**1. The problem statement, all variables and given/known data**

A molecular bond can be modeled as a spring between two atoms that vibrate with simple harmonic motion. Figure P14.63 shows an SHM approximation for the potential energy of an HCl molecule. For E < 4 * 10^-19 J it is a good approximation to the more accurate HCl potential-energy curve that was shown in Figure 10.31. Because the chlorine atom is so much more massive than the hydrogen atom, it is reasonable to assume that the hydrogen atom (m = 1.67 * 10^-27 kg) vibrates back and forth while the chlorine atom remains at rest. Use the graph to estimate the vibrational frequency of the HCl molecule.

**2. Relevant equations**

= (1/2PI) * sqrt(k/m)

Umax = (1/2)*k*A^2

**3. The attempt at a solution**

I figured from the graph that the Amplitude (max. displacement) equals 0.17nm – 0.13 nm = 0.04 nm, which equals 4 x 10^-11 meters.

It also appears from the graph that the max. potential energy is 4 x 10^-19 J. So:

Umax = (1/2)*k*A^2

4×10^-19 J = (1/2)*k*A^2

k = (4×10^-19 J) * 2 / A^2 = (4×10^-19 J) * 2 / (4×10^-11 m)^2 = 500 N/m

So, frequency = (1/2∏) * sqrt(k/m) = (1/2∏) * sqrt(500 N/m / 1.67×10^-27 kg) = 8.71×10^13 Hz

But the answer key says 7.9×10^13 Hz. I’m not sure where I went wrong. I would much appreciate any help.

**1. The problem statement, all variables and given/known data**

**2. Relevant equations**

**3. The attempt at a solution**

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