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Derive expressions for the rotational contribution to the molar entropy (smolar) for a diatomic molecule
Accepted Answer
The rotational contribution to the molar entropy (Smolar) for a diatomic molecule can be derived using statistical mechanics. Here's a simplified explanation:
1. Partition Function: The rotational partition function (qrot) for a diatomic molecule is given by:
qrot = Σ(2J + 1)
exp(-BJ(J+1)/kT)
where J is the rotational quantum number, B is the rotational constant, k is the Boltzmann constant, and T is the temperature.
2. Entropy: The molar entropy (Smolar) is related to the partition function by:
Smolar = R
[ln(qrot) + T
(d(ln(qrot))/dT)]
where R is the gas constant.
3. Approximations: For high temperatures (kT >> B), the summation in the partition function can be replaced by an integral. This leads to the following expression for the rotational entropy:
Smolar = R
[ln(T/B) + 1.5]
This expression assumes the molecule is a rigid rotor and that there are no vibrational or electronic contributions to the entropy.
For more complex molecules or specific scenarios, additional terms may be needed in the partition function and entropy calculation.
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