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1. A model for the potential energy interaction between the two nitrogen atoms in the N2 molecule is proposed that has the form: 12 V(+) = 4E. [(÷)“ – (-)"] a. Find the position of the potential minimum and its value there, in terms of o and Eo, respectively. What do these parameters represent physically about the molecule? b. Draw a hand sketch of V(r) showing rmin, V(rmin) and where V crosses the r axis. c. If the atom is disturbed from its equilibrium by a small amount, show that the 7.56 frequency of oscillation is w = where m is the mass of a nitrogen atom.

1. A model for the potential energy interaction between the two nitrogen atoms in the N2 molecule is proposed that has the form: 12 V(+) = 4E. [(÷)“ – (-)"] a. Find the position of the potential minimum and its value there, in terms of o and Eo, respectively. What do these parameters represent physically about the molecule? b. Draw a hand sketch of V(r) showing rmin, V(rmin) and where V crosses the r axis. c. If the atom is disturbed from its equilibrium by a small amount, show that the 7.56 frequency of oscillation is w = where m is the mass of a nitrogen atom.

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1. A model for the potential energy interaction between the two nitrogen atoms in the N₂ molecule is proposed that has the form: \[ V(r) = 4E_0 \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^6 \right] \] a. Find the position of the potential minimum and its value there, in terms of σ and E₀, respectively. What do these parameters represent physically about the molecule? b. Draw a hand sketch of \( V(r) \) showing \( r_{\text{min}} \), \( V(r_{\text{min}}) \), and where \( V \) crosses the r axis. c. If the atom is disturbed from its equilibrium by a small amount, show that the frequency of oscillation is \( \omega = \frac{7.56}{\sigma} \sqrt{\frac{E_0}{m}} \) where m is the mass of a nitrogen atom. d. For molecular nitrogen, the bond length is 1.1 x 10⁻¹⁰m, the bond (binding) energy is 9.79 eV (15.66 x 10⁻¹⁹ J), the mass is 14 amu = 23.38 x 10⁻²⁷ kg. In the spectroscopy laboratory, this vibration is measured to be 8.8 x 10¹³ Hz. Is this a good model for the interatomic potential? (Recall \( \omega = 2 \pi \) times frequency in Hz.) **Graph Explanation:** The potential energy function \( V(r) \) is typically sketched as a curve with a minimum point, representing the stable equilibrium position \( r_{\text{min}} \). The curve usually shows a sharp drop to a deep minimum and gradually rises again, indicating the potential energy's dependency on the distance \( r \) between two atoms. The crossing of the curve with the x-axis symbolizes the reference point for zero potential energy.
Transcribed Image Text:1. A model for the potential energy interaction between the two nitrogen atoms in the N₂ molecule is proposed that has the form: \[ V(r) = 4E_0 \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^6 \right] \] a. Find the position of the potential minimum and its value there, in terms of σ and E₀, respectively. What do these parameters represent physically about the molecule? b. Draw a hand sketch of \( V(r) \) showing \( r_{\text{min}} \), \( V(r_{\text{min}}) \), and where \( V \) crosses the r axis. c. If the atom is disturbed from its equilibrium by a small amount, show that the frequency of oscillation is \( \omega = \frac{7.56}{\sigma} \sqrt{\frac{E_0}{m}} \) where m is the mass of a nitrogen atom. d. For molecular nitrogen, the bond length is 1.1 x 10⁻¹⁰m, the bond (binding) energy is 9.79 eV (15.66 x 10⁻¹⁹ J), the mass is 14 amu = 23.38 x 10⁻²⁷ kg. In the spectroscopy laboratory, this vibration is measured to be 8.8 x 10¹³ Hz. Is this a good model for the interatomic potential? (Recall \( \omega = 2 \pi \) times frequency in Hz.) **Graph Explanation:** The potential energy function \( V(r) \) is typically sketched as a curve with a minimum point, representing the stable equilibrium position \( r_{\text{min}} \). The curve usually shows a sharp drop to a deep minimum and gradually rises again, indicating the potential energy's dependency on the distance \( r \) between two atoms. The crossing of the curve with the x-axis symbolizes the reference point for zero potential energy.
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