Source: CEM 141 Recitation 02 Key
Tags: gravitational force, electrostatic force, Coulomb's law, potential energy, kinetic energy, inverse square law, isolated systems, attractive force, repulsive force, CEM 141, general chemistry
Difficulty: Introductory | Prerequisites: Basic algebra, familiarity with the concept of force
This recitation sits near the start of CEM 141, where the course shifts from atoms and elements to the forces that hold matter together. The two fundamental long-range forces in chemistry, gravitational and electrostatic, both follow an inverse-square law, but they differ in one critical way: gravity only pulls, while electrostatic forces can push or pull depending on the charges involved. Understanding how these forces change potential and kinetic energy in an isolated system is the foundation for everything that comes next, from ionic bonding to intermolecular interactions. If you missed the first week on atomic structure, you will still follow this material, but you should already be comfortable with the idea that atoms carry charge.
Gravitational and electrostatic forces both depend on 1/r², are mediated by fields, and require two objects, but gravity is always attractive (proportional to mass) while electrostatic force can attract or repel (proportional to charge). In an isolated system, total energy stays constant: when an attractive force pulls objects closer, potential energy drops and kinetic energy rises; when a repulsive force is involved, the opposite happens.
Gravitational force
The attractive force between any two objects that have mass, proportional to the product of their masses and inversely proportional to the square of the distance between them.
In simple terms, this means every object with mass pulls on every other object with mass, and the pull gets much stronger as they get closer.
Electrostatic force (Coulomb force)
The force between two charged particles, proportional to the product of their charges and inversely proportional to the square of the distance between them. Can be attractive (opposite charges) or repulsive (same charges).
Think of it as the electrical version of gravity, except it can push as well as pull.
Inverse-square law (1/r² dependence)
A mathematical relationship where the strength of a force decreases with the square of the distance between two objects. Doubling the distance cuts the force to one quarter.
In simple terms, this means small changes in distance produce large changes in force strength.
Potential energy (PE)
The energy stored in a system due to the positions of its objects relative to each other and the forces acting between them.
Think of it as stored energy waiting to be converted into motion. A ball held above the ground has gravitational PE; two opposite charges held apart have electrostatic PE.
Kinetic energy (KE)
The energy an object possesses because of its motion, equal to ½mv².
In simple terms, this means the faster something moves, the more kinetic energy it has.
Isolated system
A system that does not exchange energy or matter with its surroundings. In an isolated system, the total energy (PE + KE) remains constant.
Think of it as a sealed box where nothing gets in or out, so whatever energy is inside just shifts between potential and kinetic forms.
Field (force field)
A region of space surrounding an object where another object would experience a force. Both gravitational and electrostatic forces are mediated by fields.
In simple terms, this means the force does not require the objects to touch; it acts across empty space through the field.
Both are mediated by fields, meaning the force acts at a distance without physical contact.
Both follow the inverse-square law: force is proportional to 1/r². As the distance between objects doubles, the force drops to one quarter of its original value.
Both require two objects. A single mass or a single charge sitting alone in space does not experience either force.
Gravitational force depends on the masses of the two objects. Electrostatic force depends on the charges of the two objects.
Gravitational force is always attractive. There is no such thing as "negative mass," so two masses always pull toward each other.
Electrostatic force can be attractive or repulsive. Opposite charges (+ and −) attract. Like charges (+ and +, or − and −) repel.
The central idea here is conservation of energy. In an isolated system, total energy (PE + KE) is constant. When one form increases, the other must decrease by the same amount.
As two masses move closer together, the gravitational force (attractive) increases because force is proportional to 1/r².
The potential energy decreases (becomes more negative) as the masses approach, because the attractive force does work on the system.
The kinetic energy increases. The objects accelerate toward each other, gaining speed, so KE rises to compensate for the falling PE.
The PE vs distance graph starts near zero at large separation and curves downward as distance shrinks, producing the classic negative well shape.
This scenario behaves identically to gravity in terms of energy changes. As opposite charges approach, the attractive electrostatic force increases.
Potential energy drops as the charges get closer, for the same reason: the attractive force does work.
Kinetic energy increases to keep total energy constant. The charges accelerate toward each other.
The PE vs distance graph looks the same as the gravitational case: starting near zero at large separation and curving downward.
As two same-sign charges are pushed closer together, the repulsive electrostatic force increases.
Potential energy increases as the like charges approach. Work must be done against the repulsive force to bring them closer, storing energy in the system.
Kinetic energy decreases. The repulsive force decelerates the objects, reducing their velocity.
The PE vs distance graph is the mirror image of the attractive cases: it starts near zero at large separation and curves upward as distance shrinks.
The PE graph for opposite charges (attractive force) curves downward because the system releases energy as the objects approach. The PE graph for like charges (repulsive force) curves upward because energy must be added to bring them closer. The shape of the curve directly reflects whether the force is attractive or repulsive.
F_{grav} = G \frac{m_1 \cdot m_2}{r^2}where G is the gravitational constant, m₁ and m₂ are the masses, and r is the distance between centres.
F_{elec} = k \frac{q_1 \cdot q_2}{r^2}where k is Coulomb's constant, q₁ and q₂ are the charges, and r is the distance between the charges.
E_{total} = PE + KE = \text{constant (in an isolated system)}Attractive forces (gravity, opposite charges): PE vs r starts at zero for large r and curves downward (into negative values) as r decreases.
Repulsive forces (like charges): PE vs r starts at zero for large r and curves upward (into positive values) as r decreases.
Both curves approach zero asymptotically at large distances, reflecting that the force effectively vanishes when objects are far apart.
The inverse-square relationship for electrostatic force is what determines the stability of ionic compounds like table salt (NaCl): the attraction between Na⁺ and Cl⁻ ions holds the crystal lattice together, and the energy required to pull those ions apart is directly related to the PE curve for opposite charges.
The gravitational PE curve is the basis for orbital mechanics. Satellites stay in orbit because their kinetic energy and gravitational PE balance in a way that keeps total energy constant, exactly the same conservation principle explored in this recitation.
Students often think gravitational force can be repulsive. It cannot. Gravity is always attractive because mass is always positive.
Students often confuse "force increases" with "potential energy increases." For attractive forces, a stronger force as objects approach means PE is decreasing, not increasing. The force does work, which lowers PE.
Students sometimes assume that if KE increases, energy is being created. In an isolated system, KE rises only because PE falls by the same amount. Total energy is conserved.
Students frequently draw the PE vs r graph for like charges curving downward. Like charges repel, so bringing them closer requires work, which stores energy. The curve goes up, not down.
⚠️ Expect questions asking you to compare gravitational and electrostatic forces in a table or short-answer format. Know the three similarities and the two key differences cold.
⚠️ Graph-drawing questions are very likely. You should be able to sketch PE vs r for attractive forces (curves down) and repulsive forces (curves up) from memory, with properly labelled axes.
⚠️ Conservation of energy in isolated systems is tested repeatedly throughout CEM 141 and beyond. If PE goes down, KE goes up, and vice versa. Be ready to explain why in words, not just state it.
⚠️ Questions may ask you to explain the physical reason behind the PE change. The answer ties back to the nature of the force: attractive forces lower PE as objects approach; repulsive forces raise it.
True or False: Gravitational force can be repulsive if one of the masses is negative.
False. Mass is always positive; gravity is always attractive.
Fill in the blank: Both gravitational and electrostatic forces are proportional to ______.
1/r² (the inverse square of the distance)
True or False: In an isolated system with two attracting objects moving closer together, kinetic energy decreases.
False. KE increases because PE decreases and total energy is constant.
Fill in the blank: The PE vs distance graph for two like charges curves ______ as the charges approach.
Upward.
True or False: Electrostatic force requires two charged objects, just as gravitational force requires two masses.
True.
Q: List three similarities between gravitational and electrostatic forces.
A: Both are mediated by fields, both are proportional to 1/r², and both require two objects.
Q: Two masses in an isolated system move closer together. Describe what happens to the gravitational force, the potential energy, and the kinetic energy.
A: The gravitational force increases (stronger attraction at shorter distance). The potential energy decreases because the attractive force does work. The kinetic energy increases to keep total energy constant, as the objects accelerate toward each other.
Q: Sketch and describe the PE vs distance graph for two oppositely charged particles approaching each other from a great distance.
A: The graph starts near zero at large r (where the charges barely interact) and curves downward as r decreases. The curve dips into negative PE values, reflecting that the attractive electrostatic force lowers the system's potential energy.
Q: How does the PE vs distance graph for two like charges differ from the graph for two opposite charges, and why?
A: For like charges, the graph curves upward as r decreases, because the repulsive force means work must be done to push them closer together, storing energy. For opposite charges, the graph curves downward because the attractive force releases energy as they approach.
Q: In an isolated system of two like charges moving toward each other, what happens to the kinetic energy? Explain.
A: The kinetic energy decreases. As the charges approach, potential energy increases (repulsive force stores energy), so KE must decrease to keep total energy constant. Physically, the repulsive force decelerates the charges.
This material connects directly to ionic bonding (later in CEM 141), where the electrostatic attraction between cations and anions determines lattice energy. The PE curve for opposite charges is essentially the energy diagram you will use to understand why ionic compounds form.
The concept of energy conservation in isolated systems reappears in thermochemistry, where the total energy of the universe (system + surroundings) is constant, and in kinetics, where the potential energy surface describes reaction pathways.
The 1/r² dependence also sets up your understanding of electron shielding and effective nuclear charge, because the distance between the nucleus and outer electrons governs how strongly they are held.
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