This paper gives a critical review of the physical meaning of the chemical potential, perhaps the most abstract of all thermodynamic quantities. To show its basic behavior, thereby to illustrate its physical significance, we have derived the chemical potential of a system of free electrons as a function of the density and temperature in different spatial dimensions. We have shown how to obtain the isothermal compressibility given the chemical potential. To emphasize the usefulness of the knowledge of dimensional dependence, both the compressibility and average kinetic energy are expressed as simple dimensional relationships of the density and, hence, the chemical potential. Finally, there is a certain temperature at which the chemical potential should identically vanish. Physical implications of zero chemical potential are discussed.