equipartition of energy 5. (a) State the principle of equipartition of energy (b
ID: 3281205 • Letter: E
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equipartition of energy
5. (a) State the principle of equipartition of energy (b) For the complete specification of a molecule containing N atoms in space, 3N coordinates are required, three Cartesian coordinates per atom. Of these 3N coordinates, 3 can be used to specify the centre of mass of the molecule. These correspond to there being 3 translational degrees of freedom. For a linear molecule, two coordinates are required to specify its orientation about its centre of mass. These correspond to there being 2 rotational degrees of freedom. The remaining 3N (3+2)- 3N-5 coordinates specify the relative positions of the N atoms. This coresponds to there being 3N-5 vibrational modes of a linear molecule for N> 1. Each vibrational mode has a kinetic energy and a potential energy degree of freedom Therefore, a linear N-atomic molecule has 2(3N-5) vibrational degrees of freedom. How many vibrational modes does a non-linear molecule consisting of N atoms have? State the number of degrees of freedom that contribute a squared term to the expression for the energy of a non-linear triatomic molecule. Include a list of the umber of different types of degree of freedom in your answer. (c) What value of Cy is predicted classically for a non-linear triatomic gas such as sulphur dioxide? [2] (d) The molar heat capacity of sulphur dioxide gas at constant volume is experimentally found to be Cy = 24.981 K-1 mol-1 at 160K and Cv 4973] K-1 mol-1 at 1900K. Explain these experimental C, values at the two different temperatures in relation to the classically predicted value.Explanation / Answer
(a) 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.
eg.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.
(b) Cartesian coordinates are the most direct way to specify the locations of atoms in a molecule. For N atoms, there are 3N Cartesian coordinates. A non-linear molecule with N atoms has 3N 6 internal degrees of freedom. The other 6 DOFs correspond to translation in the three Cartesian directions and rigid rotation about the three axes. (For linear molecule, becomes 3N 5 internal degrees of freedom.) These six/five can be specified any way, and codes usually will translate atoms to the center of mass and orient principle rotation axis in z direction. Called “standard orientation.” When reading output, like direction of dipole vector or contributions to molecular orbitals, important to know what orientation it refers to.
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