Chapter 2Atomic Structure

• HW: 1, 3, 11, 13, 17, 20, 24, 25, 30, 32, 33, 39, 40

• The Periodic Table• The Bohr Atom• The Schrodinger Equation• Orbitals• Shielding• Periodic Properties

of Atoms

Subatomic Particles• 1885 - Balmer derived a formula to calculate

the energies of visible light emitted by the hydrogen atom

n = integer, > 2R = Rydberg constant for hydrogen = 1.097 x 107 m–1

• General version of the equation: n = principal quantum number, nl < nh

• Origin of energy unknown until Bohr’s atomic theory (1913) derived same equation. R = fundamental constant = Connection between experiment and theory

E R 122 - 1

n2

E R 1nl

2 - 1nh

2

2 2Z 2e4

(40)2 h2

E = h =

hc

= hc

Bohr’s Atomic Theory• Negatively charged electrons orbit the

positively charged nucleus• When energy is absorbed, electrons move

to higher orbits• When electrons move to lower

orbits, energy is emitted• Equation predicted line spectra only

for single-electron atoms• Adjustments were made to use

elliptical orbits to better fit data• Ultimately failed - did not incorporate wave

properties of electrons. Still a useful theory.

Quantum Mechanics• Particles as waves

(de Broglie)• Uncertainty principle

(Heisenberg)

Electrons - energy can be measured very accurately, therefore cannot know position (x) with any certainty

• Probability of finding an electron at any position(electron density = probability)

• Both Schrodinger and Heisenberg proposed ways to treat electrons as waves, Schrodinger’s math was easier

=

hm

xpx

h4

Wave Functions• H= E , where H operates on .

• Solutions to equation are wave functions, each corresponding to an atomic orbital

• The conditions for a physically realistic solution:-One value for electron density/point-Continuous (does not change abruptly)-Must approach zero as r approaches infinity-Normalized (total probability = 1)-Orthogonal

z)y,(x,E = z)y,(x,z)y,V(x, + z

+ y

+ xm8

h-2

2

2

2

2

2

2

2

Atomic wave functions• Solving equations requires 3 quantum numbers:

n, , m

• n - principal (size and energy) - angular momentum (shape, contributes to energy)m - magnetic (orientation)ms - spin (orientation of electron spin)

• Plot in 3-D space (spherical coordinates), need 3 variables: r

• Break wavefunction into radial function (R), electron density at distances from nucleus, and angular function (, ), shape of orbital and orientation in space

• R(r)·Y(, ) = R(r)· Y(x,y,z)

Angular Functions, Y(x,y,z)• Table 2.3• Determine how probability changes at a

given distance from the center of the atom (shape and orientation in space)

• Look at real wave functions, in Cartesian coordinates

• Where do orbital labels come from?• Why are some regions shaded?

Radial Functions, R(r)• Table 2.4• 1s: n = 0, = 0

a0 = Bohr radius (radius of first “orbit” for H atom)= 52.9 pm

• Three ways to look at radial function:R vs. rR2 vs. r (probability)4r2R2 vs. r (radial probability density)

= h2

4 2me2

R = 2 Za0

3/2

e- ( = Zr/a0)

Radial Probability Density• 4r2R2: probability of finding electron at a given

distance from nucleus, summed over all angles• Probability of finding the electron at a certain

distance from the nucleus is not equal to the probability of finding the electron at a certain point at that distance from the nucleus.

• There is a whole surface of a sphere on which we can find the electron at that distance, r.

• 1s orbital: radial probability function has a maximum at r = a0 (Bohr radius). This is the distance from the nucleus where the electron in a 1s orbital is most likely found.

Homework Assignment• Orbital plots (use Excel) for 1s, 2s, 2p, 3s,

3p, 3d, 4s, 4p, 4d, 4f orbitals (you have to find the equations for the n=4 orbitals) (Orbitron!)

• Plot R, R2, and r2R2 (each as a function of )Print all plots, showing function approaching zero as increases

• See Figure 2.7 for n = 1 to n = 3, R and r2R2 plotsDue date?