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The vibrational motion of a diatomic molecule may be considered to be that of a simple harmonic oscillator with angular frequency ω. If a gas of these molecules is at a temperature T, what is the probability that a randomly picked molecule will be found in its lowest vibrational state?
  • a)
  • b)
  • c)
  • d)
Correct answer is option 'B'. Can you explain this answer?
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The vibrational motion of a diatomic molecule may be considered to be ...
The probability P of finding a diatomic molecule in its lowest vibrational state is given by the Boltzmann distribution:

where E0 is the energy of the lowest vibrational state, K is the Boltzmann constant, and T is the temperature of the gas.
For a simple harmonic oscillator, the energy of the nth vibrational state is given by:

where is the Planck constant.
Therefore, the energy of the lowest vibrational state (n = 0) is:

Substituting this into the Boltzmann distribution, we get:

Simplifying this expression, we have:

So, the probability of finding a diatomic molecule in its lowest vibrational state is exponentially dependent on the inverse temperature and decreases as the temperature increases.
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The vibrational motion of a diatomic molecule may be considered to be that of a simple harmonic oscillator with angular frequency ω. If a gas of these molecules is at a temperature T, what is the probability that a randomly picked molecule will be found in its lowest vibrational state?a)b)c)d)Correct answer is option 'B'. Can you explain this answer?
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