Chapter 10: Q40E (page 469)

Starting with equation (10-4), show that if $Δn$ is $- 1$ as a photon is emitted by a diatomic molecule in a transition among rotation-vibration states, but $Δℓ$ can be $\pm 1$ . Then the allowed photon energies obey equation (10-6).

Short Answer

Expert verified

The photon’s energy is equal to $E_{photon} = h \sqrt{\frac{κ}{μ}} \pm 1 \frac{h^{2}}{{μa}^{2}}$ .

Step by step solution

Calculate the energy change for a molecule

Let us calculate the energy change for a molecule changing from state n,l to state n-1, l+1, this is the photons energy.

$\begin{matrix} E_{nl} - E_{n - l, l + 1} = (n + \frac{1}{2}) h \sqrt{\frac{κ}{μ}} + \frac{h^{2} \times l \times (l + 1)}{2 {μa}^{2}} - ((n - 1 + \frac{1}{2}) h \sqrt{\frac{κ}{μ}} + \frac{h^{2} \times (l + 1) \times (l + 2)}{2 {μa}^{2}}) \\ = h \sqrt{\frac{κ}{μ}} + \frac{h^{2} \times (l - (1 + 2)) \times (l + 1)}{2 {μa}^{2}} \\ = h \sqrt{\frac{κ}{μ}} - \frac{h^{2} \times (l + 1)}{{μa}^{2}} \end{matrix}$

Then let us calculate the energy change for a molecule changing from state n,l to state n-1,l-1. This is the photon’s energy.

$\begin{matrix} E_{nl} - E_{n - l, l - 1} = (n + \frac{1}{2}) h \sqrt{\frac{κ}{μ}} + \frac{h^{2} \times l \times (l + 1)}{2 {μa}^{2}} - ((n - 1 + \frac{1}{2}) h \sqrt{\frac{κ}{μ}} + \frac{h^{2} \times (l - 1) \times (l)}{2 {μa}^{2}}) \\ = h \sqrt{\frac{κ}{μ}} + \frac{h^{2} \times (l + 1 - (l - 1)) \times (1)}{2 {μa}^{2}} \\ = h \sqrt{\frac{κ}{μ}} - \frac{h^{2} \times l}{{μa}^{2}} \end{matrix}$

Conclusion.

Thus the photon’s energy is equal to

$\begin{matrix} E_{photon} = h \sqrt{\frac{κ}{μ}} \pm 1 \frac{h^{2}}{{μa}^{2}} \\ I = 1, 2, 3, .......... \end{matrix}$

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Starting with equation (10-4), show that if $Δn$ is $- 1$ as a photon is emitted by a diatomic molecule in a transition among rotation-vibration states, but $Δℓ$ can be $\pm 1$ . Then the allowed photon energies obey equation (10-6).

Short Answer

Step by step solution

Calculate the energy change for a molecule

Conclusion.

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Starting with equation (10-4), show that if Δn is-1 as a photon is emitted by a diatomic molecule in a transition among rotation-vibration states, but Δℓcan be±1 . Then the allowed photon energies obey equation (10-6).

Short Answer

Step by step solution

Calculate the energy change for a molecule

Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

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Study anywhere. Anytime. Across all devices.

Starting with equation (10-4), show that if $Δn$ is $- 1$ as a photon is emitted by a diatomic molecule in a transition among rotation-vibration states, but $Δℓ$ can be $\pm 1$ . Then the allowed photon energies obey equation (10-6).