What happens when translational energy levels are quantized?
International Journal of Quantum Chemistry, 111 (14):
3475--3481 (November 2011)DOI: 10.1002/qua.22780
Abstract
The conventional translational partition function (CTPF) that is widely used in textbooks is essentially semiclassical, whose form is found by using the particle-in-a-box model to represent the particles motion of an ideal gas. This form assumes continuum translational energy levels, thereby replacing the sum over the energy levels with an integral at high temperature. This is only valid if de Broglie wavelength is much shorter than the container dimension in which the particle is placed. Additionally, de Broglie wavelength must also be much smaller than the mean separation of the constituent particles of the gas. To ensure this, one will have to assume large mass and container size and high temperature (T). This assumption is a restriction in itself. Should de Broglie wavelength be larger than the microscopic size of interest, the CTPF will be invalid to use which in turn will lead to erroneous conclusions, and a pure quantum mechanical treatment must then be employed for accurate results. A perfect example of the failure of using the CTPF would be the conduction electrons in a metal or in conjugated dienes, whereby the translational energy levels are significantly quantized. This quantization manifests itself in the quantum translational partition function (TPF) as its curve starts at zero and remains zero at low T till further increase in T stimulates accessible translational thermal states to be populated, at which point a rise in quantum TPF is observed, whereas the CTPF is only zero at T = 0, and then starts rising as T increases since energy levels are continuum and their discretization is insensitive to the CTPF. Therefore this article explores quantum mechanical treatment of the TPF and its effects on thermodynamic functions using our closed-form expression of quantum TPF developed in M. Toutounji, Int J Quantum Chem Early View (unpublished) (IJQC) which is valid at all Ts, sizes, and masses. Although this TPF appeared in IJQC in an abstract form before, there was not any discussion of its applicability and physical aspects therein. Also, some model calculations are presented to show the failure of the CTPF in the current literature in case of a purely quantum conditioned environment. Thermodynamic functions such as Helmholtz free energy, entropy, and heat capacity are explored using the herein closed form quantum partition function. A closer look at the quantum translational heat capacity shows a curve starts at zero at low T, and then starts sharply rising as T increases going through a maximum after which it levels off at its classical value k/2, where k is Boltzmann constant.