Having problems figuring out problem (b). The potential energy of a diatomic mol

Question

Having problems figuring out problem (b).

The potential energy of a diatomic molecule may be modeled by the Lennard-Jones potential 4 r12 r6 where A and B are constants and r is the distance between the ions. For a particular molecule, A = 0.195 × 10-120 eV . m12 and B 1.50 x 10-60 eV . m (a) Determine the amount of energy (in eV) needed to break up the diatomic molecule when the separation of the ions is such that it has a minimum amount of potential energy 2.88 ev (b) Determine the force (in nN) needed to break up the diatomic molecule. 4.33e+11 For a conservative force, how is the force related to the potential energy? If you have an expression for the attractive force the ions exert on each other as a function of their separation, how can you determine the separation where this force is a maximum? How does the force required to separate the molecule into two ions compare to this maximum force of attraction?. nN

Explanation / Answer

a) when U in minimum,

dU/dr = 0

-12*A*r^(-12-1) - (-6)*B*r^(-6-1) = 0

-12*A/r^13 + 6*B/r^7 = 0

12*A/r^13 = 6*B/r^7

2*A/r^6 = B

==> r^6 = 2*A/B

r = (2*A/B)^(1/6)

= (2*0.195*10^-120/(1.5*10^-60))^(1/6)

= 7.989*10^-11 m

U_min = A/r^12 - B/r^6

= 0.195*10^-120/(7.898*10^-11)^12 - 1.5*10^-60/(7.898*10^-11)^6

= -2.87 eV

a) Energy required to to breakup = Uf - Ui

= 0 - (-2.87)

= 2.87 J

b) F = -dU/dr

= -(-12*A*r^(-12-1) - (-6)*B*r^(-6-1))  

= -(-12*A/r^13 + 6*B/r^7)

= 12*A/r^13 - 6*B/r^7

= 12*0.195*10^-120*1.6*10^-19/(7.898*10^-11)^13 - 6*1.5*10^-60*1.6*10^-19/(7.898*10^-11)^7

= 5.34*10^-9 N

= 5.34 nN <<<<<<<<<---------------Answer

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