Understanding Covalent Bonds

Recently, I began preparing for my organic chemistry course and was reintroduced to the concept of chemical bonds, specifically covalent bonds. While taking my previous chemistry classes, my understanding of covalent bonds had been that they are formed when atoms "share" their electrons. While this is a useful way to understand the structure of a covalent bond, it does not explain why the atoms remain bonded to one another. Because covalent bonds are one of the foundational concepts of organic chemistry, I decided to try to develop a more intuitive understanding of the covalent bond.

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I began my quest with the below equation, known as the Lennard-Jones potential:

Source: https://en.wikipedia.org/wiki/Lennard-Jones_potential

The above equation is an adequate approximation of the absolute electrical potential energy of a system of two atoms. It can define absolute potential energy because it defines the system's electrical potential energy as 0 when the two atoms are infinitely apart. One situation the equation can model is that of two hydrogen atoms, the motion of which is restricted to a straight line. The internuclear distance of two covalently bonded hydrogen atoms is 7.4 × 10^(-11) m. The electrical potential energy of a system of two covalently bonded hydrogen atoms is -7.24 × 10^(-19) J. Therefore in the case of two hydrogen atoms, a = 7.4 × 10^(-11) m and b = -7.24 × 10^(-19) J. After inserting these values into the above equation, the following graph can be made:

Source of a, b values: https://www.youtube.com/watch?v=QFzZdcMnlK4&t

This graph represents the varying electric potential energy of the system of hydrogen atoms as their internuclear distance varies. The point (a, b), which represents the moment at which the two hydrogen atoms are covalently bonded, turns out to be the local minimum of this graph. This point is special because its instantaneous slope is 0. But what does the instantaneous slope of this graph represent?

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I will begin by considering two objects, denoted as objects A and B, whose motion is restricted to a straight line. I will also overlay a one-dimensional coordinate system (denoted as the s axis) on the two objects which moves along with object A. I will also assume that the two objects exert a constant conservative force on one another. The change in the potential energy of the system can then be calculated with the following equation:

Because the coordinate system (the s axis) is one dimensional, the change in potential energy formula can be simplified in the following way:

Notice that the sign of Fₛ indicates which direction the conservative force is being exerted along the s axis. If Fₛ is positive, this corresponds to the conservative force being exerted in the +s direction; if Fₛ is negative, the -s direction. It is important to realize that this is not an arbitrary choice, but instead due to the derivation above.

The above derivation was made with the assumption that the conservative force being exerted is constant. If the conservative force is not constant, the change in potential energy formula must be modified as shown below:

Because of the Fundamental Theorem of Calculus (FTC), the following derivation can be made:

The above derivation reveals that if I am given a potential energy vs s axis position graph, the negative instantaneous slope of the graph is the conservative force exerted by object A on object B at a certain position on the s axis.

I will begin by choosing one of the atoms as object A, and the other, as object B. I will also orient the s axis as shown above and choose object A's location to be at the origin. Because of these choices, the internuclear distance equals the s axis position of object B. Recall that the negative instantaneous slope of a graph of the s axis position of object B and the absolute potential energy of the system is the conservative force exerted on object B at the certain s axis position. Because the internuclear distance equals the s axis position of object B, the negative instantaneous slope of the potential energy vs internuclear distance graph then equals the conservative electric force exerted by object A on object B at a certain internuclear distance.

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The instantaneous slope of the below graph can be used to explain why hydrogen atoms remain bonded to one another in a covalent bond.

The graph above shows that when the hydrogen atoms are covalently bonded, there is no electrical force exerted by object A on object B and due to Newton's 3rd law, object B on object A. When the hydrogen atoms have an internuclear distance greater than that of their covalent bond, object A exerts a negative force on object B and consequently, object B exerts a positive force on object A. In other words, the hydrogen atoms experience an attractive force between themselves. The opposite is true when the hydrogen atoms have an internuclear distance less than that of their covalent bond.

In short, when two hydrogen atoms are in a covalent bond, they minimize their electrical potential energy. Because of the way the potential energy formula is defined, when a two object system minimizes its potential energy, the two objects do not exert a force on one another. This is why two hydrogen atoms in a covalent bond remain bonded!

I created the below program on Javascript (khan academy) to demonstrate the physics of a covalent bond. Notice that after entering the parameters corresponding to two certain atoms, if the initial internuclear distance is set to the covalent bond length of the two atoms, the atoms in the simulation will remain motionless, and consequently, remain bonded. This is because at the covalent bond length, the atoms exert no force on one another.