An Ideal Bose Gas Thermodynamic Properties at Low Temperature

Abstract

In this study, the grand canonical partition function has been employed to estimate the chemical potential, internal energy, equation of state, specific heat and entropy of the non-interacting Bose gas in low temperature limit where the quantum effects become more prominent. Approximate expressions of these thermodynamic properties of ideal Bose gas were obtained below the critical temperature using Bose-Einstein integrals, and above the critical temperature using virial expansion method and series expansion of the Bose-Einstein function. Above the critical temperature, the internal energy per particles of Bose gas linearly increases with temperature. However, near and below the critical temperature, because of the quantum effects, the internal energy per particles of Bose gas exhibits a non-linear dependence on temperature; as the temperature reaches absolute zero, the entire boson collapse into a zero energy state. Similarly, the pressure and entropy vanishes at absolute zero for non-interacting bosons which indicate the absence of collisions and occurrence of greatest order amongst the constituents, respectively. On the other hand, the specific heat capacity exhibits a cusplike singularity at the critical temperature where transition occurs between condensate and normal phases of the ideal Bose gas; and such phenomenon is purely of quantum mechanical origin. Most importantly, these thermodynamic properties clearly display non-linear characteristics near and below the critical temperature as opposed to their classical predictions. In addition, our estimation of the chemical potential near the critical temperature approaches to zero. However, it starts to deviate from zero as the temperature decreases towards absolute zero revealing the fact that the approximate chemical potential is valid only near the critical temperature.