For a molecule, the constant is approximately .(This number is measured using microwave spectroscopy, that is, by measuring the microwave frequencies needed to excite the molecules into higher rotational states.) Calculate the rotational partition function for a molecule at room temperature , first using the exact formula 6.30 and then using the approximate formula 6.31
The rotational partition function of a heterogeneous diatomic molecule
The equation is
Here, is the rotational constant, is the Boltzmann constant, and is the absolute temperature.
At higher temperatures, for , the rotational partition function becomes as follows:
Substitute for for , and in the equation
Therefore, the rotational partition function of a molecule is
The equations are
Expand the above summation from to :
Substitute for in the above equation.
Therefore, the exact value of rotational partition function of a molecule is
The dissociation of molecular hydrogen into atomic hydrogen, can be treated as an ideal gas reaction using the techniques of Section 5.6. The equilibrium constant K for this reaction is defined as
where is a reference pressure conventionally taken to be and the other P's are the partial pressures of the two species at equilibrium. Now, using the methods of Boltzmann statistics developed in this chapter, you are ready to calculate K from first principles. Do so. That is, derive a formula for K in terms of more basic quantities such as the energy needed to dissociate one molecule (see Problem 1.53) and the internal partition function for molecular hydrogen. This internal partition function is a product of rotational and vibrational contributions, which you can estimate using the methods and data in Section 6.2. (An molecule doesn't have any electronic spin degeneracy, but an H atom does-the electron can be in two different spin states. Neglect electronic excited states, which are important only at very high temperatures. The degeneracy due to nuclear spin alignments cancels, but include it if you wish.) Calculate K numerically at Discuss the implications, working out a couple of numerical examples to show when hydrogen is mostly dissociated and when it is not.
The analysis of this section applies also to liner polyatomic molecules, for which no rotation about the axis of symmetry is possible. An example is , with . Estimate the rotational partition function for a molecule at room temperature. (Note that the arrangement of the atoms is , and the two oxygen atoms are identical.)
Imagine a world in which space is two-dimensional, but the laws of physics are otherwise the same. Derive the speed distribution formula for an ideal gas of nonrelativistic particles in this fictitious world, and sketch this distribution. Carefully explain the similarities and differences between the two-dimensional and three-dimensional cases. What is the most likely velocity vector? What is the most likely speed?
Consider a hypothetical atom that has just two states: a ground state with energy zero and an excited state with energy 2 eV. Draw a graph of the partition function for this system as a function of temperature, and evaluate the partition function numerically at T = 300 K, 3000 K, 30,000 K, and 300,000 K.
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