Question:
If levels 1 and 2 are separated by an energy \(E_2 - E_1\), such that the corresponding transition frequency falls in the middle of the visible range, calculate the ratio of the populations of two levels in thermal equilibrium at room temperature.
If levels 1 and 2 are separated by an energy \(E_2 - E_1\), such that the corresponding transition frequency falls in the middle of the visible range, calculate the ratio of the populations of two levels in thermal equilibrium at room temperature.
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Use \(\lambda = 5500 \text{ \AA}\) for middle of visible range.
Updated On: Apr 20, 2026
- \(1.1577 \times 10^{-38}\)
- \(2.9 \times 10^{-35}\)
- \(2.168 \times 10^{-36}\)
- \(1.96 \times 10^{-20}\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
Use Boltzmann distribution: \[ \frac{N_2}{N_1} = e^{-(E_2 - E_1)/kT} \]
Step 2: Detailed Explanation:
Take wavelength at middle of visible range: \(\lambda \approx 550 \text{ nm}\). Energy difference: \[ E_2 - E_1 = \frac{hc}{\lambda} = \frac{6.63\times10^{-34} \times 3\times10^8}{550\times10^{-9}} = 3.616\times10^{-19} \text{ J} \] Thermal energy: \[ kT = 1.38\times10^{-23} \times 300 = 4.14\times10^{-21} \text{ J} \] Population ratio: \[ \frac{N_2}{N_1} = \exp\left(-\frac{3.616\times10^{-19}}{4.14\times10^{-21}}\right) = e^{-87.34} = 1.1577\times10^{-38} \]
Step 3: Final Answer:
\[ \boxed{1.1577 \times 10^{-38}} \]
Use Boltzmann distribution: \[ \frac{N_2}{N_1} = e^{-(E_2 - E_1)/kT} \]
Step 2: Detailed Explanation:
Take wavelength at middle of visible range: \(\lambda \approx 550 \text{ nm}\). Energy difference: \[ E_2 - E_1 = \frac{hc}{\lambda} = \frac{6.63\times10^{-34} \times 3\times10^8}{550\times10^{-9}} = 3.616\times10^{-19} \text{ J} \] Thermal energy: \[ kT = 1.38\times10^{-23} \times 300 = 4.14\times10^{-21} \text{ J} \] Population ratio: \[ \frac{N_2}{N_1} = \exp\left(-\frac{3.616\times10^{-19}}{4.14\times10^{-21}}\right) = e^{-87.34} = 1.1577\times10^{-38} \]
Step 3: Final Answer:
\[ \boxed{1.1577 \times 10^{-38}} \]
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