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

Translational Dynamics

Conservation of Energy and Momentum

Magnetism and Electromagnetic Induction

Linear Momentum

Thermodynamics

Electric Charge Field and Potential

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).