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The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances; see partition function (mathematics) for
In statistical mechanics, the translational partition function, is that part of the partition function resulting from the movement (translation) of the center of mass. For a single atom or molecule in a low pressure gas, neglecting the interactions of molecules , the canonical ensemble q T {\displaystyle q_{T}} can be approximated by: [ 1 ]
Vibrational partition function, partition function for the vibrational modes of a molecule; Partition function (quantum field theory), partition function for quantum path integrals; Partition function (mathematics), generalization of the statistical mechanics concept; Partition function (number theory), the number of possible partitions of an ...
This is almost the partition function for the -ensemble, but it has units of volume, an unavoidable consequence of taking the above sum over volumes into an integral. Restoring the constant C {\displaystyle C} yields the proper result for Δ ( N , P , T ) {\displaystyle \Delta (N,P,T)} .
The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution.
Rotational energies are quantized. For a diatomic molecule like CO or HCl, or a linear polyatomic molecule like OCS in its ground vibrational state, the allowed rotational energies in the rigid rotor approximation are = = (+) = (+). J is the quantum number for total rotational angular momentum and takes all integer values starting at zero, i.e., =,,, …, = is the rotational constant, and is ...
The terms in the bracket give the total partition function of the adsorbed molecules by taking a product of the individual partition functions (refer to Partition function of subsystems). The 1 / N A ! {\displaystyle 1/N_{A}!} factor accounts for the overcounting arising due to the indistinguishable nature of the adsorbates.
The Maxwell–Boltzmann distribution is a mathematical function that describes about how many particles in the container have a certain energy. More precisely, the Maxwell–Boltzmann distribution gives the non-normalized probability (this means that the probabilities do not add up to 1) that the state corresponding to a particular energy is ...