Consider two adjacent water molecules in an ice crystal. Assume that the binding energy connecting them follows a Lennard-Jones potential with a potential well depth of 9.6x10-21 J. How much energy must you put into the system to increase the separation of the two molecules by 10% from the equilibrium spacing?
A)0.2x10-21 J
B)0.4x10-21 J
C)0.9x10-21 J
D)1.8x10-21 J
E)9.6x10-21 J

Question

Quizplus · Accepted Answer

The Lennard-Jones potential is given by:$$U(r) = 4\epsilon \left[\left(\frac{\sigma}{r}ight)^{12} - \left(\frac{\sigma}{r}ight)^6ight]$$where $\epsilon$ is the potential well depth and $\sigma$ is the equilibrium spacing.The force between the two molecules is given by the negative gradient of the potential:$$F(r) = -\frac{dU}{dr} = 4\epsilon \left[\frac{12}{r^{13}} - \frac{6}{r^7}ight]$$The work done to increase the separation of the two molecules by 10% from the equilibrium spacing is given by:$$W = \int_{0.9\sigma}^{\sigma} F(r) dr = 4\epsilon \left[\frac{12}{r^{13}} - \frac{6}{r^7}ight]_{0.9\sigma}^{\sigma}$$$$W = 4\epsilon \left[\frac{12}{0.9^{13}\sigma^{13}} - \frac{6}{0.9^7\sigma^7} - \frac{12}{\sigma^{13}} + \frac{6}{\sigma^7}ight]$$$$W = 4\epsilon \left[\frac{12}{0.9^{13}} - \frac{6}{0.9^7} - 12 + 6ight]$$$$W = 4\epsilon \left[\frac{12}{0.729} - \frac{6}{0.478} - 6ight]$$$$W = 4\epsilon \left[16.47 - 12.55 - 6ight]$$$$W = 4\epsilon (2.92)$$$$W = 1.8	imes10^{-21} J$$

Consider Two Adjacent Water Molecules in an Ice Crystal