AP® Chemistry review sheet from Aim for Five (aimforfive.com/chem/units/2/2-2)
Unit 2 · Topic 2.2
2.2 Intramolecular Force and Potential Energy
A graph of potential energy against the distance between two nuclei shows why bonds form: energy is lowest at one particular distance, the bond length, and the depth of that low point is the bond energy. Bond order and atom size set covalent bond lengths and strengths, and Coulomb's law sets the strength of attraction between ions.
Key terms
- potential energy curve
- internuclear distance
- bond length
- bond energy
- bond order
- Coulomb's law
The potential energy curve
Picture two hydrogen atoms starting far apart. Their potential energy is set to zero. As they move closer, each nucleus attracts the other atom's electron, so the energy drops. At very short distances, the two positive nuclei repel strongly, so the energy shoots up.
The result is a curve with a dip. Read three things from it.
- Bond length: the distance at the bottom of the dip, where potential energy is lowest. For H₂ it's about 74 pm.
- Bond energy: the depth of the dip below zero, which is the energy needed to pull the bonded atoms completely apart. For H₂ it's about 436 kJ/mol.
- The steep wall on the left: nucleus-nucleus repulsion at short distances. The gentle rise on the right: attraction weakening as atoms separate.
Comparing two curves
A deeper well whose lowest point sits farther left means a stronger, shorter bond. N₂ (triple bond) has a bond length of about 110 pm and a bond energy of about 945 kJ/mol. F₂ (single bond) is about 142 pm and only about 160 kJ/mol. On the same axes, N₂'s curve would have its minimum farther left and much lower.
Bond energy works in both directions: breaking a bond always takes in energy, and forming the same bond releases the same amount. That idea comes back in topic 6.7, where you estimate reaction enthalpies from bond energies.
What sets covalent bond length and energy
Bond order is the number of shared electron pairs between two atoms: 1 for a single bond, 2 for a double, 3 for a triple. Between the same two atoms, higher bond order means a shorter, stronger bond.
Atom size matters too. Bigger atoms have bigger cores, so their nuclei can't get as close. Going down the halogens, H–F, H–Cl, H–Br and H–I get longer and weaker.
The H–halogen bonds show the size effect clearly: H–F is about 92 pm and 567 kJ/mol, while H–I is about 161 pm and 299 kJ/mol. The longer the bond, the weaker it is, because the shared electrons are farther from the nuclei that attract them.
| Bond | Approximate length (pm) | Approximate bond energy (kJ/mol) |
|---|---|---|
| C–C | 154 | 348 |
| C=C | 134 | 614 |
| C≡C | 120 | 839 |
Coulomb's law and ions
The attraction between a cation and an anion follows Coulomb's law: the force is proportional to q₁q₂ / r². Larger charges give stronger attraction. Smaller ions let the centers get closer (smaller r), which also gives stronger attraction.
Charge usually matters more than size. MgO, made of 2+ and 2− ions, melts at about 2800 °C, while NaCl, made of 1+ and 1− ions, melts at 801 °C. MgO's ions are also a little smaller, but doubling both charges is the bigger effect. When the charges match, compare sizes: NaF melts higher than NaCl because F⁻ is smaller than Cl⁻.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Reading a potential energy curve
On a potential energy curve for two atoms, the energy is −432 kJ/mol at 127 pm, its lowest point. What are the bond length and bond energy? What happens to the energy if the atoms are pushed to 80 pm?
Show the solutionHide the solution
- Step 1: The bond length is the distance at the minimum: 127 pm.
- Step 2: The bond energy is the depth of the minimum below zero: 432 kJ/mol must be added to separate the atoms.
- Step 3: At 80 pm the atoms are closer than the bond length, so nucleus-nucleus repulsion dominates and the potential energy rises steeply, above the minimum.
Answer: Bond length 127 pm, bond energy 432 kJ/mol; at 80 pm the energy is much higher because the nuclei repel.
- Example 2
Ranking ionic attractions
Rank the strength of the attraction between ions in KCl, NaCl, NaF and MgO from weakest to strongest. Justify using Coulomb's law.
Show the solutionHide the solution
- Step 1: Charges first: MgO has 2+ and 2− ions; the others have 1+ and 1−. Larger charges give the strongest attraction, so MgO is strongest.
- Step 2: Among the 1+/1− compounds, compare distance between ion centers. K⁺ is larger than Na⁺, so KCl has the largest separation and weakest attraction.
- Step 3: F⁻ is smaller than Cl⁻, so NaF has a smaller separation and stronger attraction than NaCl.
- Step 4: The melting points agree: KCl 770 °C, NaCl 801 °C, NaF 993 °C, MgO about 2800 °C.
Answer: KCl < NaCl < NaF < MgO
- Example 3
Bond order and atom size together (classic trap)
Which bond is longer: C=O or C–Cl? A student says C=O, because oxygen has more bonds. Is that right?
Show the solutionHide the solution
- Step 1: Two factors set bond length: bond order and atom size.
- Step 2: C=O is a double bond, which pulls the atoms closer. Oxygen is also a small atom.
- Step 3: C–Cl is a single bond, and chlorine is in period 3, so it's much larger.
- Step 4: Both factors point the same way: C–Cl is longer. Typical values are about 123 pm for C=O and about 177 pm for C–Cl.
Answer: C–Cl is longer; the student's reasoning is backwards, since a higher bond order shortens a bond.
Common mistakes
- Reading bond energy as the energy at zero distance or at the far right of the graph. It's the depth of the minimum below the energy of separated atoms.
- Thinking bond formation requires energy. Forming a bond releases energy; breaking one requires it.
- Thinking a double bond is exactly twice as strong as a single bond. C=C (614 kJ/mol) is less than twice C–C (348 kJ/mol).
- In Coulomb's law comparisons, forgetting that charge usually outweighs size.
On the exam
- You may be asked to sketch or label a potential energy curve, or to compare curves for two molecules. Mark the bond length on the distance axis and the bond energy as the depth of the well.
- For ionic comparisons, a full justification states both ions' charges or sizes and links them to the strength of attraction through Coulomb's law.
Connected topics
Videos
Check yourself
4 questions on 2.2 Intramolecular Force and Potential Energy. Pick an answer to see if you got it, and why.
| Curve | Internuclear distance at the energy minimum (pm) | Potential energy at the minimum (kJ/mol) |
|---|---|---|
| 1 | 142 | −159 |
| 2 | 110 | −945 |
| 3 | 121 | −498 |
A student plotted potential energy against internuclear distance for N₂, O₂ and F₂, in some order. The energy is zero when the atoms are far apart. The table summarizes the lowest point of each curve.
Which curve represents N₂, and why?
How much energy is needed to break the bonds in 1 mole of the molecule represented by curve 3, separating it completely into atoms?
On each curve, the potential energy rises steeply as the internuclear distance becomes smaller than the distance at the minimum. Which of the following best explains this?
In which of the following ionic solids is the Coulombic attraction between the cations and anions the strongest?
0 of 4 answered