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Answer :
The relative population of the [tex]\(v = 3\)[/tex] vibrational state is [tex]\( e^{-\frac{E}{kT}} \)[/tex], where [tex]\( E \)[/tex] is the energy difference between the [tex]\(v = 3\)[/tex] and [tex]\(v = 0\)[/tex] states, [tex]\( k \)[/tex]is Boltzmann's constant, and [tex]\( T \)[/tex] is the temperature.
The relative population of the [tex]\(v = 3\)[/tex] vibrational state can be determined using the Boltzmann distribution formula. This formula states that the relative population is proportional to [tex]\(e^{-\frac{E}{kT}}\)[/tex].
where [tex]\(E\)[/tex] is the energy difference between the [tex]\(v = 3\)[/tex] and [tex]\(v = 0\)[/tex] states, [tex]\(k\)[/tex] is Boltzmann's constant, and [tex]\(T\)[/tex] is the temperature. Calculating this expression yields the relative population of the [tex]\(v = 3\)[/tex]state.
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