How to Calculate the Mass of a Proton

Atoms are mysterious things, appearing in all sorts of unrelated ways in everyday language. Even if you are not a chemistry expert, you probably know that at atom is an extremely tiny part of matter and that all matter is made up of at least one kind of atom.

"Atomic" as an adjective in chemistry and physics is literal, referring to a property of the entity called an atom. In casual contexts, thanks almost solely to events in the second World War, it means "explosive," which is misleading.

Semantics aside, atoms are interesting because, despite how tiny they indeed are, they consist of even tinier things (helpfully called subatomic particles). Until late in the 20th century, it was unknown for sure whether these three primary subatomic particles (protons, neutrons and electrons) themselves could be separated into discrete structural elements. Spoiler alert: They can.

The proton is of great interest to physicists and physical chemists for a number of reasons. It is one of the two subatomic structures known as nucleons, and it is the one that carries a positive electrical charge, contrasted with its similarly-sized companion in the atomic center.

Meanwhile, electrons, though tiny and impossibly distant from the nucleus in relation to the size of the atom, experience force interactions with protons as well. Prepare to learn about the various distinguishing features of these fundamental entities.

Overview of the Atom

You may already be familiar with atoms generally, but it's never a bad idea to have the essentials in the front of your mind when you start to explore parts of it in more detail.

As of 2020, there were 118 known elements, or individual "varieties" of atoms. Each atom has one to 118 protons, which is also the atomic number on the periodic table of elements and the number that determines the identity of the element. All elements besides hydrogen also include neutrons, which are very close in mass to protons. The number of neutrons is the same or close to that as the number of protons, with these variations of elements known as isotopes.

The mass of an atom's protons and neutrons account for almost all of the atom's mass, because the third kind of subatomic particle has only about 1/1,800th the mass of either a proton or a neutron.

But the particles called electrons are vitally important to the organization of the periodic table, for it is the number and arrangement of these negatively charged particles that give individual elements their bonding properties, i.e., the manner in which they connect (or fails to connect) to other atoms.

The protons and neutrons are packed together in the nucleus, with the total number of these particles ranging from 1 to over 200 for the heaviest elements. Interestingly, the nucleus does not increase much in size when more protons and neutrons are added, but the atom as a whole does.

This is because the electrons, identical in number to protons, lie far outside the nucleus in "probability clouds" corresponding to energy, and the size of these grows with atomic number even as the nucleus remains close to the same size.

Proton Essentials

Protons sit in the nuclei of atoms and may be thought of as spherical for conceptual purposes. The same is true of neutrons, and if you were to make a three-dimensional model of a simple atom, you could choose different-colored but same-sized balls for the protons and neutrons.

The mass of a proton is about 1.67 × 10–27 kilograms (kg). That of a neutron is very slightly greater, about 1.69 × 10–27 kg, and that of an electron is 9.11 × 10–31 kg. Also, the mass of a proton is assigned 1 atomic mass unit (amu) for convenience. This unit is used for other subatomic particles as well; the mass of electrons in amu (atomic mass units) is 0.00055.

The charge of a proton is called "plus one," or +1, in relation to other physical particles, since it was once believed that protons (and electrons) represented the smallest units of charge anything in nature can have. The magnitude of this value (positive for protons, negative for electrons, making these particles therefore attracted to one another by the electrostatic force) is 1.6 × 10–19 C.

It is worth noting, just for the sake of appreciating the work of physicists and chemists, that protons for a long time were not considered to exhibit decay (meaning they basically exist "forever" once formed), are believed to have a half-life of about 1032 to 1033 years. Considering that the age of the universe itself is around 1.4 × 1010 years, seeing a proton decay radioactively would be quite the lottery-level feat!

The Structure of the Proton

Protons, as minute as they are, are also composed of their own building blocks. Both protons and neutrons, in fact, consist of three individual particles that represent types of quarks (more on those soon). Both protons and neutrons consist of some combination of three "up" quarks and "down" quarks. But if the proton has a +1 charge, and the neutron is neutral, how can this be?

The answer lies in the fact that the +1 "unit" or "fundamental" charge turns out to be divisible, after all, at least in the special circumstance of quarks. If a proton consists of 2 up quarks and 1 down quark while a neutron has 1 up quark and 2 down quarks, assigning a charge of +(2/3) to the up quark and –(2/3) to the down quark resolves the issue.

  • There are six quarks known in all: up, down, top, bottom, charm and strange. (Scientists sometime have strange naming conventions). 

Protons and neutrons are considered baryons, the heaviest class of particles thrown together from quarks. Along with mesons, they belong to a group of particles known as hadrons, which are subject to the strong nuclear force or the "glue" that holds protons and neutrons together.

Proton Spin

While summing the charges of the quarks that make up a proton gives the proton's total charge of +1, it is not that simple when it comes to angular momentum, a property related to "spin."

A proton does not actually rotate like the Earth does around its axis, but "spin" is a good way to envision the property of intrinsic, or built-in, angular momentum of a proton (given the value 1/2), which comes mainly from interactions between quarks and particles called leptons that also make up certain subatomic particles.

The interesting thing about proton spin is that physicists arrived at the right value (1/2) for the wrong reasons, but in the 21st century have been able to harmonize longstanding theoretical ideas with experimental results.

"Magical" Contributions to Proton Mass

The mass of a proton should be less than it is; adding up the masses of the individual quarks gives a result only about 9 percent of that of the measured proton mass of 1.67 × 10–27 kg. What is going on to add mass without adding matter?

In 2018, a group of physicists used an emerging and mathematically complex technique called quantum chromodynamics (QCD), or more specifically lattice QCD, to determine the mass of a proton using nonstandard means. As with proton spin, these results were encouraging, offering insights into where the proton's mass "comes from."

  • Mass for subatomic particles is often given in electron-volts, or eV. 

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