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Molecular Statistics ※ Degrees of freedom: Degrees of freedom: the total number of independent variables whose values have to be specified for a complete description of the system. (A) One atom: In the context of molecular motion these are the spatial coordinates of all the particles. Since we need three coordinates to describe the position of an atom, we say the atom has three degrees of freedom. (B) Diatomic molecule: If the atoms are not bound to one another, there will be no relation among the coordinates of the two atoms. On the other hand, when the two atoms are bound, the displacement of each other is coupled to the other. The result is to give three translational, one vibrational, and two rotational degrees of freedom for the 1 molecule. The vibrational and rotational degrees of freedom are also referred to as internal degrees of freedom. (C) Polyatomic molecule with N atoms: (i) Linear molecule: The 3N degrees of freedom of the atoms become three translational degrees of freedom and (3N-3) internal degrees of freedom. By analogy with diatomic molecules, we expect two rotational degrees of freedom for any linear polyatomic molecule (e.g. CO2 and C2H2). The remaining (3N-5) internal coordinates must correspond to vibrations. (ii) Bent molecule: A bent molecule loses a vibrational degree of freedom while gaining a rotational degree of freedom. Therefore, a nonlinear polyatomic molecule has three rotational degrees of freedom, hence (3N-6) vibrational degrees of freedom. 2 e.g. vibrational motions for CO2 and H2O: Degrees of freedom Atom Linear Bent polyatomic polyatomic Translational 3 3 3 Vibrational 0 3N-5 3N-6 Rotational 0 2 3 3 ※Molecular energies: (A) Translational energies: n 2h 2 En 8m for one axis. Where n = 1, 2, 3…., is the translational quantum number, m is the mass of the particle, and l is the distance between the walls. Since a particle has three translational degrees of freedom, the total translational energy is given by 2 h 2 nx2 n y nz2 En 8m 2x 2y 2z Where lx, ly, and lz denote the dimensions of the box. (B) Vibrational energies: For a diatomic molecule, the vibrational energy is given by: E h 1 2 V Where h called Planck’s constant, is 6.626 10 J sec 34 Andνis the frequency of oscillation. The symbolυ denotes the vibrational quantum number which can have only integer values, 0, 1, 2, …. (C) Rotational energies: Rotational energies of diatomic molecules are represented quite accurately by the quantum formula: E J 4 h 2 8 I 2 J J 1 where J, which is restricted to values of 0, 1, 2, .. is called the rotational quantum number. The moment of inert I is related to the bond length r of the diatomic molecule by I = mrr2. The mr is called the reduced mass, which is obtained from the mass of the two atoms: m r mm m m 1 2 1 2 ※ The origin of quantum mechanics: The origin of atomic structure: Three great discoveries: J. J. Thomson’s discovery: To measure e m value using magnet to deflect e- beam in 1897. R. A. Millikan’s discovery: To determine electron charge using oil drop experiment in 1909. E. Rutherford’s discovery: To define atomic structure usingαparticle to strike a thin foil of gold in 1911. Three important discoveries lead to following summary: 5 Each atom consists of a very small nucleus composed of protons and neutrons, which is encircled by moving electron. Both electrons and protons are electrically charged, the charge magnitude being 1 . 6 10 C , which is negative in 19 sign for electrons and positive for protons; neutrons are electrically neutral. The masses for proton and neutrons have approximately the same value, 1 . 67 10 kg . 2 7 The mass of electron is 9 . 11 10 31 kg . Each chemical element is characterized by the number of protons in the nucleus, or the atomic number (Z). For an electrically neutral, the atomic number is equal to the number of electrons. The atomic mass (M) of a specific atom may be expressed as the sum of the masses of protons and neutrons within the nucleus, i.e. M=Z+N. The quantization of energy: Two important phenomena: Black-body radiation: 6 A black-body radiator consists of a block of material having an internal cavity connected to the outside surface of the block by a small hole. The radiation emerging from the hole when the block is raised to a uniform temperature is called black-body radiation. The spectra radiation depends on wavelength. Planck proposed the concept of energy quantized in 1900. Photon with energy in h . 7 Photoelectric effect: The electrons come out when a light is shining onto a metal. The frequency of light controls whether or not electrons are emitted, but not light intensity. The number of electrons emitted is proportional to the intensity. Photon with energy in h and the kinetic energy of the electron emitted is determined by h 1 mv w . 2 2 Confirm Planck’s concept of energy quantized. ※Boltzmann distribution: The populations of energy levels: Since any molecule can only posses certain energies at a given temperature, each of molecules can distribute over all the available energy levels. Although we cannot keep track of the energy state of a single molecule, we can speak of the average numbers of molecules in each state, and these average numbers are constant in time so long as the temperature remains the same. The average number of molecules in a state is called the population of the state. 8 Only the lowest energy level is ocupied at T = 0 K. Raising the temperature excites some molecules into higher energy levels, and more and more levels become accessible as the temperature is raised further. Nevertheless, whatever the temperature, there is always a higher population in a state of low energy than one of high energy. Boltzmann distribution: The population in state n with energy En is determined by the temperature T according to the Boltzmann distribution: N Ne n En k BT 0 This formula was derived by Ludwig Boltzmann towards the end of the 19th century. According to this formula, the population ratio of the numbers of particles in various states with energies Ei and Ej is determined by Ni Nj e Ei E j k BT The fundamental constant kB is called Boltzmann’s constant, .k 1.38 10 B 23 J K Boltzmann’s constant is replaced by the gas constant, R=NAkB, if the energies in the Boltzmann distribution are replaced by molar energies. 9 The possibility fi of molecules occupying the i-th state is determined to be f i Ei k BT N N e N N e i i i Ei k BT In consideration of degeneracy, the equation is rewrriten to be: f i Ei k BT N N ge N N g e i i i i Ei k BT i The sum in the denominator is called the partition function, which means as “sum over states”. The Boltzmann distribution law is written as g e N g Ni i j j Ei E j k BT If fi are the fractions of molecules in different quantum states, the average energy per particle <E> is given by E fE i i 10