the same length in both bars. A label that long does not fit inside a drawing, and a drawing's text does not reflow on a narrow screen, so it is written here.

What it shows, in words

The first row is a group of 4 with 1 filled, and the bar under it is shaded for a quarter of its length. The second row is a group of 8 with 2 filled, and the bar under it is shaded for the same length as the first. Doubling how many moved and doubling the size of the group together leave the share where it was, at 0.25.

The dots are counted and the bars are measured, so the two rows agree by two different routes. A fraction records how much of the group moved. The size of the group is a separate fact, and the fraction has already divided it out.

Where this lesson stops

A ratio of two quantities in the same unit is a plain number. A ratio of different units, such as micrometres per second, is a new quantity and keeps its unit.

Explore the equations in this step

Read each operation, check its units and assumptions, or open the mathematical step behind it. These are modern teaching equations, not the equations as printed.

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Avogadro's number from the spectrum

N_A = \frac{B}{A}\,\frac{8\,\pi\,R}{c^{3}}

follows from over , and .

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Each term and operation, in words

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N A equals B over A times eight pi R over the speed of light cubed.

Solving the coefficient match for N gives Avogadro's number from the constants of the radiation law. The paper's value, from Planck's printed constants, is 6.17 times ten to the 23 per gram-molecule.

Model assumptions and every term’s meaning

Planck's radiation law, taken as given, with A and B for its constants.
These are modern teaching equations in SI notation, not a transcription of the printed paper.

Avogadro's number: Here a result: an estimate from the radiation constants, not an input. Read the prerequisite
Spectral constant B: Measured, through the spectrum. Read the prerequisite
Spectral constant A: Measured, through the spectrum. Read the prerequisite
Gas constant: Measured, through gases. Read the prerequisite
Speed of light: Measured independently. Read the prerequisite
An estimate from radiation: Every quantity on the right is measured, so the spectrum gives N. The estimate is only as good as the fit of Planck's constants. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Explore the equation · Modern model notation

Two coefficients that must agree

\frac{A}{B} = \frac{8\,\pi\,R}{N_A\,c^{3}}

over must equal eight pi times over and cubed.

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Explain this equation

Each term and operation, in words

No live values

This equation states a relation. No instrument on this page computes its terms.

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A over B equals eight pi R over N A times the speed of light cubed.

The low-frequency limit of Planck's law and the classical allocation of energy both grow as the frequency squared times the temperature, so their coefficients can be set equal.

Model assumptions and every term’s meaning

Planck's radiation law, taken as given, with A and B for its constants.
These are modern teaching equations in SI notation, not a transcription of the printed paper.

Spectral constant A: From fitting Planck's law to measurement. Read the prerequisite
Spectral constant B: From the same fit. Read the prerequisite
Gas constant: Known from gases. Read the prerequisite
Avogadro's number: The unknown this match can determine. Read the prerequisite
Speed of light: Cubed, from counting the modes of the radiation per unit volume. Read the prerequisite
Coefficients set equal: Two laws with the same form at low frequency must have the same coefficient if both hold there. This uses the classical result only in the limit where it agrees with Planck's. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Explore the equation · Modern model notation

Planck's law at low frequency

\frac{A\,\nu^{3}}{\exp\left(\frac{B\,\nu}{T}\right) - 1} \approx \frac{A}{B}\,\nu^{2}\,T

At low frequency, Planck's law is close to over times squared times .

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Explain this equation

Each term and operation, in words

No live values

This equation states a relation. No instrument on this page computes its terms.

Read the equation aloud in words

A times the frequency cubed over the exponential of B frequency over temperature minus one is approximately A over B times the frequency squared times the temperature.

Where B times the frequency is small compared with the temperature, the exponential is close to one plus its argument, and Planck's law becomes proportional to the temperature.

Model assumptions and every term’s meaning

Planck's radiation law, taken as given, with A and B for its constants.
These are modern teaching equations in SI notation, not a transcription of the printed paper.
B times the frequency is small compared with the temperature.

Spectral constant A: Planck's alpha. Read the prerequisite
Frequency: Cubed in the numerator. Read the prerequisite
Spectral constant B: Planck's beta; B times the frequency over the temperature is the small number here. Read the prerequisite
Frequency: In the exponent. Read the prerequisite
Temperature: In the exponent's denominator. Read the prerequisite
Spectral constant A: The same A, now over B. Read the prerequisite
Spectral constant B: The same B, divided into A. Read the prerequisite
Frequency: Squared: one power of the frequency cancelled against the expansion. Read the prerequisite
Temperature: Now a plain factor: the low-frequency spectrum rises in proportion to the temperature. Read the prerequisite
Approximately, not equal: Replacing the exponential by one plus its argument is an approximation that holds only for small B nu over T. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Modern qualifications

The paper writes the spectral constants as $\alpha$ and $\beta$ ; $A$ and $B$ here are editorial names for them. In §2 Einstein derives $N = 6.17 \times 10^{23}$ from Planck’s printed constants. The historical constant set reproduces that number and the modern set does not, so the two are kept apart.

Assumptions and limits: A spectral fit is not yet a free-light hypothesis

Assumed here

Both sides use the same spectral density and the same units.
The spectral constants and the gas constant are separate inputs to any inference drawn from them.

What this does not establish

Modern exact SI constants give a consistency check, not a new measurement of N.
A fit to the Planck spectrum does not by itself establish independently moving light elements.

Earlier step: A finite window cannot cure an infinite total

Source context: German source · English · Interlinear gloss · Facsimile

References

A. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik (4), 17, 132–148 (1905).

A. Einstein, On the motion of particles suspended in liquids at rest required by the molecular-kinetic theory of heat. Annalen der Physik (4), 17, 549–560 (1905), §§1–5. Bibliographic pointer; this preview is not a source transcription or translation.

A. Einstein, Zur Elektrodynamik bewegter Körper. Annalen der Physik (4), 17, 891–921 (1905).

A. Einstein, Does the inertia of a body depend upon its energy content? Annalen der Physik (4), 18, 639–641 (1905). External 1923 Perrett–Jeffery translation, electronically transcribed by John Walker; its notation was modernized. A reference for this explanatory preview, not this edition’s translation or its pinned facsimile.

R. A. Millikan, A Direct Photoelectric Determination of Planck's "h". Physical Review (2), 7 (3), 355–388 (1916).

26 foundation readings sit behind this argument.

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§2 · what spectral constants determine

Original journal scan

A spectral fit is not yet a free-light hypothesis

Avogadro's number from the spectrum

Two coefficients that must agree

Planck's law at low frequency

Modern qualifications

What this relies on

Assumed here

What this does not establish

References

Reading a graph

Fractions and ratios

A sign records direction

A probability, a frequency and a single trial

Powers of ten and physical units

Squares and square roots

Adding and averaging

Rates of change and derivatives

Density is not probability

Charges, fields and light

Entropy and the number of ways

Entropy and temperature

Exponential change

Fields and waves

Functions and graphs

Adding continuously

Logarithms: turning products into sums

The push of light

Orders of magnitude

Partial derivatives and held-fixed quantities

Probability and independence

Quantities and units

Approximating a curve near a point

Temperature and thermal energy

The units of 1905

Energy of motion and inertia

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