An approximation for the potential energy of a KCl molecules U = A[(RJ/8r8 ) - l/r], where Ro = 2.67 X l0-l0 m and A = 2.31 X 10-28 J. m. Using this approximation:
(a) Show that the radial component of the force on each atom is F, = A[(RJ/r9 ) - l/r2].
(b) Show that Ro is the equilibrium separation.
(c) Find the minimum potential energy.
(d) Use r = Ro + x and the first two terms of the binomial theorem (Eq. 13.28) to show that F, = -(7A/RJ)x, so that the molecule's force constant is k = 7A/RJ.
(e) With both the K and atoms vibrating in opposite directions on opposite sides of the molecule's center of mass. m,m2/(m, + m2) = 3.06 X 10-26 kg is the mass to use in calculating the frequency (see Problem 13.86). Calculate the frequency of small-amplitude vibrations.