Chapter 7 The Quantum-Mechanical Model of the Atom

Electron Energy electron energy and position are complimentary because KE = ½mv2 for an electron with a given energy, the best we can do is describe a region in the atom of high probability of finding it – called an orbital a probability distribution map of a region where the electron is likely to be found distance vs. y2 (wave function) many of the properties of atoms are related to the energies of the electrons

Wave Function, y calculations show that the size, shape and orientation in space of an orbital are determined be three integer terms in the wave function added to quantize the energy of the electron these integers are called quantum numbers principal quantum number, n angular momentum quantum number, l magnetic quantum number, ml

Principal Quantum Number, n characterizes the energy of the electron in a particular orbital corresponds to Bohr’s energy level n can be any integer ³ 1 the larger the value of n, the more energy the orbital has energies are defined as being negative an electron would have E = 0 when it just escapes the atom the larger the value of n, the larger the orbital as n gets larger, the amount of energy between orbitals gets smaller –The negative sign means that the energy of the electron bound to the nucleus is lower than it would be if the electron were at an infinite distance (n = ∞) from the nucleus, where there is no interaction. for an electron in H

Principal Energy Levels in Hydrogen

Electron Transitions in order to transition to a higher energy state, the electron must gain the correct amount of energy corresponding to the difference in energy between the final and initial states electrons in high energy states are unstable and tend to lose energy and transition to lower energy states energy released as a photon of light each line in the emission spectrum corresponds to the difference in energy between two energy states

Hydrogen Energy Transitions

Predicting the Spectrum of Hydrogen the wavelengths of lines in the emission spectrum of hydrogen can be predicted by calculating the difference in energy between any two states for an electron in energy state n, there are (n – 1) energy states it can transition to, therefore (n – 1) lines it can generate both the Bohr and Quantum Mechanical Models can predict these lines very accurately Since the energy must be conserved, the exact amount of energy emitted by the atom is carried away by the photon ΔEatom = - ΔEphoton

Example Determine the wavelength of light emitted when an electron in a hydrogen atom makes a transition from an orbital in n= 6 to an orbital in n=5 As electron in the n=6 level of the hydrogen atom relaxes to a lower energy level, emitting light of λ= 93.8 nm. Find the principle level to which the electron relaxed

The angular Momentum Quantum number (l) Is an integer that determines the shape of the orbital. Quantum number n (shell) Value of l Letter designation (subshell) n = 1 l = 0 s n = 2 l = 1 p n = 3 l = 2 d n = 4 l = 3 f the energy of the subshell increases with l (s < p < d < f).  

The magnetic quantum number (ml) Specifies the orientation in space of an orbital of a given energy (n) and shape (l). This number divides the subshell into individual orbitals which hold the electrons; there are 2l+1 orbitals in each subshell. Thus the s subshell has only one orbital, the p subshell has three orbitals, and so on n l # Orbitals ml 1 2 0, 1 -1, 0, 1 3 0, 1, 2 -2, -1, 0, 1, 2 4 0, 1, 2, 3 -3, -2, -1, 0, 1, 2, 3, 4

Examples Give the possible combination of quantum numbers for the following orbitals 3s orbital 2 p orbitals Give orbital notations for electrons in orbitals with the following quantum numbers: n = 2, l = 1 ml = 1 n = 3, l = 2, ml = -1

Probability & Radial Distribution Functions y2 is the probability density the probability of finding an electron at a particular point in space for s orbital maximum at the nucleus? decreases as you move away from the nucleus the Radial Distribution function represents the total probability at a certain distance from the nucleus maximum at most probable radius nodes in the functions are where the probability drops to 0

Probability Density Function

The Shapes of Atomic Orbitals the l quantum number primarily determines the shape of the orbital l can have integer values from 0 to (n – 1) each value of l is called by a particular letter that designates the shape of the orbital s orbitals are spherical p orbitals are like two balloons tied at the knots d orbitals are mainly like 4 balloons tied at the knot f orbitals are mainly like 8 balloons tied at the knot

l = 0, the s orbital each principal energy state has 1 s orbital lowest energy orbital in a principal energy state spherical number of nodes = (n – 1)

2s and 3s 2s n = 2, l = 0 3s n = 3, l = 0

l = 1, p orbitals each principal energy state above n = 1 has 3 p orbitals ml = -1, 0, +1 each of the 3 orbitals point along a different axis px, py, pz 2nd lowest energy orbitals in a principal energy state two-lobed node at the nucleus, total of n nodes Tro, Chemistry: A Molecular Approach

p orbitals

l = 2, d orbitals each principal energy state above n = 2 has 5 d orbitals ml = -2, -1, 0, +1, +2 4 of the 5 orbitals are aligned in a different plane the fifth is aligned with the z axis, dz squared dxy, dyz, dxz, dx squared – y squared 3rd lowest energy orbitals in a principal energy state mainly 4-lobed one is two-lobed with a toroid planar nodes higher principal levels also have spherical nodes

d orbitals

l = 3, f orbitals each principal energy state above n = 3 has 7 d orbitals ml = -3, -2, -1, 0, +1, +2, +3 4th lowest energy orbitals in a principal energy state mainly 8-lobed some 2-lobed with a toroid planar nodes higher principal levels also have spherical nodes

f orbitals

Now we know why atoms are spherical