Lecture 22 10/26/05

s/m10998.2light of speed c

sJ1062.6constant sPlanck'h

m100974.1constant RydbergR

2n and 1n

n

1

n

1Rhc

n

Rhc

n

RhcEEE

8

34

7

2initial

2final

2initial

2final

initialfinal

Moving between energy levels

1. Hydrogen atom has only certain allowable energy levels

1. Stationary states

2. Atom does not radiate energy while in stationary state3. Electron moves between stationary states by

absorbing or emitting a photon of energy

Good for 1 electron H atom, but failed to predict the spectrum for any other atom

Bohr Model of the Hydrogen Atom(Recap)

de Broglie (1924) proposed that if light can have both wave and particle properties, then perhaps so could

particles, such as electrons.

mv

hλ

λ

hmv

light) of c(speed for particle) of v(velocity subsitute particle, a For

λ

hmc

λ

hcmc

mcE and λ

hcE :light For

2

2

De broglie’s equation

Equation is only useful for very small particles.

For example, consider a 114-g baseball thrown at 110 mph.

mv = 5.6 kg-m/s

)range m10 the in are rays (gamma measure to smallinsanely Too

m 102.1λ

smkg 6.5

smkg 10626.6λ

mv

hλ

16

34

234

nm 15.0m 1

nm 10m 105.1λ

)s/m 100.5)(kg 1011.9(

smkg 10626.6λ

mv

hλ

910

631

234

Calculate the de Broglie wavelength of an electron (9.11 x 10-31 kg) moving at a velocity of 5.0 x 106 m/s.

Quantum Mechanics or Wave Mechanics

Theoretical approach to understanding atomic behavior

Heisenberg Uncertainty Principle

For an electron, it is impossible to simultaneously determine:

The exact position AND

The exact energy

WAVE FUNCTIONS (

Schrödinger developed mathematical models of electron

1. Behavior of the electron in the atom is best described as a standing wave

1. Only certain wave functions are allowed 2. Each is associated with an allowed En

3. Energy of electron is quantized4. 2 is proportional to the probability of finding an e- at

a given point5. Each corresponds to an Orbital

1. Region of space within which an electron is probably found

6. Quantum numbers are part of the mathematical solution (address of each electron)

(Principal Quantum Number) n

n = 1, 2, 3 … infinity

Designates the electron shell

Value of n determines the energy of electron Remember the En = -Rhc/n2

Value of n also measures size of orbital Greater n larger orbital size

(Angular Momentum Quantum Number) l

l = 0, 1, 2, 3, ….n-1

Each l corresponds to a different subshell with a different shape

Value of l Subshell label

0 s

1 p

2 d

3 f

(Magnetic Quantum Number) ml

n = ±1, ± 2, ± 3, . . ., ±l

Orientation of the orbital within the subshell

All have the same energy

Schrödinger equation does not explain closely spaced lines in some spectra of elements Red line at 656 nm in Hydrogen spectrum is

really a pair of lines: 656.272 nm and 656.285 nm

Called doublets

Magnetic Field

Proven experimentally that electron has a spin. Two spin directions are given by ms

ms = +1/2 and -1/2 Each orbital no more than 2 electrons!

4th quantum number (ms) electron spin quantum number

ml orbital -l ... 0 ... +l

orientation

ml orbital -l ... 0 ... +l

orientation

Summary: Quantum numbers

n shell 1, 2, 3, 4, ... size and energy

l subshell 0, 1, 2, ... n – 1 shape

ms electron spin +1/2 and -1/2 spin