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Chapter 3.2 Bohr Model
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the topic of this video is the bohr model
model
the learning objectives are on the
screen so before niels bohr
developed what we now refer to as the
bohr model uh
there was no application of quantized
energy states
to build the model of
of an atom such as the hydrogen atom
there is no
uh way to explain how there could be
discrete energy states there
so what the physicist kneels bohr
proposed is that the energy quantization
that we observe in different experiments
like the photoelectric effect
and also in black body radiation
was a result of discrete energy states
that are in orbit around a nucleus so
what niels bohr
derived independently of rydberg
is an equation that looks like this one over
over
lambda which is wavelength is equal to lowercase
lowercase
k this is a constant that i will give
the numerical value for in a second
that constant is divided by planck's
constant multiplied
by the speed of light another constant
and as i write this out it should
in that it has the overall
same form as the rydberg equation that i
previously discussed
the value for the lowercase k here this
particular constant
is 2.179
times 10 to the negative 18
joules okay so
uh the bohr model or niels bohr
in in developing the model came up with
this equation
using first principles and
what ended up helping get the
attention of other scientists at the
time was the fact that
niels bohr had um come up with a
an independent way of calculating what
was known as the
redbird constant and we still refer to
it as the rydberg constant
and the rydberg constant had been
experimentally measured
numerous times it was a very well known value
value
um and so the fact that niels bohr came
up with a way of using
um fundamental constants to just to to
calculate the
rydberg constant um was considered a
major success
and brought a lot of attention to the
bohr model so what does the bohr model uh
uh
look like essentially we have
a positively charged nucleus and there are
are
orbits around the nucleus
in this particular case of the hydrogen atom
atom
um maybe i'll draw one more orbit
these orbits as you get further and
further away from the nucleus
become higher and higher in energy so to
distinguish orbits from another
one another we give them numerical
integer values
one two and three the first
numerical value n equals one this is
what's referred to as the ground state
this is the lowest possible energy state of
of
an electron orbit now what can happen is
you can excite an electron from the
ground state up into an excited state
and when this happens it happens at a
very discrete energy
that that is the exact energy difference
between the n equals two
excited state and the n equals one
ground state so
anything above n equals one
that is these electrons in these uh
orbits that are further and further away
from the nucleus are at higher energies
they are excited relative to the ground state
state
okay so this can explain line spectra
and how might it explain
line spectra or specifically the line
um rather than just arbitrarily
assigning integer values like the
rydberg equation did it just you know
the rydberg equation just said okay well
if we just
use integer values not really knowing
what those integer values correspond to
we can recreate the the line spectrum of
the hydrogen atom
but what the bohr model says and what
bohr said is well those integer values
do have a meaning
they have a meaning of discrete
quantized energy states available to electrons
electrons
in orbit around a nucleus
so if we look over here in the case
where electrons move to higher energy as
light is absorbed this is the case where
we start with an electron down here at
the ground state n equals one
light comes in and excites an electron
to make and the electron can jump up
it can jump up to the n equals two state
or it could
skip the n equals two state if the
energy is high enough and it can go to
the three
four five in theory up infinitely
many states uh uh
but at some point the electron will escape
escape
the the the atom but that's a
we don't need to go there yet um it's
also important to note that you can also
have excitations uh from not just the
ground state but
in excited state to other excited states okay
okay
and these are very discreet energy jumps now
now
when we think about the line spectrum
that is light that is emitted
not light that was absorbed so in the
line spectrum of hydrogen what we see
are discrete energy bands and that
is because the um electrons have been
excited by electricity
uh electrical energy up to a higher
energy state
and then they can relax back down
so when the energy when the electron
is in an excited state and drops back
down to the ground state
light is emitted and that light is
emitted at an energy that corresponds to
the difference
between the excited state and whatever
state it relaxed to so you can have
a an electron go from any excited state
directly to the ground state
or you can also have these processes
where an excited state
will relax down to another excited state
and again this will release energy
that corresponds to the energy
associated with that transition
the other thing to point out here is an equation
equation
um that sort of uh captures this process
so if we want to ever consider
the the change in energy associated
between these transitions all we have to
do is
uh consider this delta e this is just
sort of a reworking
of the equation i already put up is
equal to
that k constant that i mentioned uh
multiplied by the this whole term which
is 1
over n 1 squared minus 1 over
n2 squared and this also by the way
since this is an energy value is equal
to hc
over lambda okay so let's go ahead and
put this uh
to to practice and i'm going to scroll
over to a practice problem
that i have here so go ahead and pause
the video now and you can write this out
in your notes if you need to
and i will go ahead and start solving it
okay so
what we need to do is find the energy in
joules and the wavelength in meters
of the line in the spectrum of hydrogen
that represents the movement of an
electron from the bohr orbit with n
equals four to the bohr orbit with n
equals six so we're going from
an excited state of n equals four to an
even higher excited state of n equals six
six
so um the uh since we're start our
starting point
is n equals four this will be our n1 and
our ending point
and two is is going up to that n equals 6
6
level so i'm going to just go ahead and
start by writing
let's calculate the the change in energy
for this jump from n equals 4 to n
equals 6.
so this is going to be equal to k times
1 over n 1 squared minus 1 over
n 2 squared so this is going to be equal
to 2.179
times 10 to the negative 18 joules
and this is uh n one i said it's going
to be four so that's four
squared minus um one over n two squared
so this is going to be six
squared so uh this will give us
2.179 times 10 to the negative
18 joules
multiplied by 1 over 16 minus 1 over
36 okay and if you do this and i
encourage you to practice
doing this in your calculator you should
get an energy value of seven point five
times ten to the negative twentieth
joules so this answer does make sense
just based off of the order of magnitude
of that scientific notation it's a very
small energy value
um so now the that's the first part of
the question uh the second part is
what is the wavelength in meters and in
what part of the electromagnetic
spectrum do we find this radiation
so let's go ahead and use the
relationship between
energy wavelength
planck's constant and the speed of light
so we know that
wavelength is equal to hc
so this is going to be equal to planck's
constant 6.626
times 10 to the negative 34 joule seconds
seconds
multiplied by the speed of light 2.998
times 10 to the 8th
meters per second all over this
energy value that we just calculated
five six six
times ten to the negative twenty joules
always do a unit check to make sure this
is cancel out joules we can sell
joules seconds will cancel out with
seconds we'll be left with units of
meters which is exactly what we want
and we should come up with a value of 2.626
2.626
times 10 to the negative 6 meters
that's our answer now if we look back to what
what
uh a figure that we uh that that shows
different parts of the electromagnetic
spectrum we'll see which region we find
this value in so we're at about 2.6
times 10 to the negative sixth meters
so this is a nice figure showing
different parts of the electro
electromagnetic spectrum we are at about
right here we are at about an order of
magnitude of 10 to the negative 6
meters for our wavelength so this puts us
us
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