found by adding up the for all orbitals. Indistinguishable classical particles In classical mechanics it is also possible to consider indistinguishable particles (in fact, indistinguishability is a prerequisite for defining a chemical potential in a consistent manner; all particles of a given kind must be interchangeable). We again consider placing multiple particles of the same kind into the same microstate of single-particle phase space, which we again call an "orbital". However, compared to quantum mechanics, the classical case is complicated by the fact that a microstate in classical mechanics does not refer to a single point in phase space but rather to an extended region in phase space: one microstate contains an infinite number of states, all distinct but of similar character. As a result, when multiple particles are placed into the same orbital, the overall collection of the particles (in the system phase space) does not count as one whole microstate but rather only a fraction of a microstate, because identical states (formed by permutation of identical particles) should not be overcounted. The overcounting correction factor is the factorial of the number of particles. The statistics in this case take the form of an exponential power series the value corresponding to Maxwell–Boltzmann statistics. Ionization of an isolated atom The grand canonical ensemble can be used to predict whether an atom prefers to be in a neutral state or ionized state. An atom is able to exist in ionized states with more or fewer electrons compared to neutral. As shown below, ionized states may be thermodynamically preferred depending on the environment. Consider a simplified model where the atom can be in a neutral state or in one of two ionized states (a detailed calculation also includes the degeneracy factors of the states): charge neutral state, with electrons and energy . an oxidized state ( electrons) with energy a reduced state ( electrons) with energy Here and are the atom's ionization energy and electron affinity, respectively; is the local electrostatic potential in the vacuum nearby the atom, and is the electron charge. The grand potential in this case is thus determined by The quantity is critical in this case, for determining the balance between the various states. This value is determined by the environment around the atom. If one of these atoms is placed into a vacuum box, then , the work function of the box lining material. Comparing the tables