We investigate the isobar of an ideal Bose gas confined in a cubic box within the grand canonical ensemble for a large yet finite number of particles,
N. After solving the equation of the spinodal curve, we derive precise formulas for the supercooling and the superheating temperatures that reveal an ${N}^{\ensuremath{-}1/3}$ or ${N}^{\ensuremath{-}1/4}$ power correction to the known Bose-Einstein condensation temperature in the thermodynamic limit. Numerical computations confirm the accuracy of our analytical approximation, and further show that the isobar zigzags on the temperature-volume plane if $N\ensuremath{\ge}14\phantom{\rule{0.16em}{0ex}}393$. In particular, for the Avogadro's number of particles, the volume expands discretely about
105 times. Our results quantitatively agree with a previous study on the canonical ensemble within
0.1 error.
The sympathetic cooling of an initially degenerate Fermi gas by either an ideal Bose gas below
Tc or an ideal Boltzmann gas is investigated. It is shown that the efficiency of cooling by a Bose gas below
Tc is by no means reduced when its heat capacity becomes much less than that of the Fermi gas, where efficiency is measured by the decrease in the temperature of the Fermi gas per number of particles evaporated from the coolant. This contradicts the intuitive idea that an efficient coolant must have a large heat capacity. In contrast, for a Boltzmann gas a minimal value of the ratio of the heat capacities is indeed necessary to achieve
T=0 and all of the particles must be evaporated.