How big are all the electrons on Earth stacked up? How big is it compared to an ant? Many people would argue that, although a single electron is small, there are so many electrons on Earth that their total volume should be larger than that of an ant.

This is an intuitive assumption, to say the least, but scientists have made it clear over the years that if all the electrons on Earth were piled up, they would be much smaller than an ant. What exactly is going on? See the following analysis.

How big is an electron?

Although scientists have long proved experimentally that electrons have a radius, there is still no definite value for how big the radius is.

In the beginning, scientists used the scattering of electrons in electron bombardment experiments to measure an electron radius of no more than 10^(-18) meters. This value was used for a long time until 1989, when the physicist Hans Demert used the "ion well technique" to measure an even lower radius of no more than 10^(-22) meters. We might as well just take this value.

A 2014 study called ACME (Advanced Cold Molecule Electron EDM) showed that the probability of an Electron's symmetry breaking is no more than a billionth of a billionth of a billionth of a billionth of a billionth. In other words: An electron is a kind of almost perfect sphere.

Using the sphere formula, we can calculate that the volume of an electron is about 4.2 x 10^ (-66) cubic meters.

What is the number of all the electrons on Earth?

Obviously, it's not practical to count the Earth's electrons directly, but a simple way to estimate it is to multiply "the mass of the Earth" by "the fraction of the Earth's mass that contains electrons" and then divide that value by "the mass of a single electron" to get an approximate answer.

In 1798, physicist Henry Cavendish measured the gravitational constant using a "Cavendish twist scale" and calculated the Earth's mass to be 5.965 x 10^24 kg based on the law of universal gravitation.

The mass of a single electron comes from the physicists Joseph John Thomson and Robert Andrew Millikan, who measured the electron's electron-to-mass ratio (the ratio of its charge to its mass, which is a fixed number) to be 1.758×10^11 coulombs per kilogram, The latter measures a charge of 1.602176565(35) x 10^ (-19) coulomb for a single electron, which gives a mass of 9.10956 x 10^ (-31) kg.

Later, scientists measured the masses of protons and neutrons at 1.672621637 x 10^ (-27) kg and 1.674927471 x 10 (^27) kg respectively.

Since the atoms of the Earth are composed of protons, neutrons and electrons, the ratio of protons to neutrons is about 49:51, and the number of electrons is almost equal to the number of protons, we can calculate that the proportion of electrons in the mass of the Earth is about 1 in 3600.

This gives us all the numbers we need for the above calculation, and a simple calculation tells us that there are about 1.8 x 10^51 electrons on Earth.

How does the volume of all the electrons on Earth compare to that of an ant?

We multiply the number of all the electrons on Earth by the volume of a single electron, and we find that the total volume of all the electrons on Earth is about 7.56 x 10^(-13) cubic meters.

Considering the problem of sphere stacking, the volume of the electron pile is larger, for a large number of spheres, different stacking patterns, the space occupancy of the sphere is different, which occupies a larger space is a pattern known as "simple cubic stacking".

As shown above, in the simple cubic packing mode, the sphere occupies about 52% of the space. If we pile all the electrons on Earth in this mode, the volume of space they occupy is about 1.45 x 10^(-12) cubic meters. What is the volume of an ant? Take, for example, the smallest ant on Earth.

The Solenopsis molesta is the smallest known ant. Its worker is only 3 to 5 mm long. However, even this small ant has a volume of at least 5 x 10^(-10) cubic meters, which is more than 300 times the size of all the electrons on Earth.

conclusion

Yes, electrons are so surprisingly small that if all the electrons on Earth were stacked up, they would be much smaller than an ant. Of course, volume is one thing, mass is another, and from the aforementioned data we can easily calculate that the mass of all the electrons on Earth is about 1.6 x 10^21 kilograms, or 16 billion billion tons, which is nothing that ants can compete with.