how to find atomic radius

What is Atomic Radius

The atomic radius of an atom is defined as the distance from the center of the nucleus to outermost electron in its energy shells. Calculating the atomic radius can be difficult. The best approach to understand the concept of atomic radius is to compare finding atomic radius to finding the radius of a circle.

Calculating the radius of a circle is fairly straightforward. This is one of the first calculations we are taught in mathematics. To find the radius of a circle, all that is needed is to measure the distance from the center of the circle to the edge of the circle. This is sometimes done by using information that is already known about the circle (like area or circumference) and working backwards to find the radius.

Similarly, to find the atomic radius of an atom, it is necessary to measure the distance from the nucleus to the outermost electron. Because the atom is in constant motion, however, there are a few extra steps that need to be taken. Often, we use the same process of 'working backwards' to find our solution.

How to Find Atomic Radius

Unlike a circle, the outermost electron doesn't have a concrete 'edge' that can be used as a constant. The location of the outermost electron tends to vary and has the ability to move closer and farther away to the nucleus of an atom with varying probability. Despite this, there are still several methods that can be used to calculate the atomic radius of an atom.

Method 1: Using Two Atoms of the Same Element That Aren't Bonded (Van der Waals Radius)

One method of determining atomic radius is to experiment with two atoms of the same element that are not bonded together. Using these two molecules, one would try to push them closer and closer together until they were able to find the minimum distance between the atoms that could be maintained without forcing a bond.

Once this distance had been achieved, it would be possible to measure the distance from the nucleus of one atom to the nucleus of another atom. Then, all that needs to be done is to divide the distance between the nuclei of the atoms by 2. The resulting number is the atomic radius of these elements.

Method 2: Using Two Atoms of the Same Element That Are Bonded (Covalent Radius)

It is also possible to determine the atomic radius of an element by looking at two atoms of the same element that are bonded together through a covalent bond. This covalent bond means that the two atoms can share electrons – at one point in time, the majority of the atoms could be in the first atom's energy shield, in the next moment the majority of the atoms could be in the second atom's energy shield.

Although these two atoms share a covalent bond, it can be assumed that the bond will not be able to alter the distance between the nucleus and the theoretical outermost point of an electron. This means that all that needs to be done is to measure the distance between the two nuclei and then divide that number by 2. The resulting number would be the atomic radius of the element.

Van der Waals Forces and Atomic Radius

To understand Van der Waals forces and atomic radius rule, you must first understand how the forces in atoms affect their polarity. The first thing you should know is that there are two types of atoms; polar atoms and nonpolar atoms.

Every atom contains a nucleus comprised of protons and neutrons. Because the neutrons have no charge of their own to contribute, the nucleus of an atom always contains a positive charge. Outside the nucleus, there is at least one electron shell (though sometimes more) that holds a certain number of electrons. These electrons are constantly moving. It is important to note, however, that these electrons are not always evenly dispersed.

Atoms with evenly distributed electrons are nonpolar. These nonpolar atoms are the most heavily affected by Van der Waals forces (although Van der Waals forces are found in all molecules). However, because electrons are constantly moving, it is not likely that a nonpolar bond will last long. Eventually, the number of electrons will be greater on one side of the atom than the other side. When this happens, the atom becomes distorted.

When an atom becomes distorted (or polarized), it interacts with atoms differently than nonpolarized atoms do. Because there are more protons on one side and more electrons on another, one side is more positive and one side is more negative. For example, if there are more electrons on the left side of an atom compared to the right side of the atom, the left side will be negative and the right side will be positive. This make the atom a dipole.

When a nonpolar atom is placed near a dipole, its electrons become distorted as well, creating another dipole. If the first dipole is negative on the left and positive on the right, the second dipole will be positive on the left and negative on the right. The forces that cause these two atoms to be joined together are known as Van der Waals forces. Because the two dipoles will be drawn as close to each other as they can without merging, the Van der Waal forces can be used to evaluate atomic radius.

How to Use Van der Waals Forces to Calculate Atomic Radius

The forces that draw two dipoles together (Van der Waal forces) can be used to calculate the radius of an atom. To do this, it is necessary to use two atoms of the same element. It is also best to use atoms that have high polarizability because these atoms also have higher Van der Waal forces.

It is fairly easy to determine how polarizable an atom is. The more electrons an atom has, the more likely it is to be able to become polarized. Additionally, the more electrons an atom has, the higher the Van der Waal forces are between these nuclei.

To calculate the atomic radius of two dipoles of the same element, simply find the nuclei of these atoms and measure the distance in between them. Once you have this number, divide by 2. The resulting number is the atomic radius.

This method is so effective because the Van der Waal forces ensure that the atoms will be drawn as close as they can without merging. Once this happens, measuring from the nucleus of the first atom to the nucleus of the second atom is effectively the same as measuring the diameter of an atom. Because the radius is half the diameter, simply dividing by 2 solves the equation.

Coulomb's Law and Atomic Radius

Much of our understanding of calculating the atomic radius of an atom is thanks to the development of Coulomb's Law. This law helps us understand the relationship between the charges of the particles that make up an atom. In turn, understanding the relationship of charged particles helps to predict the size of an atom's radius.

How Did Coulomb Create His Law?

Coulomb's Law was first published in 1785 by a French scientist named Charles Augustin de Coulomb. Although his accomplishment was made long before we had computerized technology to aid in research, his methods were very effective. Coulomb used a torsion balance to help himself understand the relationship that different charged particles had with each other.

He took two spheres and charged them with negative energy. Once he had done this, he then attempted to push them together to see how close the spheres would get. When he did this, he noticed that two spheres that were charged with the same energy (in this case negative energy) repelled away from each other. He measured the distance between the spheres, then pulled one sphere away.

After making notes about the distance between spheres, Coulomb took a sphere that hadn't been charged (a neutral sphere) and held it next to the charged sphere. Eventually, he would discover that putting a neutral sphere next to a charged sphere would transfer half of the negative energy to the neutral sphere. Additionally, when he pushed the two original spheres together again, they were able to get closer to each other but still repelled the similar energy.

Eventually, this study would lead Coulomb to several realizations:

1. The force between charged particles is proportionate to the magnitude of the particle charges.
2. The force between charged particles is inversely proportional to the radius squared. In other words weak charges created big radii, while strong charges created small radii.
3. Opposites attract and like charges repel. This is why protons are attracted to electrons, but repel away from each other. (Think of the effect two positive magnets have on each other vs. one positive and one negative magnet).
4. Although two like charges repel against each other, neutrons enable protons to exist together in the nucleus because they help to take away half of a proton's charge.

The Significance of Coulomb's Law and Inverse Square Law

The discoveries that Coulomb made from his work with the torsion balance are highly influential – especially since Coulomb discovered that inverse square law could be applied to the particle relationships in an atom.

Inverse square law (first put forward by Isaac Newton in his law of universal gravitation), states that physical intensity is inversely proportional to the square of the distance of the physical quantity. In the case of Coulomb's law, this describes the relationships the protons have with each other and the electrons in the outermost shell. This idea can be seen in one of Coulomb's equations below.

In this equation, F is equivalent to force. q1 and q2 represent the magnitude of the charges of the atoms. Lastly, r is the distance between the charges of the atom. This equation shows that the forces of the atoms that are being studied have the potential to show the radius of the atom(s) in question.

Electron Shielding and Atomic Radius

The number of electrons in the inner shells of an atom versus the number of electrons in the outermost shell of the atom will always be in constant 'competition' with each other. The outermost electrons have a negative charge but are attracted to the nucleus of an atom which has positively charged protons. Unfortunately, the inner shell electrons have the same charge, which makes them want to repel the electrons in the outer shell.

At the same time, the protons in the center of the atom are fighting to pull away from each other. Because they all have the same charge, they want to be able to push closer to the electrons than themselves. They are balanced out by the neutrons in the core because neutrons do not have a charge of their own. These neutrons are, however, able to take half the charge from a proton which helps the protons to remain balanced enough to remain in the nucleus. The more powerful a nucleus is, the smaller the atomic radius is because of the strong attraction to the electrons.

The relationship between electrons and how they interact with each other is dependent on where these electrons are located. Although it would seem that electron in the same energy shell would have an impact on one another, they are relatively unaffected by negative charges that are parallel to them. Electrons located in different energy shells, however, have a significant impact on each other. They are constantly trying to push away from one another, but they are kept inside the atom thanks to the strong pull of the protons in the nucleus.

Why Don't the Protons and Electrons Ever Form Bonds Together?

Based on what we know of how protons and electrons interact, it would make sense if these particles were able to form bonds with one another. However, this never happens because the nucleus is powerful enough to keep the electrons in the inner and outer shells from leaving the atom. Additionally, the inner shell electrons are powerful enough to prevent the outer shell electrons from merging with the protons in the atom nucleus.

In most cases, the number of electrons in the outermost shell will always be less than the number of protons in the nucleus. Because of this, the strong pull of the protons in the nucleus will cause the electrons in the outermost shell to fight harder to reach the nucleus. The inner shell electrons will fight back, but the more electrons there are in the outermost shell, the closer they will be able to get to the nucleus. This is why elements located closer to the right side of the periodic table will always have smaller atomic radii than elements in the same group on the left side of the periodic table.

Trend for Atomic Radii on the Periodic Table

Many of the trends for atomic radius can be explained using Coulomb's law of attraction. Coulomb's law helps to demonstrate why having more electrons in the outer shell will always result in a 'tighter' alignment of atomic particles. Similarly, atoms that have fewer electrons in their outermost shell will have a more loose alignment of atomic particles.

By using these observations, it is possible to use Coulomb's law of attraction to establish trends for atomic radius on the periodic table.

Noble Gases Always Have the Smallest Atomic Radius in their Group

Each group in the periodic table corresponds to the number of energy shells all the known elements have. Elements to the left of the periodic table signify the start of the energy shell where there are fewer electrons in the outermost shell. These elements always have bigger atomic radii compared to other elements in their group. Elements to the right of the periodic table signify the end of an energy shell where the shell is almost entirely filled. The radius of elements to the right of the periodic table are always smaller than the radius of elements on the left. Because the noble gasses are the farthest to the right, it is easy to conclude that these are the elements that have their energy shells completely filled.

The fact that the noble gasses have their energy shells completely full is key to understanding why they will always have the smallest atomic radii out of all the other elements in their group. All electrons are drawn to the positive energy of the protons in the nucleus of the atom. However, they also have to combat the electron shielding effect of electrons that are located in the inner shells of the atom. Therefore, having more electrons will always result in a smaller radius.

This knowledge makes it easy to predict trends. When looking at Argon (Ar) versus Sodium (Na), it can be assumed that Ar will always have a smaller radius. This can be presumed because both atoms are in group three, but Ar has more electrons available than Na.

It is important to acknowledge, however, that this trend only applies to elements that are in the same group as a specific noble gas. If an element is in a higher or lower group, the arrangement will change depending on how many energy shells exist – not how many electrons are in the energy shell.

Example:

Potassium (K) will always have a smaller atomic radius than Xenon (Xe) because K is located in group 4 while Xe is located in group 5.

Atomic Radius Decreases from the Bottom Left to the Upper Right Hand Corner of the Periodic Table

When looking at the entire periodic table, it is easy to see which elements have the largest radii and which element have the smallest radii. Because the elements with the most energy shells and the least amount of electrons in those energy shells have the largest radii, it is easy to determine that Ununennium (Uu) has the largest atomic radius of all the known elements. This is because it has the most energy shells (8) and the least amount of electrons in its outermost shell (1).

Similarly, it can be determined that Helium (He) will have the smallest atomic radius of all the elements. This is because it has the fewest energy shells (1) and enough electrons to fill its energy shell (2). Additionally, because the number of electrons in the outermost shell is equivalent to the number of protons in a He atom, the forces between these two types of atomic particles will ensure that the radius of this atom is very small.

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how to find atomic radius

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