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physical - yr2, s1 > 4b > Flashcards

4b Flashcards

1
Q

whats H made up of

A

1 proton
1 electron

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2
Q

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

A

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

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3
Q

H’ = what

A

T’ + V’
potential and kinetic energy

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4
Q

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

A

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.

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5
Q

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

A

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.

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6
Q

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

A

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.

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7
Q

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

A

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.

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8
Q

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’

A

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.

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9
Q

born oppenheimer approx

A

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’

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10
Q

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

A

(V’nuc-el + T’. el)Y = EY

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11
Q

how many directions can the e- move in

A

the e- can move in 3 different directions

so we need a 3Dimensional

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12
Q

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

A

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.

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13
Q

for the H atom,, is V’ = 0

A

nopeeee

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14
Q

why isnt V’=0 for the Hatom

A

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

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15
Q

what is V’ for the hydrogen atom

A

= -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

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16
Q

Z for H =

A

1

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17
Q

Z for He =

A

2

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18
Q

what does the V’.nuc-el acc depend on

A

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

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19
Q

isotropic meaning

A

same in all directions

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

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20
Q

when do we use spherical // polar coordinates

A

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.

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21
Q

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

A

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.

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22
Q

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

A

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

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23
Q

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

A

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

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24
Q

Y. rad looks like what

A

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

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25
Q

Y.rad against r graph gives what

A

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

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26
Q

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

A

R(r) = (polynomial in r ) x e^-Br

solutions are dependent on the angular momentum ,, expressed in the Laguerre polynomial

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27
Q

have we solved the radial schrodinger equation

A

yes!!

we found that Y.rad = e^-Br

weve solved it and found eigenfunctions

we then get eigen values which give us energy!!

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28
Q

what gives us E in schrodinger equation

A

the eigenvalues

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29
Q

steps for schrodinger

A

find eigenfunctions
then find eigenvalues as these give us energy

30
Q

energies for radial solutions =

A

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

31
Q

okay so what have we done so far

A

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

32
Q

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

A

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

33
Q

different values of ‘n’ will give what

A

different wavefunctions and different energies

diff principle number

34
Q

n = 1 wavefunction radial isssss

A

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-

35
Q

n =2 wavefunction shape radial

A

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

36
Q

n = 3 wavefunction radial

A

starts high,, dips below x axis
gets back ubove x axis but stays close to it

exponential that is modified by a different polynomial

37
Q

all energies areeee

A

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)

38
Q

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

A

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

39
Q

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

A

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)

40
Q

larger n valueeeee(radial) in terms of energy

A

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

41
Q

angular schrodinger equation

A

H’.ang Y.ang = E.ang Y.ang

42
Q

what is the angular schrodinger

A

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

43
Q

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

A

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-

44
Q

spherical harmonics give rise to what

A

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

45
Q

l = 0 =

A

s

46
Q

l = 1

A

p

47
Q

l = 2

A

d

48
Q

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

A

shape of angular motion

0 –> n-1

49
Q

ml = + what values can it have

A

orientation around the nucleus !!

same shape tho

bc the angular motion is quantised

  • l —> l
50
Q

why are there 3 different p orbitals

A

bc p : l = 1

so ml : -1, 0 1

so 3 diff ml’s so 3 different p orbitals

51
Q

solving the angular schrodinger equation gives ussss

A

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

52
Q

spherical harmonics show us theeee

A

shape of the orbital

53
Q

E.ang

A

constant
all have the same energy

54
Q

what is an orbital

A

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)

55
Q

Y.radial needs

A

‘n’

56
Q

Y.angular needs

A

l , ml

57
Q

so the wavefunction we need for 1e- isssss

Y(1e-)

A

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.

58
Q

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

A

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.

59
Q

orbital energies are negative meaning

A

the e- is bound to the nucleus

60
Q

why does n=2 have 2 different wavefunction graphs

A

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.

61
Q

radial node:

A

where wavefunction graph passes through 0

62
Q

angular node:

A

the origin of the graph ,, where it starts

63
Q

orbitals with the same principal quantumn number =

A

same energy
they form a shell

64
Q

orbitals with the same n and l values

A

same n so same shell

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

65
Q

shell

A

same n value

66
Q

subshell

A

different l value
same n number

2p and 2s

67
Q

in a shell we should have how many subshells

A

u should have ‘n’ sub shells

68
Q

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

A

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

shell = same n value.

69
Q

ground state §

A

1s

no n = 0 no no no

everything else is an excited state. but with different energy levels of the excited states.

70
Q

are 2s and 2p degenerate

A

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

physical - yr2, s1 (17 decks)

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