The potential energy of two atoms in a molecule can sometimes be approximated by the Morse function, \[ U(r)=A\left[\left(\mathrm{e}^{(R-r) / S}-1\right)^{2}-1\right] \] where \( r \) is the distance between the two atoms and \( A, R \), and \( S \) are positive constants with \( S \ll R \) (a) Find the equilibrium separation \( r_{o} \), at which \( U(r) \) is a minimum. (b) Sketch this function for \( 0<r<\infty \). (c) Now write \( r=r_{o}+x \) so that \( x \) is the displacement from equilibrium, and show that, for small displacements, \( U \) has the approximate form \( U= \) constant \( +\frac{1}{2} k x^{2} \) (d) What is the force constant \( k \) ?
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One possible form for the potential energy $(U)$ of a diatomic molecule (Fig. 8$)$ is called the Morse Potential: $\quad U=U_{0}\left[1-e^{-a\left(r-r_{0}\right)}\right]^{2}$ (a) Show that $r_{0}$ represents the equilibrium distance and $U_{0}$ the dissociation energy. $(b)$ Graph $U$ from $r=0$ to $r=4 r_{0}$ assuming $a=18 \mathrm{nm}^{-1}, U_{0}=4.6 \mathrm{eV}, \quad$ and $r_{0}=0.13 \mathrm{nm}$
The lower-energy states in a covalently bound diatomic molecule can be found approximately from the so-called Morse potential $U(r)=U_{0}\left(e^{2\left(r-r_{0}\right) / a}-e^{-2\left(r-r_{0}\right) / a}\right),$ where $r$ is the atomic separation and $U_{0}, r_{0},$ and $a$ are constants determined from experimental data. Calculate $d U / d r$ and $d^{2} U / d r^{2}$ to show that $U$ has a minimum, and find expressions for (a) $U_{\min }$ and (b) the separation $r_{\min }$ at the minimum energy.
In some diatomic molecules, the force each atom exerts on the other can be approximated by $F=-C / r^{2}+D / r^{3}$ , where $r$ is the atomic separation and $C$ and $D$ are positive constants. $(a)$ Graph $F$ vs. $r$ from $r=0.8 D / C$ to $r=4 D / C$ . (b) Show that equilibrium occurs at $r=r_{0}=D / C$ . (c) Let $\Delta r=r-r_{0}$ be a small displacement from equilibrium, where $\Delta r \ll r_{0} .$ Show that for such small displacements, the motion is approximately simple harmonic, and (d) deter- mine the force constant. (e) What is the period of such motion? [Hint: Assume one atom is kept at rest.]
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