4b

Question 1

whats H made up of

Answer 1

1 proton
1 electron

Question 2

in a hydrogen,, how do we define where the e- is in respect to the proton

Answer 2

we draw a +1 dot for the proton and draw a ‘x’ , ‘y’, and ‘z’ axis coming out of it.

the z axis is going up,, the y axis is coming towards us and the x axis is going to the rhs

the e- we then draw in its position

proton is at the origin

Question 3

H’ = what

Answer 3

T’ + V’
potential and kinetic energy

Question 4

H’ = what more specific: energy of the system!!: total way the system has energy

Answer 4

H’ = T’.nuc + T’.el + V’. nuc-el

bc the nuc can move and so has kinetic energy

the e- can move and have kinetic energy

the potential energy between the nuc and the e-

how the nuc + el move and energy between those 2 things.

Question 5

in the H atom when we draw the 3 line axis wirth the nuc and the electron,,, can the potential energy,, V’ ,, equal to 0

Answer 5

nope
its no longer 0 bc its the nergy between the nuc and the electron

and bc they have opposite charges they have potential energy.

Question 6

what does V’ nuc-el mean and how does it change

Answer 6

energy between nuc and electron bc theyre oppositely charged this isnt 0.

the distance between the nuc and the electron alters the V’. nuc-el.

Question 7

bc the nuc is so heavy,, what does the e- think about it

Answer 7

the e- think its stood still

bc the e- moves so fast and the proton moves so slow the e- thinks its not moving.

e- just follows the proton / nuc.

Question 8

bc the nuc barely moves bc its so much heavier (made up of 1 proton) than the e- ,, in hydrogen,, what can we say about H’

Answer 8

we said H’ = T’.nuc + T’.el + V’.nuc-el

but bc we just said the nuc is sm heavier and so it looks like its not moving,, we can say the
H’ = T’.el + V’.nuc-el

nuc T’ has a tiny tiny effect so we can ignore its Ke. we just need to think about where the e- is.

Question 9

born oppenheimer approx

Answer 9

basically saying how H’ is approx equal to H’ = T’el + V’.nuc-el

due to nuc moving so slowly - we can ignore its T’

Question 10

so using the born oppenheimer approx,, how can we write the schrodinger equation:
HY = EY for a H atom.

Answer 10

(V’nuc-el + T’. el)Y = EY

Question 11

how many directions can the e- move in

Answer 11

the e- can move in 3 different directions

so we need a 3Dimensional

Question 12

okay so if movement of an e- is now 3D,, what do we need to change

Answer 12

what T’.el is equal to

T’.el = - (h.dash^2/2me x V-^2)

where m = mass of an e-.
V-^2 = del squared = d2/dx2 + d2/dy2 + d2/dz2

= 3d version due to moving e- in more directions.

Question 13

for the H atom,, is V’ = 0

Answer 13

nopeeee

Question 14

why isnt V’=0 for the Hatom

Answer 14

bc theres potential between the nuc and the e- bc theyre opposite charges.

Question 15

what is V’ for the hydrogen atom

Answer 15

= -Ze^2 / 4nEo x 1/r

Z = nuc charge
e = charge on e-
n = pi
Eo = vacuum permittivity (amount of screening//screaming done between the nuc and the e-)
r = electron - nuc distance.

the charge is a constant

dont rememeber this -

1/r = distance

Question 16

Z for H =

Answer 16

1

Question 17

Z for He =

Answer 17

2

Question 18

what does the V’.nuc-el acc depend on

Answer 18

the variable part of the equation we dont need to memorise

aka 1/r

where r is the distance between the nuc and the e-.

this is the only thing that changes the V’.nuc-el.

distance getting larger = smaller V.

distance getting closer = larger V

Question 19

isotropic meaning

Answer 19

same in all directions

V’ is the same in all directions : as long as the distance is the same

Question 20

when do we use spherical // polar coordinates

Answer 20

when a value is isotropic aka it doesnt change with direction //angle.

instead of Y(x,y,z) we use Y(r,o|, 0-)

r = distance
other things = 2 angles.

Question 21

which hamiltonian operator can we find // try to find now that we know V is isotropic

Answer 21

we try find the Y(r,o|, o-) one

we can find it bc we can split it up into radial part and angular part.

Question 22

what can we separate the isotropic wavefunction into Y(r,o|,o-)

Answer 22

radial part: R(r)
angular part: Y(o|, o-)

this is not an approx!!
we can figure out each one separately.!! due to using the polar spherical coordinatesssss

Question 23

okay so now that we can separate the angular and radial parts of the wavefunction,, how does this effect the schrodinger equation

Answer 23

we now get 2 schrodinger equation:

H’.rad Y.rad = E.rad Y.rad
radial schrodinger. energy we get from moving e- away from nuc!!! solved via analytical solutions: we can find a radial hamiltonian, find eiganfunctions and eigenvalues!!

and the angular one aswelllll

Question 24

Y. rad looks like what

Answer 24

-e^ beta.r
or smt like that
looks like an exponential!!

Question 25

Y.rad against r graph gives what

Answer 25

gives an exponential! e^-Br (beta) where r is the distance between the nuc and the e-. r is always larger or equal to 0 so its an eigenfunction,, finite, normalised, single valued and continuous its literally just a slope down

Question 26

what form do radial wavefunctions have + what are solutions dependent on + how are these expressed

Answer 26

R(r) = (polynomial in r ) x e^-Br solutions are dependent on the angular momentum ,, expressed in the Laguerre polynomial

Question 27

have we solved the radial schrodinger equation

Answer 27

yes!! we found that Y.rad = e^-Br weve solved it and found eigenfunctions we then get eigen values which give us energy!!

Question 28

what gives us E in schrodinger equation

Answer 28

the eigenvalues

Question 29

steps for schrodinger

Answer 29

find eigenfunctions then find eigenvalues as these give us energy

Question 30

energies for radial solutions =

Answer 30

Ec,, - bunch of constants that give the rhydberg constant x 1/n^2 where n = integer, 1,2,3 (principal quantum number) energy only depends on this!! so energy = - rhydberg / n^2

Question 31

okay so what have we done so far

Answer 31

we want to solve the full schrodinger equation: H'Y=EY so we broke it down into Y(r, o|,o-) = R(r) radial and Y(o|,o-) angular parts weve solved an radial schrodinger equation: H'.rad Y.rad = E.rad Y.rad to get radial wavefunction: e^-Br and energies (eigenvalue) these help explain whats going on in a hydrogen atom

Question 32

when u get sphercal polar coordinates,, what other force is occuring

Answer 32

centrifugal force bc its spinning around the proton// nuc aswell.

Question 33

different values of 'n' will give what

Answer 33

different wavefunctions and different energies diff principle number

Question 34

n = 1 wavefunction radial isssss

Answer 34

e^-Xr e^-something starts high up and just slopes down with Y as y axis and r as x axis. r being distance from nuc to e-

Question 35

n =2 wavefunction shape radial

Answer 35

starts high,, dips below x axis,, is modified by the polynomial and the laguerr polynomial and then rises but still below x axis

Question 36

n = 3 wavefunction radial

Answer 36

starts high,, dips below x axis gets back ubove x axis but stays close to it exponential that is modified by a different polynomial

Question 37

all energies areeee

Answer 37

negative bc Energy = - Rh/n^2 the most negative one is n=1 (ground state) n=2 is higher(less negative) and n = 3 is even higher(less negative)

Question 38

when looking at the radial wavefunction graphs for the different n values,, what should we look out for

Answer 38

the x axis unitssss look at the range of different values!!

Question 39

low energy way of arranging an e- around a proton =

Answer 39

n = 1 bc energy is based on n: ground state also shows how the e- spends most of its time near the nucleus: bc Y is high at a lower value of 'r'. the larger the n value,, the principle quantum number,, the higher energy way of arrangement it is. (arranging an e- around a proton)

Question 40

larger n valueeeee(radial) in terms of energy

Answer 40

the closer to 0 it is(less and less negative.,,, closer to positive)

Question 41

angular schrodinger equation

Answer 41

H'.ang Y.ang = E.ang Y.ang

Question 42

what is the angular schrodinger

Answer 42

spherical haronics used as solutions for when quantum particles move around smt // around an origin aka the e- around the proton: its not free to go anywhere,, its quantised according to spherical harmonics

Question 43

what do we need to describe the angular motion of the e- around the central proton

Answer 43

angular momentum , l and magnetic quantum number, ml so u get Y.ang(l, ml) u need 2 quantum numbers to determine the angular motion of the e-

Question 44

spherical harmonics give rise to what

Answer 44

they give rise to orbital shapes , s, p ,d orbital shapes.

Question 45

l = 0 =

Answer 45

s

Question 46

l = 1

Answer 46

p

Question 47

l = 2

Answer 47

d

Question 48

what does l do ,, angular quantum number + what values can it have

Answer 48

shape of angular motion 0 --> n-1

Question 49

ml = + what values can it have

Answer 49

orientation around the nucleus !! same shape tho bc the angular motion is quantised - l ---> l

Question 50

why are there 3 different p orbitals

Answer 50

bc p : l = 1 so ml : -1, 0 1 so 3 diff ml's so 3 different p orbitals

Question 51

solving the angular schrodinger equation gives ussss

Answer 51

info about the different orbitald and their orientation: all based on one e- moving around a proton. gives us spherical harmonics (solution to angular schrodinger equation) we get different ways of rearranging an e- around the proton but they all have the same energy

Question 52

spherical harmonics show us theeee

Answer 52

shape of the orbital

Question 53

E.ang

Answer 53

constant all have the same energy

Question 54

what is an orbital

Answer 54

a one electron wavefunction H' Y(1e-) = E Y(1e-) and that wavefunction is an orbital and that equals the radial part x the angular part (remember that we split this up to get 2 different schrodinger equations and solved them separately)

Question 55

Y.radial needs

Answer 55

'n'

Question 56

Y.angular needs

Answer 56

l , ml

Question 57

so the wavefunction we need for 1e- isssss Y(1e-)

Answer 57

product of the 2 split up ones: aka the angular x radial so u have Y(n,l,ml) we need these 3 to say which wavefunction n: tells us how its moving radially,, relative to the nuc ml, l : tells us how its moving angularly, relative to the nucleus we need all 3 to tell us what an orbital looks like.

Question 58

what do we need to tell us what an orbital looks like

Answer 58

Y(n,ml,l) n:tells us how its moving radially relative to the nuc ml,l: tells us how its moving angularly relative to the nuc.

Question 59

orbital energies are negative meaning

Answer 59

the e- is bound to the nucleus

Question 60

why does n=2 have 2 different wavefunction graphs

Answer 60

n=2 ,, l = 0,1 ml = -1,0,1 u get 2s and 2p 2 different wavefunctions bc u do radial x 2 different angular parts(different l values) 1 radial one is multiplied by 2 different angular parts to give 2 different products.

Question 61

radial node:

Answer 61

where wavefunction graph passes through 0

Question 62

angular node:

Answer 62

the origin of the graph ,, where it starts

Question 63

orbitals with the same principal quantumn number =

Answer 63

same energy they form a shell

Question 64

orbitals with the same n and l values

Answer 64

same n so same shell same l so same subshell l=0=s l=1=p l=2=d etc

Question 65

shell

Answer 65

same n value

Question 66

subshell

Answer 66

different l value same n number 2p and 2s

Question 67

in a shell we should have how many subshells

Answer 67

u should have 'n' sub shells

Question 68

how can we find out how many degenerate orbitals there are in a shell

Answer 68

n^2 tells us how many degenerate orbitals there are in a shell. shell = same n value.

Question 69

ground state §

Answer 69

1s no n = 0 no no no everything else is an excited state. but with different energy levels of the excited states.

Question 70

are 2s and 2p degenerate

Answer 70

yes, for this lecture they are degenerate: for H atom!!!! bc same shell : aka same 'n' number l = different but l gives shape not energy n^2