Michael S. answered • 13d
B.S. in Chemistry, Indiana University; Organic Chem Teaching Intern
E = +3.2 × 10−18 J
Coulomb's law for potential energy
E = k q1q2 / r, with k = 8.988 × 109 N·m2/C2
Note this is the energy form, with r to the first power. The force version has r2 in the denominator. Mixing these up is the most frequent error on this problem, and here it would give you 4.4 × 10−8 N — a number in the wrong units entirely, which is your clue.
Step 1 — convert
72 pm = 72 × 10−12 m = 7.2 × 10−11 m
Each proton: q = +1.602 × 10−19 C
Step 2 — substitute
q1q2 = (1.602 × 10−19)2 = 2.566 × 10−38 C2
k × q1q2 = (8.988 × 109)(2.566 × 10−38) = 2.307 × 10−28 N·m2
E = (2.307 × 10−28) / (7.2 × 10−11) = 3.2 × 10−18 J
The sign is part of the answer
Both charges are positive, so their product is positive and E comes out positive. That is not a formality — it means the system is higher in energy than the two protons infinitely far apart, which is the mathematical statement that they repel. Pushing them closer costs energy; releasing them lets them fly apart.
Contrast a proton and an electron at the same separation: q1q2 is negative, E = −3.2 × 10−18 J, and the negative sign says the pair is bound. Same magnitude, opposite physical meaning. Never drop the sign on these.
Is the number reasonable?
3.2 × 10−18 J is about 20 eV, or roughly 1900 kJ/mol if you scale it up by Avogadro's number. That is enormous — several times a typical covalent bond. Which is exactly the point of problems like this: the electrostatic repulsion between two protons at atomic distances is ferocious, and it gets worse as r shrinks. Squeeze them into a nucleus (~10−15 m, another 104 times closer) and this energy climbs by the same factor. Nothing chemical can hold that together, which is why the strong nuclear force has to exist.
Unit check: (N·m2/C2)(C2)/(m) = N·m = J. If your units do not land on joules, you used the force equation.
