Pchem: rotational motion/hydogenic atoms

what is angular momentum represented by and it’s equation in terms of mass?

J = Iω = (mr²)ω

What is the equation for rotational kinetic energy of a particle orbiting a nucleus in terms of it’s angular momentum?

Ek = J²/2I = J²/mr²

angular momentum is a product of…

moment of inertia * angular velocity

What is a good application for rotational motion in 2D/particle on a ring?

benzene molecule

What is the boundary condition for a particle on a ring?

⨚(Φ+2π) = ⨚(Φ)

When ml>0, roation is __________ and when ml< 0, rotaiton is _______

CW, CCW

What is the normalized wavefunction for a particle on a ring?

⨚(m_l) = (e^(im_lΦ))/sqrt(2π)

What is the eigenvalue for energy of a particle on a ring?

E_ml= m_l²*ħ²/2I

What is the z-component angular momentum operator?

-iħ (d/dΦ)

What is angular momentum eigenvalue for a particle on a ring?

J = m_lħ

What quantum number corresponds to the minimum energy required to start rotation of a particle on a ring?

ml = 1

What quantum number is nondegenerate for a particle on a ring? What does this mean physically?

ml = 0; the particle has an infinite wavelength and is stationary

For 3D rotational motion/particle on a sphere, what are the variables?

r (distance from origin to point), Φ = angle of projected shadow on the x-axis (horizontal angle), θ = angle compared to the z-axis (vertical angle)

How to construct a wavefunction of a particle on a sphere?

= (radial component of n and l) (spherical harmonics of l and m)

What is the l quantum number? What does it tell us about a wavefunction

orbital angular momentum quantum number; tells us magnitude of angular momentum and the shape of the “volume” of space

what is the ml quantum number and what it tells us?

it is the magnetic quantum number, telling us the z-component of angular momentum and the orientation of orbital in space

What is the energy eigenvalue of a particle on a sphere?

E_l,ml = l (l+1)*(ħ²/2I)

What quantum number is a particle on a sphere is energy dependent on?

it only depends on l

What is the angular momentum equation for a particle on a sphere?

J = l(l+1)*ħ

What is degeneracy equal to for a particle on a sphere?

n²

what is the number of angular nodes equal to for a particle on a sphere?

the quantum number l

What shell corresponds to quantum numbers n =,1,2,3,4

K,L,M,N

What is the equation for ionization energy of an electron?

I = (-hcZ²R_H)/n²

What does l = 0, 1, 2, 3 correspond to for subshells?

s, p, d, f

How many orbitals are in s, p, d, f

1, 3, 5, 7

What is ρ equal to?

r/a₀

What happens to the probability of an electron as the radius increases?

it decreases, but the volume of a given width at a frather radius is larger

What is the 3D probability equation for an s-orbital and for any orbital?

P = (⨚)² * 4πr²

P (r ) = r²R( r)²

What kind of orbital is this?

dz²

what is a hydrogenic atom?

at atom with one electron

Who discovered electrons by studying cathode rays

JJ Thompson

Who created the plum-pudding model? what did it say

William Thompson (lord kelvin); electrons are like seeds spread throughout postiively charged pudding

Who disproved the plum-pudding model and how?

Ernest Rutherford; directed radiation at an electron and they reflected—>detected electrons hitting ZnS screen—> this indicated that the positive ccharge was centered at the center of an atom

Who found that spectral data is not smooth and creates sharp, thin bands

Johannes Rydberg

What 3 things did the Bohr model suggest?

1) electrons can only have quantized eneryg levels

2) light is emitted druing e- transition

3) angular momentum J of orbiting electrons is quantized

What is the Bohr radius?

0.053 nm

What is the potential energy of a hydrogenic atom equal to?

V( r) = -(Ze²)/(4πε₀r)

e= elementary charge

ε₀ = vacuum permittivity

What part of the wavefunction forms sphereical/radiial nodes

the radial wavefunction

What part of the wavefunction forms angular nodes?

angualr wavefunction

What polynomial is included in the radial wavefunction? what does it do?

Laguerre; bridges the ends of the function

What part of the radial wavefunction is dominant far from the nucleus?

the decaying exponential in terms of r

What is the energy equation for a hydrogenic atom?

En = -ue^4/(32π²ε₀ħ²) * Z²/n²

Equation for finding number of radial nodes

n - 1 - l

what is the orbital approximation for a multielectron atom?

⨚₁,₂ = ⨚₁⨚₂

What experiment showed that electrons have quantized states? What was the expectation vs results?

Stern-Gerlach experiment

• expectation: tiny magnets of atoms should create a continous smear when going trhough a magnetic field , creating 2l+1 lines

• observation: 2 lines; magnetic moment of electrons must be quantized in a way that only gives 2 states (2 spins)

Who proposed electron spin?

Uhlenbeck and Goudsmit

What is the equation for electron spin?

S = sqrt(s(s+1))ħ

s= ± 1/2—> spin quantum number

What principle states that there can only be two electrons per orbital and that their spins must be paired?

Pauli exclusion principle

What is the name for particles with half integer spin?

fermions

What orbital is this?

4dxy