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Introduction · A motion the theory requires

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Letters

Show modern qualifications

Einstein's letters change 1 of this paper's 18 formulas. 1 is already in his letters, and 16 stay in today's letters, with a note saying so.

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A qualification

What the paper sets out to show, and what would decide it

What does Einstein say the molecular-kinetic theory requires of suspended bodies, how sure is he that this is the Brownian motion, and what would observing it decide?

The 2 printed paragraphs this passage explains

Introduction, paragraph 1: Einstein sets out to show that, on the molecular-kinetic theory of heat, bodies large enough to see in a microscope, suspended in a liquid, must move by amounts a microscope can detect. These motions may be the so-called Brownian molecular motion, but he finds the reports available to him too imprecise to judge.
Introduction, paragraph 2: If the motion and its laws can be observed, classical thermodynamics no longer holds exactly for spaces visible under a microscope, and the true size of atoms can be determined exactly; if the predicted motion is not found, that is a serious argument against the molecular-kinetic view of heat.

The paper opens with a claim about what a theory requires. The molecular-kinetic theory of heat takes the heat of a liquid to be the irregular motion of its molecules. From that theory, Einstein announces, it will be shown that bodies suspended in a liquid and large enough to see in a microscope must, because of the molecules' thermal motion, move by amounts large enough to be detected easily with a microscope.

Open the foundation: Temperature and thermal energy

He does not claim to have explained a motion already seen. The motions to be treated, he writes, may be the same as the so-called Brownian molecular motion; but the information he could obtain about it was 'so ungenau, daß ich mir hierüber kein Urteil bilden konnte': so imprecise that he could form no judgment. The identification is left open. The paper predicts a motion and derives its laws; whether the reported motion is that one is a question for observation.

The second paragraph says what depends on the prediction, in both directions. If the motion, together with the laws the paper expects it to follow, can really be observed, then classical thermodynamics can no longer be regarded as exactly valid even for spaces that can be told apart in a microscope, and an exact determination of the true size of atoms becomes possible. If instead the prediction proves false, that would be a weighty argument against the molecular-kinetic view of heat.

Why would a visible motion limit thermodynamics? Classical thermodynamics describes a liquid in equilibrium at one temperature by a few quantities, such as its pressure and temperature, which then stay fixed. It has no place for a part of the liquid that keeps moving of its own accord, now one way and now another. On the molecular view such restless departures from the average are always present. For a body of ordinary size they are far too small to notice; the paper argues that for a grain about a thousandth of a millimetre across, the size §5 uses, they are large enough to see.

Why would it give the size of atoms? §§4 and 5 tie how far such a grain wanders in a given time to N, the number of real molecules in a gram-molecule. With N known, the mass of a single molecule is the mass of a gram-molecule divided by N, and its size can be estimated from the volume a gram-molecule fills. The introduction announces this; the later sections carry it out.

Open the foundation: Orders of magnitude

The road there: §1 argues that suspended bodies should exert osmotic pressure just as dissolved molecules do; §2 derives that from the molecular-kinetic theory; §3 turns it into a diffusion coefficient; §§4 and 5 give the spread of a grain over time and the displacement to look for.

Show every step here: What the paper sets out to show, and what would decide it

Words first. The molecular-kinetic theory of heat is the view that heat is the motion of molecules too small to see. A body suspended in a liquid floats in it without dissolving, like a fine grain of powder stirred into water. The paper's own example, in §5, is a grain a thousandth of a millimetre across: far too small to see by eye, easy to see in a microscope. A gram-molecule is the amount of a substance whose mass in grams equals its molecular weight, what is now called a mole.

The claim. If the molecular-kinetic theory is right, the molecules of a liquid are always moving and always striking any grain suspended in it. Einstein announces that the paper will show that, for grains big enough to see in a microscope, these impacts must move the grain by amounts a microscope can detect easily.
The word 'must' belongs to the theory: the motion is required if the theory holds. The paper does not report having watched it.
The identification, left open. A jittering of small particles in liquids had been reported and was called the Brownian motion, after the botanist Robert Brown. Einstein writes that his motions may be the same, but that the information he could obtain about it was so imprecise ('so ungenau') that he could form no judgment. So the paper predicts a motion; whether the reported jitter is that motion is for observation to say.
The stakes, first direction. Suppose the motion, and the laws the paper derives for it, are observed. Then classical thermodynamics, which describes a liquid at one temperature by a few steady quantities such as its pressure and temperature, is not exactly valid for regions as small as a microscope can resolve. And the true size of atoms can be determined exactly.

A tool for this step

Temperature and thermal energy

What does a temperature say about the jostling of molecules and grains?

The air in a room at 20 °C is a crowd of molecules moving in every direction and colliding billions of times a second. Warm the room and, on average, they move faster. The kinetic theory of heat reads temperature this way: through the average energy of motion of the particles, taken over very many of them.

For any particle in equilibrium with its surroundings, the average energy of motion along each direction is the same:

\left\langle \frac{1}{2}\,m\,v_x^{2}\right\rangle = \frac{1}{2}\,k_B\,T

The average of one half m v x squared equals one half k B T.

Here m is the particle's mass, $v_x$ its speed along x, T the absolute temperature, and $k_B$ = 1.38 × 10⁻²³ joules per kelvin is Boltzmann's constant. It equals R/N, the gas constant shared out per molecule. The Brownian paper writes RT/N where a modern text writes $k_BT$ ; the paper's own letter k means the viscosity, not this constant.

The average is the point. One molecule's energy of motion changes at every collision, from almost nothing to several times the average. The temperature fixes how energy is shared out over many particles. It is not the energy of any one of them.

Nothing in the rule mentions size. A grain ten billion times heavier than a molecule has the same average energy of motion, so it moves more slowly, by the square root of the mass ratio. That is why a suspended grain belongs to the same thermal story as a dissolved molecule, as §1 of the paper insists.

Worked example: A nitrogen molecule and a half-micrometre grain at 20 °C

At T = 293 K, $k_BT$ = 1.381 × 10⁻²³ × 293 = 4.05 × 10⁻²¹ J, so the average energy of motion along one direction is half of that, 2.02 × 10⁻²¹ J.
A nitrogen molecule has a mass of 4.65 × 10⁻²⁶ kg. Setting $\frac{1}{2} m v_x^2$ equal to $\frac{1}{2} k_BT$ gives a typical speed along one direction of √(4.05 × 10⁻²¹ ÷ 4.65 × 10⁻²⁶), about 295 metres a second.
A grain of radius 0.5 μm and density 1200 kg/m³ has a mass of 6.3 × 10⁻¹⁶ kg, about 1.4 × 10¹⁰ times the molecule's. The same average energy gives it about 2.5 millimetres a second.
In water that motion is turned about within some 70 nanoseconds: the grain's mass divided by its drag coefficient 6πηa. That is far too brief to follow, so what a microscope sees is the net result of the zigzag.

This is why the Brownian paper asks for a displacement over a time rather than a speed: the thermal speed is real, but it turns about far faster than any observer can watch.

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: one spread of speeds, two scales

At any moment, the speeds of particles along one axis are spread in a bell curve. The temperature sets its width, and the width depends on the particle's mass. Label the same curve for a molecule or for a grain.

Label the axis for

At 293 K, a nitrogen molecule moves along one axis with a spread of about 295 m/s. The curve does not change; only the numbers on its axis do, by a factor of about 120 000.

Shaded in the middle, under half a width either way: the particles that have, at that moment, less than a quarter of the average energy of motion along the axis, about 38 per cent of them. Outlined at the two ends, beyond two widths: those with more than four times the average, about 5 per cent. Collisions move each particle about the whole curve.

What it shows, in words

At 293 kelvin a nitrogen molecule's speed along one axis is spread with a width of about 295 metres a second, and a half-micrometre grain's with a width of about 2.5 millimetres a second. Measured in its own width the spread is the same bell curve for both, with the same average energy of motion along the axis, 2.02 × 10⁻²¹ joules. At any moment about 38 per cent of the particles have less than a quarter of that average and about 5 per cent more than four times it. The temperature fixes the spread, not the energy of any one particle.

Where this lesson stops

This lesson stops at the average energy of motion. How the jostling of many molecules makes a grain wander, and how far, is §4 of the Brownian paper.

Why thermodynamics would fail there. A grain that keeps moving, now left, now right, in a liquid at one temperature is an unevenness that those steady quantities do not describe. On the molecular view such unevenness is always present. For an ordinary body it is far too small to notice; for a grain a thousandth of a millimetre across, the paper argues, it is large enough to see.
Why atoms could be measured. §§4 and 5 connect the grain's wandering over a given time to N, the number of real molecules in a gram-molecule, what is now called Avogadro's number. A gram-molecule of a substance is a known mass of it, so once N is known, one molecule's mass is that mass divided by N, and its size can be estimated from the volume a gram-molecule fills.
The stakes, second direction. Suppose the predicted motion is looked for carefully and is not there. Einstein says that would be a weighty argument against the molecular-kinetic view of heat. He calls it an argument, and a weighty one: a failed prediction would count heavily against the theory.
What is not said. The introduction does not claim that the existence of molecules had been settled, that the motion had been measured with these laws, or that Einstein had studied the Brownian reports in detail. Each consequence is conditional: if the motion is observed, then one thing follows; if the prediction fails, then another.
The plan of the paper. §1 argues that suspended bodies should exert osmotic pressure just as dissolved molecules do. §2 derives that from the molecular-kinetic theory. §3 turns it into a diffusion coefficient. §§4 and 5 give the spread of a grain over time and the displacement a microscope should show.

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.

Modern qualifications

Robert Brown described the irregular motion of small particles suspended in water in 1828, from observations made in 1827. Later students of the motion included Christian Wiener (1863) and Louis Georges Gouy, who argued in 1888 (Journal de Physique (2) 7, p. 561) that it comes from the thermal agitation of the liquid; Carl Nägeli had argued in 1879 that single molecular impacts are far too weak to move such particles visibly. The 1905 paper cites none of this literature, which fits the introduction's statement that the information available to Einstein was too imprecise for a judgment.

Einstein's next paper on the subject, 'Zur Theorie der Brownschen Bewegung' (Ann. d. Phys. 19, p. 371, 1906), opens by saying that after this paper appeared, Siedentopf told him that Gouy and other physicists had become convinced by direct observation that the Brownian motion comes from the irregular thermal motion of the liquid's molecules.

In 1905 the molecular-kinetic theory was in wide use, and it also had critics who treated atoms as a useful hypothesis rather than an established fact. The second paragraph speaks to that disagreement by naming an observation that could count against the theory as well as for it. Jean Perrin's measurements of 1908 and 1909 on suspended grains are later evidence, dated after this paper; they do not appear in it.

A modern lens: the motion the introduction predicts is what is now called a thermal fluctuation, and the spaces that can be told apart in a microscope are where such fluctuations become visible. The term is later usage; the paper speaks of classical thermodynamics ceasing to hold exactly.

Assumptions and limits: What the paper sets out to show, and what would decide it

Assumed here

The molecular-kinetic theory of heat: the heat of a liquid is the irregular motion of its molecules.
A body suspended in the liquid, large enough to see in a microscope, is struck by those molecules.

What this does not establish

The introduction states what the later sections derive; it derives nothing itself.
Einstein does not identify his motion with the Brownian motion; he says the reports available to him were too imprecise for a judgment.
Both consequences are conditional. The passage does not say the motion had been observed with its laws, nor that the existence of molecules was settled in 1905.

Source context: Annalen der Physik (4), 17, 549–560 (1905), §§1–5. Bibliographic pointer; this preview is not a source transcription or translation.

Explanation

Einstein's displacement argument: the typical distance grows with the square root of time, not with time itself. Every explanation stays on the page when reading-only is on.

Static worked case

For radius 0.5 μm, viscosity 1.35×10⁻³ Pa·s, and T = 290.15 K, the RMS displacement is about 0.8 μm in one second. This static worked case stays in the markup; loading the live ensemble does not replace it.

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