Answer:
T = 525K
Explanation:
The temperature of the two-level system can be calculated using the equation of Boltzmann distribution:
[tex] \frac{N_{i}}{N} = e^{-\Delta E/kT} [/tex] (1)
where Ni: is the number of particles in the state i, N: is the total number of particles, ΔE: is the energy separation between the two levels, k: is the Boltzmann constant, and T: is the temperature of the system
The energy between the two levels (ΔE) is:
[tex] \Delta E = hck [/tex]
where h: is the Planck constant, c: is the speed of light and k: is the wavenumber
[tex] \Delta E = 6.63\cdot 10^{-34} J.s \cdot 3\cdot 10^{8}m/s \cdot 4 \cdot 10^{4}m^{-1} = 7.96 \cdot 10^{-21}J [/tex]
Solving the equation (1) for T:
[tex] T = \frac{-\Delta E}{k \cdot Ln(N_{i}/N)} [/tex]
With Ni = N/3 and k = 1.38x10⁻²³ J/K, the temperature of the two-level system is:
[tex] T = \frac{-7.96 \cdot 10^{-21}J}{1.38 \cdot 10^{-23} J/K \cdot Ln(N/3N)} = 525K [/tex]
I hope it helps you!
