Potential Energy Graphs and Motion

Where are you the most stable? At the top of a building that is a thousand meters tall, or just above the surface on the ground floor? The answer seems logical and obvious. When you are on the ground floor, you have low potential energy. Low potential energy systems are stable systems. Most chemical reactions occur because they are thermodynamically favorable. This means that the process goes from a state of high energy to low energy, from being less stable to more stable. Here, we will discuss the relationship between potential energy and stability, as well as all the information that can be revealed from a system just by analyzing potential energy and graphs.

Potential and Kinetic Energy Graphs

By plotting the potential energy as a function of position, we can learn various physical properties of a system. First, we take a look at the most simple case, a free-falling object. We know that the change of potential energy \(\Delta{U}\) of this system will be given by the expression below,

$$\Delta{U}=mg\Delta{y},$$

where \(m\) is the object's mass in kilograms, \(\mathrm{kg}\), \(g\) is the acceleration due to gravity in meters per second squared, \(\frac{\mathrm m}{\mathrm s^2}\), and \(\Delta{y}\) is the object's position or altitude in meters, \(\mathrm{m}\).

Fig. 1 - Potential energy as a function of position for an object that is free-falling. The force mg is the slope, above the slope, we have kinetic energy, and below we have potential energy. The total energy of the system is a constant horizontal line.

We know that the total mechanical energy of an isolated system is conserved and is constant. It is represented by a horizontal line on the graph, as know that the potential energy \(U\) and the kinetic energy \(K\) are interchanging values such that the total mechanical energy \(E\) remains constant. For example, take a look at the point \(y_A\). The energy below the line corresponds to potential energy, while the energy above the line is kinetic energy.

When you throw an object and it reaches its highest position, we know that its velocity will be zero as its motion changes direction and it begins to fall. In the graph, we see that when the object reaches \(y_max\), the potential energy equals the total energy of the system, meaning that the kinetic energy at this moment will be zero. The kinetic energy will always be zero or positive, such that the potential energy will be always equal to or less than the total energy,

$$\begin{align*}K&=E-U,\\K&\geq 0,\\U&\leq E.\end{align*}$$

The velocity of the object can also be determined by knowing its potential energy and the total energy of the system:

$$\begin{align*}E&=K+U,\\E&=\frac12mv^2+U,\\v&=\pm\sqrt{\frac2m(E-U)}.\end{align*}$$

Now we will consider the case of a spring-mass system. If we examine the energy of the system, we see that the potential energy looks like a parabola, as it depends on the square of the position,

$$U=\frac12kx^2,$$

where \(k\) is the spring constant that determines the stiffness of the spring in Newtons per meter, \(\frac{\mathrm N}{\mathrm m}\), and \(x\) is the object's displacement from the equilibrium position in meters \(\mathrm m\).

Fig. 2 - Potential energy as a function of position for a spring-mass system. The force \(kx\) is the slope, above the slope, we have kinetic energy, and below we have potential energy. The turning points indicate points where the potential energy is maximum. This happens when the spring is fully compressed or stretched. The total energy of the system is a constant horizontal line.

Stable and Unstable Equilibrium

The points on a potential energy against position graph where the slope is zero are considered equilibrium points. The locations with local maximums are locations of unstable equilibrium, while local minimums indicate locations of stable equilibrium.

Fig. 3 - The graph of potential energy against position indicates the different types of stability.

An equilibrium position for any object is one in which the object would be at rest naturally when there are no net forces on it. For example, an equilibrium position for a marble allowed to roll around the sides of a glass bowl would be at the bottom of the bowl. It would naturally come to rest at the bottom if no other external forces move the marble or bowl around. This position is known as a stable equilibrium.

An object is in stable equilibrium if it is given a displacement from the equilibrium position and a force acts on it, in the opposite direction, to return it to that equilibrium position. The potential energy of the object increases momentarily, before returning to its value at equilibrium. Lift the marble slightly and release it, and it will return to the equilibrium position at the bottom of the bowl.

An object is in unstable equilibrium if it is given a slight displacement from the equilibrium position and a force acts on it, in the same direction, pushing it further away from that equilibrium position. The potential energy of the object changes rapidly once displaced. Imagine the marble is made to rest on the lip of the bowl in a position of unstable equilibrium. The marble is then given a slight nudge, which will cause it to roll down the side of the bowl into the center, or oppositely, it's forced out of the bowl entirely if pushed in the other direction.

An object is in neutral equilibrium if it is given a slight displacement from the equilibrium position and this does not affect its equilibrium. The potential energy of the object is unchanged after it is displaced. Imagine the marble has been displaced by a few centimeters on a flat, horizontal surface for an example of this.

Potential Energy and Force in Graphs

When we visualize the potential energy as a function of the object's position in a graph, we find that the force is the negative of the slope. This is due to the relationship between potential energy and work (recall that work is equal to the product of force and displacement):

$$\begin{align*}\Delta U&=-W,\\\Delta U&=-F\Delta x,\\F&=-\frac{\Delta U}{\Delta x},\\F&=\lim\limits_{\Delta x\to 0}-\frac{\Delta U}{\Delta x},\\F&=-\frac{\operatorname dU}{\operatorname dx}.\end{align*}$$

Alternatively, we can use calculus and integrals to find the expression for the potential energy. The following equation applies to all conservative forces, forces that only depend on the initial and final position of the object. In other words, conservative forces are independent of the path taken by the object,

$$\Delta U=-\int_{x_i}^{x_f\;}\vec{F}_{cons}\cdot\operatorname d\vec{x}.$$

The relationship between the potential energy and force tells us a lot about the stability of the system. By definition, if the potential energy is increasing then \(\frac{\operatorname dU}{\operatorname dx}\) is positive, which means that the force would be negative. Similarly, if the potential energy is decreasing, then the force is positive. When \(x=0\), we see that the slope, the force, and the acceleration are zero. This is what an equilibrium point looks like. On either side of stable equilibrium points, there is a force that points back to equilibrium. We see that local minimums indicate locations of stable equilibrium.

Fig. 4 - Visual representation of how forces point back to equilibrium around a point of stable equilibrium.

On the other hand, if the force points away from the equilibrium point, there is an unstable equilibrium. Points of unstable equilibrium are located in a potential energy graph as local maximums. In the image below, we see the potential energy graph for a system that has stable and unstable equilibrium points.

Fig. 5 - If the potential energy is increasing, then the force must be negative. If it is decreasing, the force must be positive. A point of unstable equilibrium is a maximum. We see that around unstable equilibrium points, the forces point away from the equilibrium point. In stable equilibrium points, forced point back to the equilibrium point.

Graph Between Potential Energy and Internuclear Distance

Graphs of potential energy as a function of position are useful in understanding the properties of a chemical bond between two atoms. The energy of a system made up of two atoms depends on the distance between their nuclei. At large distances, the energy is zero, meaning that the two atoms are not bonded and are separate from each other. If the two atoms are very close, there is a repulsive force, but at a distance of an atomic diameter, there are attractive forces that bond them. This happens because, at a distance of an atomic diameter, the electromagnetic force is overcome by the strong nuclear force. The local minimum in the curve represents the distance where attractive and repulsive forces are balanced. This distance between atoms is called the bond length. The energy at this distance is called the bond energy.

Fig. 7 - Potential Energy as a function of the internuclear distance (picometers or \(10^{-12}\,\mathrm{m}\)) between two atoms. The minimum indicates the bond energy and the distance between atoms at the point where repulsive and attractive forces balance each other.

Potential Energy and Reaction Coordinate Graph

A reaction coordinate graph shows how the energy of a system changes during a chemical reaction. During a reaction, reactants transform into products. Energy must be added to the system in order to reach the transition state.

This transition state is represented as a maximum in the potential energy as a function of the reaction coordinate graph. The difference between the maximum and the energy of the reactant at the beginning of the reaction is called the activation energy \(E_act\). For a reaction to reach the transition state, bonds in the reactants must be stretched or broken. The energy required to provoke these changes is the activation energy. The difference between the reactant's energy and the product's energy is \(\triangle E\). This energy difference lets us know if the reaction is exothermic (releases heat) or endothermic (absorbs heat). In the above case, we see that the product's energy is lower than the reactant's energy, so the excess energy is released as heat and the reaction is exothermic.