Ammonia, ( )

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Atomic, Molecular & Optical Physics · Linear algebra and vector spaces

Invariance, Ch Point groups and crystallographic symmetry

Problem

Ammonia, NH3\mathrm{NH_3} (C3vC_{3v}), has measured dipole moment 1.47 D1.47\,\mathrm{D} and H-N-H bond angle α=106.7∘\alpha=106.7^\circ. Model the molecular dipole as the vector sum of three equal N-H bond dipoles of magnitude μb\mu_b, each lying along a bond. Take the C3C_3 axis as zz and let each bond make the angle β\beta with it, so that the C3C_3 symmetry puts the unit vectors along the bonds at (sin⁡βcos⁡ϕk, sin⁡βsin⁡ϕk, cos⁡β)(\sin\beta\cos\phi_k,\ \sin\beta\sin\phi_k,\ \cos\beta) with ϕk=0∘,120∘,240∘\phi_k=0^\circ,120^\circ,240^\circ. ((a)) From the dot product of two of these unit vectors, show that cos⁡α=32cos⁡2β−12\cos\alpha=\tfrac32\cos^2\beta-\tfrac12, and show that the sum of the three bond dipoles lies along zz with magnitude 3μbcos⁡β3\mu_b\cos\beta. Find β\beta. ((b)) Estimate the N-H bond dipole μb\mu_b from the measured molecular dipole. ((c)) Estimate what the molecular dipole would be if the bond angle were instead the ideal tetrahedral angle 109.5∘109.5^\circ, keeping μb\mu_b fixed at the value found in (b).

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