the grand canonical ensemble
Now open the walls to particles as well. The system exchanges both energy and particles with a reservoir that fixes the temperature T and the chemical potential mu. Picture a small region drawn imaginarily inside a larger body of gas, with molecules wandering freely in and out across its boundary. Both energy and particle number now fluctuate.
The constants are mu, V and T. The probability of a microstate with energy E_i and particle number N_i is p_i = e^(-(E_i - mu N_i)/k_B T) / Xi, where Xi, the grand partition function, is sum_i e^(-(E_i - mu N_i)/k_B T) = sum over N of z^N Z_N, with z = e^(mu / k_B T) the fugacity. The thermodynamic bridge is the grand potential Phi = -k_B T ln Xi = -PV, and the mean particle number is <N> = k_B T (d ln Xi / d mu)_{T,V}.
The ensemble is indispensable for quantum gases: because the occupation numbers of the single-particle states are then statistically independent, the Fermi-Dirac and Bose-Einstein distributions drop out almost trivially, whereas fixing N exactly would entangle them. Honestly, it is a mathematical convenience even for closed systems; in the thermodynamic limit its results match the fixed-N answer, and mu is simply tuned to give the desired average <N>.
Adsorption of gas onto a surface: each site exchanges molecules with the surrounding gas, so the grand canonical ensemble at the gas's chemical potential mu gives the coverage, the Langmuir isotherm.
Open to particles: the reservoir's mu sets how eagerly they enter.
The chemical potential is set by the reservoir, and the ensemble fixes only the average particle number <N>, not N itself; it is the natural setting for indistinguishable quantum particles.