Bohr Radius

Q: Does the Bohr radius change for other atoms?

For a one-electron ion with nuclear charge $Z$, the typical size shrinks to $a_0 / Z$. Heavier atoms with many electrons are more complicated, and the simple formula does not describe them exactly.

In one sentence: The Bohr radius is the typical size of a hydrogen atom, about 0.0529 nanometers, and it comes from combining a few fixed constants of nature.

$a_0 = \frac{4\pi \varepsilon_0 \hbar^2}{m_e e^2}$

Read it out loud: “a sub 0 equals four pi, times epsilon sub 0, times h-bar squared, all over m sub e times e squared. a sub 0 is the Bohr radius, the typical size of a hydrogen atom, about 0.0529 nanometers. Pi (rhymes with pie) is the circle number, 3.14159. Epsilon (EP-sih-lon) sub 0 is the vacuum permittivity. H-bar is Planck's constant divided by two pi. m sub e is the mass of the electron, and e is the elementary charge.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Powers of ten, like $10^{-11}$ and $10^{-34}$: The Bohr radius is about $5.29 \times 10^{-11}$ meters. The exponent $-11$ means divide by ten eleven times, which moves the decimal point eleven places to the left: 0.0000000000529 m. The other constants here use powers like $10^{-34}$ and $10^{-19}$ , and they are read the same way. To multiply powers of ten, add the exponents. To divide, subtract them.
Greek letters $\varepsilon$ (epsilon), $\pi$ (pi) and $\hbar$ (h-bar): The Greek letter $\varepsilon$ is said EP-sih-lon. Its small subscript 0 marks the value for empty space, a vacuum, so $\varepsilon_0$ is the vacuum permittivity. The letter $\pi$ is said ‘pie’ and equals 3.14159…, the number of diameters that fit around a circle. The symbol $\hbar$ is read ‘h-bar’. It is Planck’s constant $h$ divided by $2\pi$ .
Electric charge, the coulomb (C) and $e$: Electric charge is a property that makes particles push or pull on each other. Protons carry positive charge, and electrons carry negative charge. The unit of charge is the coulomb, written C. The elementary charge $e$ is the size of one proton’s charge, $1.602176634 \times 10^{-19}$ C. An electron has the same size of charge, but it is negative.
The farad (F) and the unit F/m: A farad, written F, is the unit of capacitance. A capacitor is a device that stores charge between two plates, and one farad means it stores one coulomb for each volt of push. The unit F/m means farads per meter. The vacuum permittivity $\varepsilon_0$ is measured in F/m and sets how strong the electric force between charges is in empty space. A larger $\varepsilon_0$ would make that force weaker.
Mass in kilograms and the unit J s: Mass is the amount of matter, and its unit is the kilogram, written kg. The electron mass $m_e$ is $9.1093837 \times 10^{-31}$ kg, which is extremely small. Planck’s constant is measured in joule seconds, written J s. A joule is a unit of energy, and one joule per second is one watt. Since J s is the same as kg m $^2$ /s, its units can be checked against mass, length and time.
The hydrogen atom: nucleus, proton and electron: An atom has a central nucleus made of protons and, usually, neutrons, with electrons moving around it. Hydrogen is the simplest atom. Its nucleus is a single proton, and one electron moves around that proton. The Bohr radius is about how far that electron typically sits from the proton. An atom is mostly empty space, and its size comes from the balance between electric attraction and the wave nature of the electron.

What each symbol means

Symbol	Meaning
$a_0$	The Bohr radius, a length of about $5.29 \times 10^{-11}$ m, which is 0.0529 nanometers. It is a natural size for the hydrogen atom, and its unit is meters (m). For scale, a human hair is about 0.07 mm wide, which is more than a million times wider than the Bohr radius.
$\varepsilon_0$	The vacuum permittivity, $8.8541878128 \times 10^{-12}$ F/m, read as farads per meter. It is a fixed number that sets how strong electric forces are in empty space. A larger value makes the electric pull between charges weaker.
$\hbar$	The reduced Planck constant, $1.054571817 \times 10^{-34}$ J s, read ‘h-bar’. It equals Planck’s constant $h$ divided by $2\pi$ . Its unit, joule seconds, is energy multiplied by time, and quantum effects are tiny at everyday scales because this number is so small.
$m_e$	The mass of the electron, $9.1093837 \times 10^{-31}$ kg, which is far smaller than the mass of a proton. The subscript $e$ means electron. A heavier electron would give a smaller atom, because the mass sits in the bottom of the formula.
$e$	The elementary charge, $1.602176634 \times 10^{-19}$ C, the size of the charge of one proton. It is measured in coulombs (C). The electron has the same amount of charge, but negative. Because the charge is squared in the formula, the sign does not matter.

The formula piece by piece

$\varepsilon_0$: Start with the vacuum permittivity $\varepsilon_0$ . It sets how strong the electric pull between the electron and the proton is. A smaller $\varepsilon_0$ means a stronger pull, which keeps the electron closer and makes the atom smaller.
$\hbar^2$: Next is $\hbar^2$ , the reduced Planck constant multiplied by itself. It stands for the wave nature of the electron. The electron’s wave has to fit around the atom, and a bigger $\hbar$ spreads that wave out, so it makes the atom bigger.
$m_e e^2$: The bottom of the fraction is $m_e e^2$ , the electron mass times the elementary charge squared. A heavier electron or a stronger charge makes this bottom bigger, which makes the whole fraction smaller and the atom smaller.
$4\pi \varepsilon_0 \hbar^2$: The top of the fraction is $4\pi\varepsilon_0\hbar^2$ . The factor $4\pi$ comes from the geometry of an electric field that spreads out in all directions around a point charge. Together with $\varepsilon_0$ and $\hbar^2$ , it sets the numerator of the Bohr radius.
$a_0 = \frac{4\pi \varepsilon_0 \hbar^2}{m_e e^2}$: The whole formula in one sentence: the Bohr radius equals four pi times the vacuum permittivity times the reduced Planck constant squared, divided by the electron mass times the elementary charge squared.

What the formula is doing, and why

In the Bohr model of the hydrogen atom, the electron circles the proton the way a planet circles the sun, and the smallest allowed orbit has radius $a_0$ . What goes in are the constants that set the size of atoms: the vacuum permittivity $\varepsilon_0$ , the electron’s charge $e$ and mass $m_e$ , and the quantum scale $\hbar$ . What comes out is a length in meters.

Plugging in numbers. The measured constants give $a_0 = 5.29 \times 10^{-11}$ m. That is 0.0529 nanometers, about one twentieth of a nanometer. Atomic sizes are often given in picometers, where one picometer is $10^{-12}$ m, so the Bohr radius is about 53 picometers.

Why does it have this shape? The electric force pulls the electron in, and the wave nature of the electron pushes it out. The Bohr radius is where the two balance. The constant $\varepsilon_0$ is on top, because a weaker electric pull lets the electron sit farther out. The charge $e$ and the electron mass $m_e$ are on the bottom, because a stronger pull or a heavier electron makes the atom smaller. The $\hbar^2$ is on top, because the wave nature of the electron spreads the atom out.

The orbits in the old picture. In the Bohr model, the orbit for level $n$ has radius $r_n = n^2 a_0$ . The level 2 orbit is four times as wide as the first, and the level 3 orbit is nine times as wide. This is a useful picture, but it is only a model.

The modern picture. In quantum mechanics, the electron in the lowest state has no fixed orbit. Its distance from the proton is spread out like a cloud. The most probable distance is exactly $a_0$ , while its average distance is $1.5 a_0$ . So $a_0$ is still the natural size of hydrogen, but the electron is not a planet on a circle.

An everyday picture

Think of a ruler sized to the atom. The Bohr radius is the unit of length for hydrogen, and other distances in the atom are counted in multiples of it, in the same way that centimeters measure a table. A level 2 orbit in the old model is 4 of these units wide, and a level 3 orbit is 9 of them.

Worked example: Size of the n = 2 orbit in the Bohr model

Find the radius of the second Bohr orbit in hydrogen, where $n = 2$ . We use $a_0 = 5.29 \times 10^{-11}$ m, the Bohr radius, rounded to three digits. This is a model size, not a measured orbit of an electron.

Write the orbit rule. In the Bohr model, the orbit for level $n$ has radius $r_n = n^2 a_0$ . The level number is squared, so for level 2 we use $n^2 = 2 \times 2 = 4$ . That gives $r_2 = 4 a_0$ , four times the Bohr radius.
Multiply the front numbers. Multiply 4 by 5.29 to get 21.16. The power of ten belongs to the Bohr radius, so it comes along unchanged: $4 \times 5.29 \times 10^{-11}\ \text{m} = 21.16 \times 10^{-11}$ m.
Tidy to one digit before the decimal point. Moving the decimal point one place to the left makes 21.16 into 2.116. The power of ten goes up by one to balance it: $21.16 \times 10^{-11} = 2.116 \times 10^{-10}$ m. Written out, that is 0.0000000002116 m, which is hard to read without scientific notation.
Convert to nanometers. To convert meters to nanometers, multiply by $10^{9}$ , because one nanometer is $10^{-9}$ m. The exponent changes from $-10$ to $-10 + 9 = -1$ : $2.116 \times 10^{-10}$ m $= 2.116 \times 10^{-1}$ nm $= 0.2116$ nm.
Compare with the first orbit. The level 1 orbit is $a_0 = 0.0529$ nm, and the level 2 orbit is four times that, $4 \times 0.0529 = 0.2116$ nm. The factor is $n^2$ , so the level 3 orbit would be nine times the Bohr radius, which is about 0.476 nm.
Read the answer as a model size. The Bohr orbit is the old picture of the electron as a planet on a circle. What the answer tells us is the scale of the atom, not the exact spot where the electron is found in the modern picture.

Answer: In the Bohr model the second orbit of hydrogen has a radius of about 0.2116 nm, which is four times the Bohr radius. That is about two tenths of a nanometer, the scale of an atom. It is a model size, not a measured orbit of an electron.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Treating the Bohr radius as a fixed orbit that the electron travels on. In quantum mechanics the electron’s distance from the proton is spread out, and $a_0$ is only the most likely distance for the lowest state.
Confusing $\hbar$ with $h$ . The $\hbar$ in this formula is $h / 2\pi$ , which is about $1.05 \times 10^{-34}$ J s. Using $h$ instead makes the answer $(2\pi)^2 \approx 39$ times too large, because $a_0$ depends on $\hbar$ squared.
Misplacing the level number. The orbit radius is $n^2 a_0$ , so $n = 2$ gives $4 a_0$ , not $2 a_0$ .
Forgetting that the answer is in meters. $5.29 \times 10^{-11}$ m is 0.0529 nm, and nanometers are the more convenient unit for atoms.

Where you will meet it

Estimating the size of atoms and the spread of electron clouds in chemistry and solid-state physics.
Atomic units, a system in which distances in hydrogen are measured in multiples of $a_0$ , which makes many formulas simpler.
Estimating the size of exotic atoms, such as positronium or muonic hydrogen, by scaling $a_0$ with the mass of the orbiting particle.
Textbook quantum mechanics, where the hydrogen wave functions are written in terms of $a_0$ .

Questions people ask

Is the Bohr radius a measured size?

It is calculated from fixed constants of nature, not measured directly as an orbit. Measured sizes of hydrogen agree with it, so it is a meaningful scale, even though the Bohr orbit picture itself is not exact.

Why does the formula have so many constants?

Atomic size comes from a balance between the electric attraction, set by $\varepsilon_0$ and $e$ , and the wave nature of the electron, set by $\hbar$ . The electron mass $m_e$ sets how heavy the electron is in that balance.

Does the Bohr radius change for other atoms?

For a one-electron ion with nuclear charge $Z$ , the typical size shrinks to $a_0 / Z$ . Heavier atoms with many electrons are more complicated, and the simple formula does not describe them exactly.

Hydrogen Energy Levels: An electron in a hydrogen atom can sit only on certain energy steps, and when it jumps between steps it gives off or takes in light of one exact color.
Hydrogen-Like Atom Energy: An atom or ion with only one electron has energy levels like hydrogen’s, but a larger nuclear charge pulls every level deeper, so the electron is held more tightly.
Schrödinger Equation: The Schrödinger equation says how a quantum particle’s wave changes from moment to moment, the way Newton’s laws say how a moving ball changes its position.