Bohr Model of the Atom Bohr made three assumptions (postulates) 1. The electrons move only in certain circular orbits, called STATIONARY STATES. This motion can be described classically 2. Radiation only occurs when an electron goes from one allowed state to another of lower energy. The radiated frequency is given by hf = E m - E n where E m and E n are the energies of the two states 3. The angular momentum of the electron is restricted to integer multiples of h/ (2 ) m e vr = n
The Schrödinger equation The hydrogen atom The potential energy in spherical coordinates (The potential energy function is spherically symmetric.) Partial differential equation with three independent variables
The spherical coordinates (alternative to rectangular coordinates)
For all spherically symmetric potential-energy functions: ( the solutions are obtained by a method called separation of variables) Radial function Angular function of and The hydrogen atom The functions and are the same for every spherically symmetric potential-energy function. Thus the partial differential equation with three independent variables three separate ordinary differential equations
The solution The solution is determined by boundary conditions: - R(r) must approach zero at large r (bound state - electron localized near the nucleus); and must be periodic: (r, and (r, describe the same point, so and must be finite. Quantum numbers: n - principal l – orbital m l - magnetic
Principal quantum number: n The energy E n is determined by n = 1,2,3,4,5,…; E = - 13.6 eV - 3.4 eV Ionized atom n = 1 n = 2 n = 3 reduced mass
Quantization of the orbital angular momentum. The possible values of the magnitude L of the orbital angular momentum L are determined by the requirement, that the function must be finite at and There are n different possible values of L for the n th energy level! Orbital quantum number
Quantization of the component of the orbital angular momentum
Quantum numbers: n, l, m l – orbital quantum number l - determines permitted values of the orbital angular momentum n – principal quantum number n – determines permitted values of the energy l = 0,1,2,…n-1; m l - magnetic quantum number m l – determines permitted values of the z-component of the orbital angular momentum n = 1,2,3,4...
Wave functions n,l,m l = 1m = ±1 l = 0 n = 1 n = 2 n = 3 l = 0,1 l = 0,1,2 polynomial ~
Quantum number notation Degeneracy : one energy level E n has different quantum numbers l and m l l = 0 : s states n=1 K shell l = 1 : p states n=2 L shell l = 2 : d states n=3 M shell l = 3 : f states n=4 N shell l = 4 : g states n=5 O shell.
Spin angular momentum and magnetic moment Electron posseses spin angular momentum L s. With this momentum magnetic momentum is connected: where g e is the gyromagnetic ratio For free electron g e =2
Allowed values of the spin angular momentum are quantized : spin quantum number s = ½ Własny moment pędu - spin The z – component of the spin angular momentum: Spin angular momentum and magnetic moment
To label completely the state of the electron in a hydrogen atom, 4 quantum numbers are need:
Many – electron atoms and the exclusion principle Central field approximation: - Electron is moving in the total electric field due to the nucleus and averaged – out cloud of all the other electrons. - There is a corresponding spherically symmetric potential – energy function U( r). Solving the Schrodinger equation the same 4 quantum numbers are obtained. However wave functions are different. Energy levels depend on both n and l. In the ground state of a complex atom the electrons cannot all be in the lowest energy state. Pauli’s exclusion principle states that no two electrons can occupy the same quantum – mechanical state. That is, no two electrons in an atom can have the same values of all four quantum numbers (n, l, m l and m s )
Shells and orbitals N max - maximum number of electrons occupying given orbital n shell orbital 1K0s 2L0s L1p 3M0s M1p M2d 4 N N N N 0 1 2 3 s p d f N max 2 2 2 6 6 6 2 10 14
Shells K, L, M N : number of allowed states state with m s = +1/2 state with m s = -1/2 1s 2 2s 2 2p 2 1s 2 2s 2 2p 4 carbon oxygen Hund’s rule - electrons occupying given shell initially set up their spins paralelly
Example: l = 1, s = ½ j = 3/2j = 1/2 Possible two magnitudes of j : Total angular momentum - J
NMR ( nuclear magnetic resonance) Like electrons, protons also posses magnetic moment due to orbital angular momentum and spin ( they are also spin-1/2 particles) angular momentum. Spin flip experiment: Protons, the nuclei of hydrogen atoms in the tissue under study, normally have random spin orientations. In the presence of a strong magnetic field, they become aligned with a component paralell to the field. A brief radio signal flips the spins; as their components reorient paralell to the field, they emit signals that are picked up by sensitive detectors. The differing magnetic environment in various regions permits reconstruction of an image showing the types of tissue present.