Translational partition function for monatomic gas?
Video answer: Translational partition function
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18.1: The Translational Partition Function of a Monatomic Ideal Gas. which is the product of translational partition functions in the three dimensions. We postulate therefore that the observed energy of a macroscopic system should equal the statistical average over the partition function as shown above.
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• THE TRANSLATIONAL MOTION: mm→+1m2 so that 3/2 12 trans 2 2(mm)kT h ⎡π + ⎤ qV= ⎢⎥ ⎢⎣ ⎥⎦ • THE ELECTRONIC PARTITION FUNCTION will be similar to that for a monatomic gas, except the definition of the q q
Let consider the translational partition function of a monatomic gas. Consider a molecule confined to a cubic box. A molecule inside a cubic box of length L has the translational energy levels given by. (18.1.1) E t r = h 2 ( n x 2 + n y 2 + n z 2) 8 m L 2.
18.1: Translational Partition Functions of Monotonic Gases Since the levels are very closely spaced for translation, a large number of translational states are accessible available for occupation by the molecules of a gas. This result is very similar to the result of the classical kinetic gas theory
Where can we put energy into a monatomic gas? Only into translational and electronic modes! The total partition function is the product of the partition functions from each degree of freedom: = trans elec q V T q V T q V T
18.1: The Translational Partition Function of a Monatomic Ideal Gas. 18.2: Most Atoms Are in the Ground Electronic State at Room Temperature. 18.3: The Energy of a Diatomic Molecule Can Be Approximated as a Sum of Separate Terms. 18.4: Most Molecules Are in the Ground Vibrational State at Room Temperature.
For the moment we assume it is monatomic; the extra work for a diatomic gas is minimal. Remember the one-particle translational partition function, at any attainable temperature, is From this we can obtain the average energy per particle, , and since the particles are non-interacting, the energy of particles in a box is just .
The same procedure applies to polyatomic ideal gases as to diatomic ideal gases. The translation partition function for polyatomic ideal gases has the same exact form as that for diatomic ideal gas or the monatomic ideal gas. (18.7.1) q t r a n s = (2 π M k T h 2) 3 / 2 V N
The partition function is the sum of the Boltzmann factor over all possible states, where is the energy of state . Classically, we can approximate the summation over cells in phase-space as an integration over all phase-space