7. Two carbon atoms have a non-covalent interaction whose potential energy varies
with distance as 𝐸 𝑟 = 32000000/𝑟!” − 729/𝑟!, where r is measured in
Angstroms. The equilibrium inter-atomic distance will be
a. 𝑟 = 0
b. 𝑟 = 0.667
c. 𝑟 = 6.67
d. 𝑟 = 66.7
Equilibrium Inter-atomic Distance: 6.67 Ångstroms (Option c)
The equilibrium inter-atomic distance for the given non-covalent interaction potential is 6.67 Ångstroms. This corresponds to option c.
Potential Energy Function
The potential energy follows the Lennard-Jones form: E(r) = 32000000/r12 – 729/r6, where r is in Ångstroms. The repulsive r-12 term dominates at short distances, while the attractive r-6 term (van der Waals dispersion) prevails at longer ranges.
Equilibrium Condition
Equilibrium occurs where dE/dr = 0. Differentiating gives:
dE/dr = -12 × 32000000/r13 + 6 × 729/r7 = 0
Simplifying:
384000000/r13 = 4374/r7 ⟹ 384000000 = 4374 r6 ⟹ r6 = 384000000/4374 ≈ 8778.7 ⟹ r ≈ 6.67
Option Analysis
- a. r = 0: Physically impossible; potential diverges to +∞ as r → 0 due to repulsion. Derivative undefined.
- b. r = 0.667: Too short; repulsive term dominates (E > 0, steep positive slope). Actual minimum at larger distance.
- c. r = 6.67: Correct; matches solved r = 6.67 where forces balance.
- d. r = 66.7: Too distant; attractive term weak, no minimum (approaches zero asymptotically).
Understanding Lennard-Jones Potential
Non-covalent interactions between carbon atoms model van der Waals forces using the Lennard-Jones potential E(r) = 32000000/r12 – 729/r6, common in CSIR NET Life Sciences for molecular simulations. Equilibrium inter-atomic distance occurs at energy minimum, found by setting dE/dr = 0.
Step-by-Step Calculation
Solve dE/dr = -12 × 32000000/r13 + 6 × 729/r7 = 0, yielding r6 = 8778.7 so equilibrium inter-atomic distance r = 6.67 Å. This matches typical C-C van der Waals radii sums (~3.4-4 Å doubled).
CSIR NET Exam Options Explained
Option c (r = 6.67) is correct; a (r = 0) impossible due to repulsion; b (0.667) too close; d (66.7) too far from minimum. Practice such derivatives for molecular biology, biochemistry sections.
