PHY232 - Remco Zegers - 2 models of the atom Newtonian era: the atom is a solid sphere Thompson (~1900): atom is a sphere of positive charge with negatively charged electrons in it Rutherford (1911) devised a ‘planetary model’ with a positively charged nucleus in the core and electrons orbiting around it. where do atomic transitions come from continuous centripetal acceleration and thus emission of photons Bohr (1913): model for Hydrogen(-like) atoms (we’ll look at it) Quantum-mechanical description of the atom
4 atomic spectra if an Hydrogen atom gets ‘excited’ (for example by heating it), light is emitted of certain specific wavelengths following the equation: 1/ =R H (1/2 2 -1/n 2 ) R H =1.097x10 7 m -1 n=3,4,5,6... Light of these wavelengths gets absorbed if white light (consisting of all wave lengths) is shone on a Hydrogen gas. Balmer series, after its discoverer Johann Balmer
PHY232 - Remco Zegers - 5 atomic spectra Similar spectra are observed for other elements, but the patterns are more complicated. Nevertheless, measuring such spectra allows one to identify which elements are present in a sample
PHY232 - Remco Zegers - 6 Bohr’s model for Hydrogen proton electron
PHY232 - Remco Zegers - 7 Bohr’s theory for Hydrogen assumption 1: the electron moves in circular orbits around the proton. The Coulomb force between the + nucleus and the – electron produces the centripetal acceleration. As a result, one can deduce the kinetic energy of the electron Note: this is a pure classical reasoning 1 proton
PHY232 - Remco Zegers - 8 Bohr’s theory for Hydrogen assumption 2: only certain orbits are stable, namely those in which no electromagnetic radiation is emitted in the absence of external forces. Hence, the energy of the atom is constant and the equations on the previous slide can be used. The size of the allowed orbits are those for which the orbital angular momentum of the electron is a multiple times ħ (h/(2 )): This is derived (de Broglie) from the assumption that a fixed number of electron-wavelengths must fit in the orbit: n=1n=2n=3 2
PHY232 - Remco Zegers - 10 Bohr’s theory for Hydrogen assumption 3 Radiation is emitted when an electron jumps from an outer orbit to an inner orbit. The energy of the radiation (and thus the frequency) is determined by the change in the atom’s energy due to the jump radiation is absorbed if an electron jumps from an inner orbit to an outer orbit The energy of an orbit can be calculated with: E=E kinetic +E potential E kin : 3 4
PHY232 - Remco Zegers - 11 combining assumptions 1,2 & 3: 1 & 2 give (solving for r, while eliminating v) combibe with 3 Bohr’s theory for Hydrogen 1 2 3 5
PHY232 - Remco Zegers - 12 finally, by combining 4 and 5 and using c=f and R H =1.097x10 7 m -1 Bohr’s theory for Hydrogen 4 5 with n f and n i : integers >0 6
PHY232 - Remco Zegers - 13 The hydrogen spectrum emission spectrum absorption spectrum By measuring the wave length of the light, one can determine the energy spectrum of Hydrogen n=1: ground state (energy is –13.6 eV) n= : electron is removed from atom: the atom is ionized. The ‘n’ is usually referred to as a shell; the 1 st shell, the 2 nd shell etc
PHY232 - Remco Zegers - 14 question The quantum number n can increase without limit. Therefore, the frequency of the emitted light from state n to the ground state can also increase without limit. a)true b)false
PHY232 - Remco Zegers - 15 example 1 How much energy does it take to ionize a Hydrogen atom?
PHY232 - Remco Zegers - 16 example 2 What is the wavelength of the light emitted if an electron goes from the 5 th shell to the 2 nd shell in a Hydrogen atom? What is the energy of the photon?
PHY232 - Remco Zegers - 17 example 3 What is the wavelength of the light absorbed if a hydrogen atom in its ground state is excited into its n=4 state? How much energy is absorbed (what is the excitation energy)?
PHY232 - Remco Zegers - 18 lon-capa do problem 2 from lon-capa 12
PHY232 - Remco Zegers - 19 Heavier atoms Bohr’s equation also does well for heavier atoms IF they have been ionized such that only one electron remains in its orbits. For example for Helium (2 protons (Z=2) in the nucleus), 2 electrons in the orbits would make it neutral, but only if one is missing can Bohr’s equations be applied. The equations need to be slightly modified however to take into account that the Coulomb forces/energies are different. Change e 2 into Ze 2 everywhere where it occurs. Z2Z2 Z2Z2 5a 6a Z2Z2
PHY232 - Remco Zegers - 20 lon-capa do problem 1 from lon-capa 12
PHY232 - Remco Zegers - 21 question. Consider a hydrogen atom and a singly ionized helium atom. Which one has the lower ground state energy? a) hydrogen b) singly ionized helium c) the same
PHY232 - Remco Zegers - 22 More general description Bohr’s classical approach breaks down if more than 1 electron is present in the atom. Instead, the problem has to be treated quantum mechanically by solving the Schrödinger equation The solutions give the distributions of electron clouds (so-called wave-functions) in the atom. The clouds describe the probability of finding an electron in a certain positions. The clouds are characterized by quantum numbers, which follow ‘simple’ rules the wave functions for Hydrogen
PHY232 - Remco Zegers - 23 atomic shells and quantum numbers The electrons are ordered according to 4 quantum numbers the principal quantum number n range: 1,2,3…. Usually referred to as K(n=1), L(n=2), M(n=3), N(n=4) shells the orbital quantum number l range: 0,1,2,…n-1 (so there are n possibilities) usually referred to as s (l=0),p (l=1) d (l=2),f (l=3),g (l=4),h,I… the orbital magnetic quantum number m l range: -l, -l+1,-1,0,1…l-1,l (there are 2l+1 possibilities) the spin magnetic quantum number m s range: -1/2 (electron spin up) or + ½ (electron spin down) in each state with given n,l,m l one can maximally place 2 electrons (m s =-1/2 and m s =+1/2). Pauli exclusion principle no two electrons in an atom can have the same set of quantum numbers n,l,m l,m s
PHY232 - Remco Zegers - 24 example: Hydrogen Hydrogen (not ionized) has only 1 electron. ground state quantum numbers: n=1 l=0,…,n-1=0,0 so only l=0 m l =-l,…,+l so only m l =0 m s =-1/2 or +1/2 one could maximally place 2 electrons in here (different m s ) this level is referred to as 1s 1 : one electron in the level with quantum numbers n=1, l=0 (s) n=2 states n=2 l=0,n-1=0,1 so l=0 or l=1 m l =-l,…+l, so m l =0 if l=0 and m l =-1,0,1 if l=1 for each m l, m s =-1/2 ot +1/2 these levels are referred to as: 2s 0 : n=2, l=0 it is empty but I could put 2 electrons in there 2p 0 : n=2, l=1 it is empty but I could put 6 electrons in there, namely: two each in n=2,l=1 with m l =-1,0,1
PHY232 - Remco Zegers - 25 more complicated example, Sodium Na: Sodium has Z=11 (11 protons), so if not ionized, it has 11 electrons. Atomic level will fill up according to lowest n, then lowest l there are exceptions to this (see also table in back of book and table 28.4) n=1, l=0, m l =0, m s =-1/2,+1/21s 2 2 electrons n=2, l=0, m l =0, m s =-1/2,+1/22s 2 2 electrons n=2, l=1, m l =-1,0,1, m s =-1/2,+1/2 2p 6 6 electrons n=3, l=0, m l =0, m s =-1/2,+1/23s 1 1 electron (2 possible, but only 1 needed to get to 11) SUM: 11 electrons So, the ground state configuration can be described as: 1s 2 2s 2 2p 6 3s 1 Note that the n=1 and n=2 shells are filled with 10 electrons (Neon) so this is sometimes referred to as: [Ne]3s 1
PHY232 - Remco Zegers - 26 energy levels (example for Li Z=3) electrons outside the last filled (sub)shell are most important for the chemical properties of an atoms.
PHY232 - Remco Zegers - 27 questions a) what is the electron configuration of Argon (Z=18) b) what is the electron configuration of Titanium (Z=22)
PHY232 - Remco Zegers - 28 For Titanium the 4s level has moved down and the 3d level moved up, so that the 4s level is lower in energy than the 3d level
PHY232 - Remco Zegers - 29 Periodic table of elements Filled sub-shell, so very stable ‘Noble gases’ alkali metals very reactive one loosely bound electron halogens: very reactive, need one more electron to fill shell see also back of book