Question

A molecule has a triply degenerate excited state lying at 300 cm-1 above the non-degenerate ground state. At what temperature will 20 percent of the molecule be in the upper (excited) level?

Answer 1

Answer on Question #75546, Physics / Molecular Physics | Thermodynamics

A molecule has a triply degenerate excited state lying at $300~\mathrm{cm}^{-1}$ above the non-degenerate ground state. At what temperature will 20 percent of the molecule be in the upper (excited) level?

Solution:

The fraction of populations of the two states of a two-level system is

$$\frac {p _ {1}}{p _ {2}} = 3 e ^ {- \frac {\Delta E}{k T}} = \frac {0 . 2 0}{1 - 0 . 2 0}$$ $$\Delta E = h c \tilde {v} = - k T \ln \frac {p _ {1}}{3 p _ {2}}$$

So,

$$T = \frac {- h c \tilde {v}}{k \ln \frac {p _ {1}}{3 p _ {2}}} = \frac {- (6 . 6 2 6 \times 1 0 ^ {- 3 4}) (2 . 9 9 8 \times 1 0 ^ {1 0}) (3 0 0)}{(1 . 3 8 1 \times 1 0 ^ {- 2 3}) \times \ln \left(\frac {0 . 2 0}{3 \times (1 - 0 . 2 0)}\right)} = 1 7 3. 7 K$$

Answer: 174 K

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