Quantum numbers and the periodic table

Development of Quantum Mechanics Neils Bohr recognized that electrons exist in definite energy states - called quantum states Electrons can move from one state to another by releasing or absorbing energy – usually in the form of photons Energy released is proportional to the frequency of the photon (E = hv) High frequency = high energy h = Planck’s constant (6.626 x 10-34 J s)

Wave Nature of Matter Louis De Broglie theorized that since energy was proportional frequency (Planck’s Law) AND also proportional mass (Einstein’s E=mc2), frequency must also be proportional to mass. mc2 = hf since v = λf f = v/λ Therefore, mc2 = hv/λ c = speed of light, so mv2 = hv/λ mλv = h or λ = h/(mv) mv is a physics quantity known as momentum

Wave Nature of Matter Enter Werner Heisenberg! Heisenberg recognizes that if you know the motion (momentum) of an object, it is impossible to know its position and vice versa If something is moving its position is changing and if you freeze an object, you change its motion So, we can only list “probabilities” for where an electron is within an atom. This concept set the stage for the idea of quantum mechanics

Schrodinger’s Wave Equation Edwin Schrodinger developed a probability equation to describe the position of an electron by treating it like a wave Equation itself is complex calculus, but involves wave density Blue dots on diagram represent high wave density  likely to find electron here Look like clouds – hence “electron cloud model”

The Quantum Numbers The Wave Equation also predicts energy state – called “orbitals” where electrons can be located These orbitals are described by the quantum numbers The principal quantum number, n, represents the energy level of the orbital The angular quantum number, l, represents the shape of the orbital The magnetic quantum number, m, represents the orientation of the orbital (direction in space) The spin quantum number, s, represents the direction of the electron’s spin

The Principal Quantum Number The principal quantum number correspond to the Bohr model’s energy level n can be any nonzero integer These are equivalent to the rows of the periodic table For example, Na (sodium) is in the 3rd row – its n is 3 What is the n for Kr?

The Angular Quantum Number The l is used to describe the shape of the orbitals l values are given by the following formula: l = 0 through n – 1 So if n = 3, l = 0, 1, and 2 Meaning the third energy level (n), has three orbitals or subshells These orbitals will have different shapes, which will be important for how they relate to bonding

The Orbitals The orbitals are usually referred to by the letters s, p, d, f (l = 0, 1, 2, 3) s orbitals (l = 0) are spherical in shape and grow larger with increasing energy level (see page 232 for explanation of WHY s is spherical) Can “hold” two electrons s orbitals correspond to columns 1 & 2 on the periodic table

The Orbitals p orbitals (l = 1) are shaped like mirror tear drops and fall along the x, y, or z axis There are 3 p orbitals in each energy level of 2 or higher Can hold 6 total electrons p orbitals correspond to columns 13 through 18 on the periodic table Orbitals are filled with one electron at a time, so if you have three electrons, one goes in each orbital The fourth electron would go into the 1st p orbital

The Orbitals There are five d orbitals (l = 2) The d orbital can hold a total of 10 electrons Corresponds to columns 3 – 12 on the periodic table

The Orbitals There are 7 f orbitals (l = 3) of complex shapes that correspond to the lanthanide and actinide series Can hold 14 electrons

The Magnetic & Spin Quantum Numbers The m quantum number describes the orientation of the orbital m can be any integer between –l through +l So if l = 1, m = -1, 0, and +1 Corresponds to the 3 p orbitals The s quantum number represents the two electrons found in each orbital Can be either +1/2 or -1/2 Odd # electrons are +; even are -

Pauli Exclusion Principle No two electrons in the same atom can have the same set of four quantum numbers. Therefore, no two electrons in the same atom can have the exact same energy. This means that every electron in an atom must differ by at least one of the four quantum number values: n, l, ml, and s.

Matching Quantum Numbers to the Periodic Table What are the quantum numbers corresponding to the last electron for Oxygen? Oxygen is in the 2nd row of the periodic table, so n = 2 Oxygen is in the p section of the periodic table, so l = 1 Oxygen is in the fourth column of the p block, so m = -1 Oxygen has an even # of electrons, so s = -1/2