Translational energy levels

Translational Energy Levels To predict the nature of translational energy levels, one assumes that the molecules are confined to a rectangular box with sides 

and L , so that the volume of the box is LXLVL . The potential energy of the particle inside the box is zero but goes to infinity at the walls. The quantum mechanical solution in the. v direction to this particle in a box problem gives 

Similar expressions can be written for elrans. r and eirans. -, the energies in the y and z directions. The total translational energy is the sum of the contributions in the three directions. 

In a cubic box with dimensions Lv = Lv = Lz = L, the total translational energy becomes 

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Unless the moment of inertia is small, the energy difference between rotational energy levels is not large, although not as small as the difference between translational energy levels. 

Since the energy difference between translational energy levels is very small and the sum is over a large number of particles, we can assume the energy... 

Translational energy levels are very close together. We can neglect the 1 compared to n. and replace the sum by an integral to give... 

We see that the size of the quantum of translation energy is around 1020 times smaller than kg T in this example. To a very good approximation we can treat translational energy levels as though they were continuous rather than discretely spaced. 

Obtain a general formula for the most probable three-dimensional translational quantum number j = jmax for a gas (assume a Boltzmann distribution). Evaluate this expression for NO2 at 1000 K (assume a cubic container 0.1 m on each side). Determine the translational energy that this corresonds to (J/mole). Find the fraction of molecules having a translational energy level greater than jmax. Hint Solution to this problem will involve the error function, erf(x). 

We shall need to know how to evaluate these separated partition functions. The translational energy levels can be derived from the quantum mechanical solution for a particle in a box they are so closely spaced that the partition function can be evaluated in closed form by integration, and has the value... 

Translational energy levels are obtained by confining the molecule to a rectangular box with sides Lx, Ly, and Lz, in which case, the energy levels in the x direction are given by... 

See Section 1.7 in the text for the particle-in-a-box translational energy levels.)... 

Beginning in about 1930 the principles of quantum mechanics were applied to this problem by Eyring, Polanyi, and their coworkers, and the result is known as the activated-complex theory In this theory reaction is still presumed to occur as a result of collisions between reacting molecules, but what happens after collision is examined in more detail. This examination is based on the concept that molecules possess vibrational and rotational, as well as translational, energy levels. 

The translational partition functions of gases are very large which tells us that there are many translational energy levels accessible to the molecules at normal temperatures. We can use the partition function to calculate the contribution of translational motion to the molar thermodynamic properties employing the relations of Table 9.3. 

Let s start with one neon atom and think through what happens as we add more atoms and open the stopcock (Figure 20.2). One atom has some number of microstates (W) possible for it in the left flask and the same number possible in the right flask. Opening the stopcock increases the volume, which increases the number of possible particle locations and, thus, translational energy levels. As a result, the system has 2, or 2, times as many microstates possible when the atom moves through both flasks (final state, Wfinai) as when it is confined to one flask (initial state, Wi Hiai). 

The derived density of states for the translations, rotations, and vibrations can be used in Eq. (6.41) to obtain the corresponding classical partition functions. This will yield an accurate translational partition function at all temperatures of chemical interest because the translational energy level spacings are so dense. It will also yield accurate rotational partition functions at room temperature because molecular rotational constants are typically between 0.01 and 1 cm k However, at the low temperatures achieved in molecular beams, the accuracy of the classical rotational partition function (especially for molecules with high rotational constants, such as formaldehyde or H2 (Bg = 60.8 cm )) is insufficient. The energy level spacing of vibrations (ca. 2000 cm ) are considerably larger than the room temperature of 207 cm " so that even at room temperature, the vibrational partition function must be evaluated by summation in Eq. (6.40). 

In the statistical-mechanical evaluation of the molecular partition function of an ideal gas, the translational energy levels of each gas molecule are taken to be the levels of a particle in a three-dimensional rectangular box see Levine, Physical Chemistry, Sections 22.6 and 22.7. 

The translational energy levels can be taken as the energy levels (3.72) of a particle in a three-dimensional box whose dimensions are those of the container holding the gas of diatomic molecules. 

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