What is the law of equipartition of energy (in detail explaination please)
Equipartition of energy, law of statistical mechanics stating that, in a system in thermal equilibrium, on the average, an equal amount of energy will be associated with each independent energy state. Based on the work of physicists James Clerk Maxwell of Scotland and Ludwig Boltzmann of Germany, this law states specifically that a system of particles in equilibrium at absolute temperature T will have an average energy of 1/2kT associated with each degree of freedom (see freedom, degree of), in which k is the Boltzmann constant. In addition, any degree of freedom contributing potential energy will have another 1/2kT associated with it. For a system of s degrees of freedom, of which t have associated potential energies, the total average energy of the system is 1/2(s + t)kT.
For example, an atom of a gas has three degrees of freedom (the three spatial, or position, coordinates of the atom) and will, therefore, have an average total energy of 3/2kT. For an atom in a solid, vibratory motion involves potential energy as well as kinetic energy, and both modes will contribute a term 1/2kT,resulting in an average total energy of 3kT.
We shall make our first assumption about how the internal energy distributes itself among N gas molecules, as follows: Each independent degree of freedom has an equal amount of energy equal to (1/ 2) kT , where the constant k is called the Boltzmann constant and is equal to k = 1.3806505×10-23 J⋅ K−1 . The total internal energy of the ideal gas is then Einternal = N(# of degrees of freedom) 1 2 kT . (29.3.2) This equal division of the energy is called the equipartition of the energy. The Boltzmann constant is an arbitrary constant and fixes a choice of temperature scale. Its value is chosen such that the temperature scale in Eq. (29.3.2) closely agrees with the temperature scales discussed in Section 29.2. According to our classical theory of the gas, all of these modes (translational, rotational, vibrational) should be equally occupied at all temperatures but in fact they are not. This important deviation from classical physics was historically the first instance where a more detailed model of the atom was needed to correctly describe the experimental observations. Not all of the three rotational degrees of freedom contribute to the energy at all temperatures. As an example, a nitrogen molecule, N2 , has three translational degrees of freedom but only two rotational degrees of freedom at temperatures lower than the temperature at which the diatomic molecule would dissociate (the theory of quantum mechanics in necessary to understand this phenomena). Diatomic nitrogen also has an intramolecular vibrational degree of freedom that does not contribute to the internal energy at room temperatures.
Equipartition of energy, law of statistical mechanics stating that, in a system in thermal equilibrium, on the average, an equal amount of energy will be associated with each independent energy state. Based on the work of physicists James Clerk Maxwell of Scotland and Ludwig Boltzmann of Germany, this law states specifically that a system of particles in equilibrium at absolute temperature T will have an average energy of 1/2kT associated with each degree of freedom (see freedom, degree of), in which k is the Boltzmann constant. In addition, any degree of freedom contributing potential energy will have another 1/2kT associated with it. For a system of s degrees of freedom, of which t have associated potential energies, the total average energy of the system is 1/2(s + t)kT.
For example, an atom of a gas has three degrees of freedom (the three spatial, or position, coordinates of the atom) and will, therefore, have an average total energy of 3/2kT. For an atom in a solid, vibratory motion involves potential energy as well as kinetic energy, and both modes will contribute a term 1/2kT,resulting in an average total energy of 3kT.
We shall make our first assumption about how the internal energy distributes itself among N gas molecules, as follows: Each independent degree of freedom has an equal amount of energy equal to (1/ 2) kT , where the constant k is called the Boltzmann constant and is equal to k = 1.3806505×10-23 J⋅ K−1 . The total internal energy of the ideal gas is then Einternal = N(# of degrees of freedom) 1 2 kT . (29.3.2) This equal division of the energy is called the equipartition of the energy. The Boltzmann constant is an arbitrary constant and fixes a choice of temperature scale. Its value is chosen such that the temperature scale in Eq. (29.3.2) closely agrees with the temperature scales discussed in Section 29.2. According to our classical theory of the gas, all of these modes (translational, rotational, vibrational) should be equally occupied at all temperatures but in fact they are not. This important deviation from classical physics was historically the first instance where a more detailed model of the atom was needed to correctly describe the experimental observations. Not all of the three rotational degrees of freedom contribute to the energy at all temperatures. As an example, a nitrogen molecule, N2 , has three translational degrees of freedom but only two rotational degrees of freedom at temperatures lower than the temperature at which the diatomic molecule would dissociate (the theory of quantum mechanics in necessary to understand this phenomena). Diatomic nitrogen also has an intramolecular vibrational degree of freedom that does not contribute to the internal energy at room temperatures.
