Question

5. (10 points) A simple function that looks like the potential well of a diatomic molecule is the Morse potential given by: U(x) = D. (1-e-Bx) (1) where, x is the displacement of the bond from its equilibrium position, and D. is the value of U(x) at large separations. D. is called the classical dissociation energy and is characterized by the depth of the potential well. We can expand U(x) in a Taylor series about x = 0 to obtain U(x) = D.BPx2 - D.B3x3 + ... + (2) a) Given that D. = 7.31 x 105 molecule and B = 1.82 x 101m' for HCI, calculate the force constant of HCl. (Hint: To do this, you will have to equate the harmonic oscillator potential energy to one term in the series expanded form of U(x) shown by equation 2.) b) Using Microsoft Excel, or some other graphing software, plot the Morse potential (equation I) for HCl and plot the corresponding harmonic oscillator potential on the same graph. (In order to clearly view the potential wells, modify the x-and y-axes on your graph so that the x axis range is -2 x 10-" to 4.0 x 10- m and the y-axis range is 0 - 2.5 x 10-18 J.)

Answer 1

(a) If k is the harmonic force constant (harmonic term), then

$\frac{1}{2}kx^{2}=D_{e}\beta ^{2}x^{2},\ or\ k=2D_{e}\beta ^{2}$

$\therefore k=2\times 7.31\times 10^{-19}\times \left ( 1.82\times 10^{10} \right )^{2}=484.27\ N/m$

(b) The graph looks like

* 10-16 Potential (J/molecule) 2 3 1 1 Displacement (x) *10-10