When a hypothetical diamotic molecule having atoms 0.8890 nm apart undergoes a rotational transition from the l=2 state to the next lower state, it gives up a photon having energy 8.850 * 10^-4 eV. When the molecule undergoes a vibrational transition from one energy state to the next lower energy state, it gives up 0.2540 eV. Find the force constant of this molecule.
When a hypothetical diamotic molecule having atoms 0.8890 nm apart undergoes a rotational transition from the l=2 state to the next lower state, it gives up a photon having energy 8.850 * 10^-4 eV. When the molecule undergoes a vibrational transition from one energy state to the next lower energy state, it gives up 0.2540 eV. Find the force constant of this molecule.
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![### Educational Text on Rotational and Vibrational Transitions in Diatomic Molecules
**Problem Statement:**
When a hypothetical diatomic molecule having atoms 0.8890 nm apart undergoes a rotational transition from the \( l = 2 \) state to the next lower state, it gives up a photon having energy \( 8.850 \times 10^{-4} \) eV. When the molecule undergoes a vibrational transition from one energy state to the next lower energy state, it gives up \( 0.2540 \) eV. Find the force constant of this molecule.
**Analysis:**
1. **Rotational Transition:**
- Given:
- Distance between atoms \( r = 0.8890 \) nm = \( 0.8890 \times 10^{-9} \) m
- Transition from \( l = 2 \) to \( l = 1 \)
- Energy of photon emitted \( \Delta E_{\text{rot}} = 8.850 \times 10^{-4} \) eV
Rotational energy levels for a diatomic molecule are given by:
\[
E_l = \frac{l(l + 1)h^2}{8\pi^2I}
\]
where \( I \) (moment of inertia) is \( \mu r^2 \) and \( \mu \) is the reduced mass of the system.
2. **Vibrational Transition:**
- Given:
- Energy difference \( \Delta E_{\text{vib}} = 0.2540 \) eV
Vibrational energy levels for a diatomic molecule are given by:
\[
E_v = \left( v + \frac{1}{2} \right) h \nu
\]
where \( \nu \) is the vibrational frequency and \( v \) is the vibrational quantum number.
**Objective:**
To find the force constant \( k \) of this molecule, which is related to the vibrational frequency \( \nu \) by:
\[
\nu = \frac{1}{2\pi} \sqrt{\frac{k}{\mu}}
\]
By solving the given problem, we can establish the relationship and find the desired force constant \( k \).
**Steps:**
1. Calculate the moment of inertia \( I \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6fcd7fab-9903-4b0b-a30c-af5d8b8fc75f%2F623f2275-1b74-4962-8a0c-9fcfe8fab786%2Fg327b0x_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Text on Rotational and Vibrational Transitions in Diatomic Molecules
**Problem Statement:**
When a hypothetical diatomic molecule having atoms 0.8890 nm apart undergoes a rotational transition from the \( l = 2 \) state to the next lower state, it gives up a photon having energy \( 8.850 \times 10^{-4} \) eV. When the molecule undergoes a vibrational transition from one energy state to the next lower energy state, it gives up \( 0.2540 \) eV. Find the force constant of this molecule.
**Analysis:**
1. **Rotational Transition:**
- Given:
- Distance between atoms \( r = 0.8890 \) nm = \( 0.8890 \times 10^{-9} \) m
- Transition from \( l = 2 \) to \( l = 1 \)
- Energy of photon emitted \( \Delta E_{\text{rot}} = 8.850 \times 10^{-4} \) eV
Rotational energy levels for a diatomic molecule are given by:
\[
E_l = \frac{l(l + 1)h^2}{8\pi^2I}
\]
where \( I \) (moment of inertia) is \( \mu r^2 \) and \( \mu \) is the reduced mass of the system.
2. **Vibrational Transition:**
- Given:
- Energy difference \( \Delta E_{\text{vib}} = 0.2540 \) eV
Vibrational energy levels for a diatomic molecule are given by:
\[
E_v = \left( v + \frac{1}{2} \right) h \nu
\]
where \( \nu \) is the vibrational frequency and \( v \) is the vibrational quantum number.
**Objective:**
To find the force constant \( k \) of this molecule, which is related to the vibrational frequency \( \nu \) by:
\[
\nu = \frac{1}{2\pi} \sqrt{\frac{k}{\mu}}
\]
By solving the given problem, we can establish the relationship and find the desired force constant \( k \).
**Steps:**
1. Calculate the moment of inertia \( I \
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