Lecture 10 - Fundamental of Statistical Thermodynamics - Mechanical Engg Video Lecture - Mechanical Engineering

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okay ah we all had a good weekend and if
you recall last time we were talking
about the fermi-dirac distribution for
the Einstein distribution and as you say
for this is the average number of
particles in one pond State right so I
certain energy and with a certain
temperature and with certain chemical
potential and we stated that for photons
and photons I thermal equilibrium the
chemical potential mule equals zero
however I caution AO if you do and in
light emitting devices or if you do
solar cells the chemical potential alpha
photons the immediate photons is not
zero and that's because in this case the
photon is interacting with the electrons
holes and you have with the electron
holes that matter at the equilibrium and
that's in this case you actually have a
mu not equal to zero and that's why the
light emitted from a light-emitting
diode or laser can be much higher than
the blackbody radiation intensity and we
and from there once we have this FG
lamber of particles we discuss the
applications essentially what we have
discussed now these quantum mechanics
tells me what's the energy of any matter
I give you write a crystal with the
electron energy levels which the photon
energy levels a molecule atom right in
principle we can solve without the
quantum mechanically allowable ground
state energy there is no temperature
there right and then we add it to the
each quantum
stay the distribution function f right
that's how many so this is the energy of
a quantum state this is how many
particles we have at that quantum state
and then we discuss the density of
states that's a per near that energy
level how many of those quantum states
right so we're doing our country and by
this time what I'm hoping is using so we
show the black body radiation in that
case is H nu and the Bose Einstein
distribution into the state of photons
and you can easily say the blackbody
radiation or based on this expressions
if you need a chair there other chairs
in the front and then if you take so
this gives the energy density right and
if you take a temperature derivative
derivative of that energy density you
get a specific heat although for photons
we seldom talk about specific heat and
that's why see for our full lungs the
lattice we can measure it so we use the
specific heat and at a low temperature
you get a key cubic power and that's
very similar to what a blackbody
radiation
normal T fourth power that's an energy
and the temporal derivative give you T
cubed power right so at the end of the
last lecture I said that I'm going to
tell you I could just stop here and move
to the next chapter but I'm going to use
half of today's lecture to tell you a
statistical foundation because there are
some very useful language I'm not going
to test you as I said but I thought it's
really beautiful so it's good that you
have some you know the terminologies
here so in statistics what we want if I
have example a system like the universe
irani I want to measure its energy or I
want to measure its temperature so what
I have
is a time average I do certain amount of
measurement over a certain period of
time right because instantaneously those
quantities are fluctuating right and so
I one way to find a quality that we
report is the time average but in
statistical mechanics the the framework
is developer most along the in-sample
average there is a time average for
example if you do molecular dynamics
simulation right you actually solve the
equation of motion and you get the
trajectory of the atoms and you see how
to do the averaging and the in sample
average is based on not considering time
history so if I have a system there are
many quantum quantum mechanics say okay
I can go to solve in principle sure the
equation I find that all quantum states
of the system right so I have a quantum
state one and all those quantum states
and this Omega is typically much much
larger than the number of particles you
have in fact it will see soon even one
atom the quantum state say allowable
quantum state in this room from one
molecule will be much much larger than
the total number of molecules in this
room so this is a huge labor Omega right
and the in sample averages actually look
at each of this quantum state the time
evolution you can but right now let's do
not think about time right each quantum
state what's the probability so if you
do a statistic averages is what's the
probability of for observing quantum
state I right so that's the and in that
quantum state what's this quantity
I for example what's the energy of that
quantum state
what's the momentum of that quantum
state microscopic array we're talking
right and if I can I know the
probability and now for that C one C of
the system in my example this is a many
state former ensemble right so if I know
the probability of observing that
quantum state in the example and then
now D is the value that I am interested
in that specific example then my average
this in sample average is then X I P I
and sum over all the quantum state so
this is the concept of in sample average
and that in statistics we do most of the
time we don't actually try to solve the
time history its time history like I say
it's only a tool that was developed out
of a computer which they become very
powerful you solve example like I said
the molecular dynamics right so there is
a very fundamental assumption when these
two different ways of averaging are
equal to each other a system when the
time averaged equals in sample average
that's called a gothic okay so our
Gothic system is when the time averaged
equals in sample average and most
systems are got it
it's a it turns out that starting longer
gothic system is actually active branch
all four sticks or physics right so
a guarding system is the one that after
some time it turns to equitable it
samples all the states okay
and but there is there i interesting
examples of a long or gothic system i
can't help to give you one example and
this example is our framers problem is
FPU problem is a fermi pasta Coulomb so
turns out in 1950s when the computer was
just becoming available and I see this
Fermi pasta Allah the poverty report it
was not a UN paper and what they
reported is they simulated
a one-dimensional atomic H in right and
this spirit is not a harmonic right
before when we started photon we started
harmonic KX F equals KX right and they
added as a higher order term long linear
term right and what that long in your
term this is a day later on were
discussing more is the cause for scatter
if I have a perfect crystal or that
Sprint just to say we have the following
picture there you have a infinite
thermal conductivity because the wave is
not interacting with each other so if I
have a crystal I create a disturbance
here they do not interact they propagate
infinite so a perfect crystal with the
perfect harmonic spring has an infinite
thermal conductivity right and most
material has a finite thermal
conductivity because those Springs are
long Hamas a high higher order terms
that start to interact different modes
that's a scattering that's we're going
to talk
all right so the reason they start a
long harmonic sprint is look at the
scattering in this atomic chain but what
they found is very interesting and they
found that if they put a one mode
putting energy into one mode after some
time the energy of this mode give to
other modes right but after a long time
of simulating so if it's a normal system
we're going to approach equilibrium so
area modes gather some energy right
that's the you product you input energy
in one mode and the you entry equal
partition in the air air mode to get
some energy but what they found is after
some time or energy came back to this
mode okay so what does that imply is
this system is another or got it it
doesn't let work approach equilibrium
and so from that what they imply I'm
very interested in this because they
also imply the this atomic a channel has
the infinite thermal conductivity energy
will ever dissipate right and that was
actually motivation when I got a call
from a friend and Intel said I will look
at the solution for his spreading right
thermal management and I looked I say
okay server management he said the
skinny know is going to Moore's Law is
going come to an end we're going to make
rather than just a shrink they gate but
that we're going to make more products
so I look at my cell phone right if you
look at your iPad it's all metal here
right in some of the old cell phone is
plastic right oh no glass there metals
this is very pretty heavy
well it's actually say iPad is aluminum
in the vacuum these all class
yeah but there should be a medal here
right okay
ah I think I say basically natto is used
to spread out heat they say why your
iPad the back side there's illuminance
spread of the head right so I say can I
turn polymer into a good heat conductor
right so I said oh I remember say a PU
problem maybe polymer chair is a
infinite heat conductor so we add that's
how we actually started someone polymer
work and we were able to turn polymer at
least emitted as good as the most metals
we achieve the 300 times improvement in
the molecular chain direction that's
just a fiber a single molecule we don't
know yet this year this calculation say
it may even be much higher maybe even
divergent we have not been able to
measure it right so this is a think
about this the fundamental aspects
sometimes give you motivation which
directions you want to look at so but
that's just that there are not many
systems that's long or gothic most
systems our organic so we can have the
in-sample average equals time average
and okay so what are with this I'm going
to say there are different samples okay
and this are there I'm going to discuss
three major assembles you can construct
the own example one also depends on the
macroscopic constraint to the solar
system there are three types one is I
give you a system and that's closed
system and I fix the total number of
particles the volume and the energy
right so this example has a fixed amount
of energy fixing on a number particular
and volume and
that's an isolated system right energy
is a constant and I miss it doesn't
interact with the surrounding volume is
constant number of particles it constant
right that's isolated system so it turns
out that the probability so what I'm
looking at is what's the probability of
observing a quantum state of this
macroscopic system which has many
quantum states right so this is a one
type of constraint and the next type of
constraint is I don't do or isolated
system rather than the energy is a
constant I have its temperature is
constant right so I have a n V key and
in this case temperature is constant so
the energy can fluctuate and you can
interact with the surrounding right
environment that Isaac let's say let's
say in ramen is a constant temperature
and the third is a system that I don't
need to fix the number of particles so
you can have mass exchange the particle
change with the surrounding right so if
the particle is in exchange just other
equilibrium when a system is that a
thermal equilibrium temperature is equal
when the system is either mass
equilibrium right the particle
equilibrium the chemical potential is
equal right so I deal with the system
the chemical potential volume
temperature fix the number of particles
of norfolk's right so now for each of
this I can take a system and I make this
make an assemble in this case I make an
example of quantum state I I the quantum
state that has am a fixed lambreaux and
way you write what's the probability of
observing this quantum state
that's the P that's what I want to find
out and similarly for this one is again
our eyes quantum state what's the
probability but in this case I have n I
have way I have T right I do not have
fixed energy and this system may have a
specific energy I rather than a given
energy what's the probability of
observing this quantum states they say
the quantum state in the whole in sample
that I have many many of the such states
right and say means for this right say
this in this case the ice quantum state
can have a eye and an eye at the
particle is also not fixed and I have mu
weighty so those are the questions we
want to answer the probability of P I
right P I and here is AE I and with the
Resta MV t and here is Pei and Ni and of
the rest of the fixed quantity mu and T
so we want to find out what's the
probability and we have specific names
for those three different constraint and
the inside ball form here is
microcanonical okay and here is
canonical canonical and here is a grand
canonical what what what's it what's
them you want to get
okay grand canonical so three different
canonical examples okay and the starting
point is the microcanonical and the
Wells you settle on that micro canonical
you can derive all the distribution
probability for all other canonical
systems so the microcanonical ensemble
and the four so if I have a total of
Omega systems in this example micro
canonical right and the probability for
micro canonical there's no proof it's a
principle is that for such a system
everybody has an equal right so equal
probability so in this case is equal
probability because I have a total of
Omega quantum state each quantum state
is 1 over Omega so this is a equal
probability assumption and the based on
this it was a Boltzmann a/c for such a
system and portsmen principle collected
the number of example in the system is a
direct measure of the entropy of this
system so the roadmap principles say the
entropy of such a system which is a
function of the macro scale
be constrained and the meal you write so
entropy is a function and they yo is kV
log Omega
that's a board's one principle first man
was a very depressed he was a great
scientist and didn't get a right
together reclamation and he actually
jumped in the lake and so if you have a
chance go to Vienna his tombstone is the
s equals K log Omega okay so what do I
say be here the entropy if this one you
have to also take a lot of probability
and by taking logarithm you actually
make the entropy a DD additive quantity
if you have two subsystems the total
entropy of the system is the entropy the
sum of the entropy of each system okay
the brick because if you have two
systems each one has a Omega the total
combination of state is Omega 1 times
Omega 2 so that makes they not give you
the summation okay so ah this the fact
that I have function S is functional
only mu this are called a thermodynamic
function thermodynamic potential
so it turns out for our microcanonical
ensemble the thermodynamic potential is
the entropy if you know entropy as a
function in video you'll know everything
about this system okay but if you know
this as a function of other variables
like a mule or temperature you do not
have that right potential you do not
have all the information okay so why is
that so if I say I know the s of the
system I can find out all other
quantities because if I look at this s
is a function of n vu so I have D ults
deal and I can tape on my end my way
constant a deal right if I run right
into partial D s D way and I write you
an end constant DV plus D s d N and I
have your way constant DN right this is
a just a partial derivatives and if you
go to think about the what you write in
tabular thermodynamics that you write
your equals TDS minus PD way right so
with that I can say really what I have
do D D SD u is 1 over T T oh right
so you find out this function another
temperature and the DSD way is a really
P over T DV so now the pressure and say
DST a is the chemical potential over T
TN and using this you could rewrite it
into your more familiar form this is a
deal equals TDS minus Fe
D way and plus mu DN right this is the
form that you're more familiar so what
it means if I know this function I can
find mule I can find P I can find T of
the system so I know everything about
the system okay but they are typically
we do not do micro canonical ensemble
you are micro canonical ensemble is
harder to work with
and then many times we like to work with
the system at a constant temperature
okay
or if you recall an idea with electron
we say okay I take a one system and that
this there could be electronic electron
out so that's how is write the a Fermi
Dirac was an instant distributions right
so that's actually the ground cannot the
result of Grand canonical example so
let's move on to say if I have here
micro canonical example will be
canonical in sample probability okay in
sample and again I want to find what's a
probability of a quantum system quantum
state that has energy i right but unlike
a micro canonical example which we every
system has equal right this one is no
longer equal right and if we want to use
what we just learned on micro canonical
to derive expression for the probability
in this canonical example what you do is
you include the environment right
because this system is not isolated but
if I include the environment
right here is the environment right and
that all together this is a closed
system okay and when I have one quantum
state of the system itself like the
energy I the environment has many many
other states right so the environment
that will have Omega I state quantum
state so I have to quantize in my
environment because each combination
gives me a system in the microcanonical
ensemble right so if might this Omega T
is the total quantum state of combined
total of combined system plus
environment together right is the system
plus environment together that gives me
the total number of states and then
Omega I is the when this system is at a
one quantum state the ice quantum state
with energy I
what's the number of states of the
environment right so if this is emeriti
in the combined is UT at the total
energy right and the entropy of the
combined emergent system and environment
is st okay and in this case the are
energy because I have to combine the
system plus environment is my isolated
system right so the when the system is
self that AEI the energy of the
environment is UT Maziar the add
together to give me constant UT right so
UT my CI
that's the environment the energy of the
environment so my probability are not so
yeah this is the day this is the state's
condom States of environment right so my
probability of counting of the system I
II III certain temperature and I of
course I can write as a volume C number
density right as my macroscopic
constraint is one that's one quantum
state of the system how many states this
would hold or carb right how much didn't
environment that also have the C when
this is system is at the energy I and
divided by Omega T let's total right so
now I need to do a little math using the
factor is a micro canonical ensemble to
derive what's this probability so let me
say here is omega I I have this energy
but say my real average energy of this
system is you AEI is just a specific
quantum states what's the energy of that
state right so the average energy
because in this case the average energy
can fluctuate there's a constant
temperature energy can fluctuate right
so I can write this u t minus EE I in
terms of U T months U which is the
average energy of the system plus your
mass e I which is deviation your my CI
is deviation from the equilibrium the
average energy right
so that that's just a function I'm going
to rewriting variables and divided by
Omega T okay and I could right now I can
go back to use the board's mind
principle omega and s relation right so
I have Omega for the environment at this
energy and Omega for the total so I can
write the in terms of s so I have really
if you look at it is the inversion so
it's furniture s when the system the
energy is a UT my co plus u my CI this
is just a variable inside divided by Kb
and I have a log already I have
exponents already you will see the
Boltzmann factor and here I have e s of
the total of the system UT 10 KB right
so I'm using the result to turn s and
log Omega and here I'm going to expand
this s I'm going to expand the reason I
write into this is I'm going to expand
this is the equilibrium the average of
the system this is the average of the
environment utmsu and this is a
deviation from that average so I can
write this in terms of equal s equals s
UT my Co plus D s do
and yo Massey I that's a Taylor
expansion right DSD yo given the
microcanonical ensemble DSD is one over
T okay so now if you go one more step
this is a one over T here we have one
over T and a um sei
so what I have then is p/e I really T I
don't need to carry anyway which is the
fixed doesn't change so is a exponential
s yo team SEO and Plus you my CI over T
and I should have KB and here I should
also have KB divided by exponential s yo
T okay right so are the entropy of
average entropy here of the total system
mass the sister this system so this is
really the entropy of the environment
right the entropy of the environment
this is a total entropy so s UT is the s
utmsu that's the entropy of environment
s yo that's the entropy of the system
the additive property of entropy right
so with that this su cancel this one and
what I have
is now exponent sure yo mass st over KB
that's one and KBT and the other is the
exponential mass e I over KBT this is a
I fly I combine using the relation here
okay so this is the probability of a
canonical system clinical example when
one quantum state has an energy I and
that the system is at temperature T
what's the probability right and that
before here is yo is a system average
energy as is the system of average
entropy and temperature so this one is
independent of each quantum state here
is the quantity for each quantum state
and I can write this as exponential my
CI KBT over Z and that's Z 1 over Z is
really this factor here ok
what is the you minus s T yeah that's
the comforts free energy right F F
equals u minus s T so my summer dynamic
potential for canonical system is
actually F is f is au minus s T if I
know F as a function of T and wait I
need to know that function f e is a
function of NV t then I know everything
about this system okay so let's
do a few more steps if if here is my
probability of course I can find out
what's the relation between Z and the e
I because my probability add up of Pei
and T that has to equal sum of all
quantum state must be one mobilisation
right and from here I can say my Z is
really sum up over my CI KBT okay and
because the F equals the Oh minus TS now
I have a Z I have this and I can find I
can also write the frame holds free
energy here is okay so from based on
whether the relation I have here the
thermodynamic potential for canonical
system which is a NV t and if I know
this function which is a log KB T log Z
this is the one I write we write this
one and I have F equals KBT log Z and if
I know this free home free energy I will
be as a function NV t I should know
everything about this system ok so you
can say now from the beauty is that this
in-sample approach was easy once we
start the microcanonical example as a
basic assumption equal probability you
can derive probability for all other
examples
okay and if you happen to do molecular
dynamics simulation and you will see
that at the end there is actually very
similar sometimes most people don't
discuss this is when you simulate a
system
you typically simulate a micro canonical
example you simulate the system with a
fixed number alpha particle and total
energy and in fact that you spend a lot
of time try to maintain your energy
constant otherwise your simulation blow
up it's not stable but when you do
analysis you actually go this is a as I
mentioned before you have to go through
like a chapter 10 on my book
and you actually use the canonical
example result and board my statistics
this is the board's mind factor here we
mentioned before right so there is
inherently constants I say inconsistency
in this approach and the people did
discuss the ways to treat it
in fact one way for example is to add a
thermostat to the molecular dynamics
simulation and there are different ways
to add a thermostat so you can actually
you add an additional equation of motion
to simulate to the environment so that I
the whole you can still be
self-consistency so there are different
thermostat constant the temperature
thermostat are there there you and
constant pressure times thermostat but
if you go to check all this paper
actually invariably come to those basic
how you can go from once and sample
result to another in sample results okay
so I spend a lot of time in talking
going from macro Chronicle to canonical
right the next one is
here is closed system by constant
temperature so energy changing
environment now if you have an open
system and at constant temperature there
is a both energy and particle exchange
with the environment and you do the same
thing so you will find the next
probability I'm not going to derive it
now the grand canonical and the
probability of a quantum system as a
either energy I has the N patent i
particle is a exponential mass e i- mule
and i KBT that's what why we use that
before and divided by another partition
function this one this Z is a coded
partition function and same Z there
partition function and see the factor
here this is what I started I say this
is a Gibbs vector right that's how we
derive from a Dirac Einstein
distribution we started with this
probability and we say localization that
will give me the partition function
there right the normalization factor I
use it to use the symbol a but here is
symbol Z and same say here so this is a
board demand factor for canonical - they
gives factor for grand canonical for
open system right so this is essentially
what I say is the statistical foundation
and the beautiful mathematical treatment
once you start with micro canonical
and it's pretty it's a deep concept so
I'm not asking you to and I said I'm not
going to test you but I wanted you to
know the terminology okay now let's look
I look at a few examples how we use this
let's apply this to few problems and I
discuss electron I just got fallen from
in Iraq for the Einstein and now let's
go back to gas molecule
right if you recording in chapter 1 we
discussed the Maxwell distribution
velocity we say okay the voila the
energy of a molecule is one-half MV x
squared plus y squared plus VZ square
and but you could also do quantum
partying box solution right we solve
this when you have a partner in a box
and that is H bar square 2 m KX square
plus KY square plus KZ square you can do
either condom or our classic or I'm
going to actually use the quantum
notation here right this is essentially
the HK gives you momentum right P
squared 2 m that's the quantum the
quantum here and of course given the
energy I can go to look at that this is
a system at a constant temperature so I
can use the canonical in-sample and the
partition function for one particle for
one particle is z1 right and that will
be summing up all possible KX KY KZ K X
K Y KZ as a sum of all energy levels as
whether we were showing the Z where did
we see
right Z is summing up all energy quantum
states right so I sum up all common
states and the is exponential e I which
is the function of K X K Y KZ K BT right
so this is my partition function Z and
and if I if I want to know what's the
average energy energy I have my average
energy e is just that they again its
summation I can write a KX KY KZ rather
than three summation symbols and that
this is the e is this combination of it
for each state was how much energy and
the probability are for the state ID of
this quantum state right that
probability is what I have here a i kb t
over Z right so I have the probability
expression I have the energy expression
I can go to find that average but there
is a simpler way rather than doing this
summation and you can actually prove
easily if I take a derivative of the of
this Z you will say this one is actually
is equals T log of Z 1 dy or y equals K
BT that's just a mathematical result and
for convenience I if I find this I can
just take the derivative of Z 1
okay right if you say because if I do
derivative I will log I say this is a in
the denominator that gives me Z and the
numerator gives me instance your games
with me probability and if I take a KBT
as a 1 over this should be that Y should
be quick now I think actually dy should
give me Y should be 1 over KBT rather
than KBT ok so if you do that you'll get
an e in the front that's a in the front
okay
so ah that means I do not need to
actually do this summation if I just
find the z1 my problem is solved and
that to do z1 I convert this summation
if this one doesn't change rapidly right
I do my summation into integration using
density of states if you change the
rapidly I cannot do that right so if
it's very the states of a far apart or a
KT I cannot do that integration to
summation summation to integration so
now if because this states are closely
spaced I can do it so you'll find out Z
1 from this I can actually do my
integration this exponential e KBT our
density of states de and 0 to infinite
this density of states for particle in
box given this dispersion energy with
vector relation you find that that is a
way for pi we we did this before so I'm
not going to redraw it h square 2m h
three half we did in the context of
electron but it could be the same for
molecule okay right so you substitute
the ten subspace energy to into it and
you find out this one equals way the
volume divided by lambda cubed well this
lambda cubic is lambda itself non lambda
cubed M by itself h squared 2 pi M KB T
so this is your R lambda and this lambda
is called the thermal de Broglie
wavelength okay it's a thermal de
boracay wavelength the Broglie
wavelength why why is the customer de
Broglie wavelength basically if you
think about your energy is KBT right you
you go from that energy to momentum
because KB T 1/2 KBT energy equals
momentum P squared divided by 2m right
from momentum you go converted into
wavelength that's what you'll get okay
let's call thermal the Bragg wavelength
so from Z if I go into here because Y is
1 over KBT so take a log and they take a
derivative you find the average energy
of this equals mass D log z dy and the
equals three-halves KBT that's a result
we found
in when we did they bought a Maxwell
distribution ok came back to the same
result but through a different this is a
more stringent statistical approach
foundation there and it says the average
energy of a particle is the three half
KBT and recall we commented the u-clip
partition right if you have quadratic
atomic energy here is one-half MV x
squared or KX squared that's quadratic
term right and if this energy do not
separate the far apart at high
temperature and each mode contribute the
one-half KBG that's the equal partition
so I have three modes here as three half
KBT okay and of course from here the
specific heat of one molecule is D Y D
DT so three half KB right
specific heat of one molecule is three
half KB and then if it's one more right
is one more is three half KB times
Avogadro's constant and this to gives
you the universal gas constant right
that's what do you know from
thermodynamics is the three half are
right but you say no that's not all
three half that's only monotonic gas
what about diatomic gas right what who
remember that was the specificity of
that humming gasp
ah
v right v hops are so why is that okay
so let's go to if I have in general okay
and this is only one molecule let me add
one more C common that when I have
dilute particles right here I have
before that was one molecule if I have n
molecules right and the energy right if
my energy is say summation of air or the
molecule AI right
and the KBT okay and the question is how
do this summation right if it's dilute
my molecules that means that the the
dilute means really the energy levels of
those molecule do not affect each other
so my individual molecule solution is
pretty much valid and all the molecule
have the same all those energy levels
right so in that case I will have this
is for each quantum state identical
quantum state I I have n molecules so I
have exposure Z 1 to the nth power okay
but there is a one trick that is the are
the molecules are indistinguishable if
you do your combination if it's in the
screen think which were objects what do
you do
the count the probability of continent u
so u divided by n factorial this is the
right so that today if you the real if
you indistinguishable but when when they
are is a dilute right so when I see that
load I say the molecule do not affect
each other energy levels right and when
I sit down load one condition I can use
the loot is the number of states of one
molecule in the box is much larger than
the number of particles the number of
quantum state of one molecule right it
is much larger than the number of
particles so the the particle do not
influence each other and so the number
of states is a really kxkykz
times 1 right each combination of kxkykz
gives me one state so i have the density
of states integrated over say all energy
and the energy is zero I can say roughly
in the three because the average energy
is three half KBT and that gives me much
larger than the total number of
molecules and this actually gives me the
condition if you do your math again we
have things upstairs MV PI over six and
one-half lambda cubic much less than 1
so that's them mathematics what it means
that they say lambda cubic lambda is the
blog of wavelets right so we divide by n
if I'm overweight
oh this side that's a volume of one
particle it is a much larger than the
thermal de Broglie wavelengths so the
wave of issue atom do not over web lap
with the other atoms right so that's the
part of your a dilute that loop
particles okay so how I go from here to
three fives KBT and if I have the total
energy here we only consider the
translation so if I have the energy of
one particle GI right here we have a
translation plus you can also have
rotation of the particle you can have
the vibration of the particle right
and the electronic so the total
partition function for n molecules is
really the of the here we have
translation and rotation and vibration
and the electronic partition function
each of this to the nth power and
divided by n factorial okay and from
this en you go to take the average
energy once you find that Z you take an
average energy take a temporal
derivative right so it really depends on
at the end a whole it because it's a
board manufactory I write my electronic
energy is a few electron volts okay
let's think about this what's the
specific heat right electron say energy
level this is a simple hydrogen model
this is a minus thirteen point six eBay
right this is a man
13 point 6 divided by 4 because the N
square that's a big gap here right so
the electronic energy normally I'm not
excited because the e I over KBT is a
huge number and this leg here right
so electrons do not go to a higher
energy level and they do not contribute
to specific heat a room temperature
because there will have your change
their energy they're still at the ground
state right and vibration energy level
what do you know about the vibration
frequency of hydrogen if you think about
the gas atoms Faragher absorption
spectrum right and in the infrared a few
micron you've got a lot of lines right
those lines are the vibrational energy
levels vibration plus rotation rotation
is a small energy vibration is is
actually this is a mini rotation
attached to vibration
okay so vibration is a few micron that
means about 0.2 0.3 to 0.5 electron
volts right few micron one micron is 1.2
for the electron volt right
so if it's 2 micron is 0.6 right 4
micron point 3 electron volts ok and
Katie is 26 and milli electron volt
right
26 and milli electron was so when you
have this is the point 3 divided by 0.3
that exponential minus 10 so low more
vibration i room-temperature not excited
okay and rotation turns out are very low
temperature just a rotation
okay the energy needed to rotate the
object and that's equivalent temperature
okay
it's about 80 Kelvin so I
room-temperature rotation the molecule
rotational modes or rotational modes are
excited and if you think about more
rotation is the two degree of freedom
right see the Phi right the energy is
the one half Omega I square now one half
I Omega square the quadratic equal
partition say there will be another two
times one half KT so if you go to look
at a specific heat of molecules a gas
it's actually not just the step function
and really depends on what the
temperature range you're dealing with
and specific heat of constant volume
specific heat is the temperature and the
translation is easily excited motion
translational so this is the
three-halves r universal gas constant or
more okay and around the 80 Kelvin the
rotations start to kick in so there is a
transition range and when you go to
higher temperature all the rotational
energy are fully excited equal partition
began to be valid it's not a sharp
transition there so here you get five to
sir are you that's where you say when
you do your gas calculation you see this
specific is the five five-halves okay
and when you go to thousands of degrees
you add another vibration
is one-half MV squared plus one-half KX
squared so two quadratic terms you add
another two times one-half KBD you can
polish it but this will require you go
to thousand degree right so that for gas
you can go to it's a diatomic gas go to
seven half right if you have three more
atoms you have to look more of those
different modes per molecule okay so
that's the discussion on specific heat
and I think what's really for you
probably most useful is understanding
gas specific heat and the previous I
spend most of today's lecture really
discuss a different example I just want
you to know their treatment this is the
statistical foundation or Goudy city and
in sample average and a different
example from microcanonical to canonical
and grand canonical examples and just
briefly where we're going now if you
look how the water we have discussed so
far I repeat this before what we know
now is the energy of atom of solid
molecules that's a ground state energy
quantum mechanical solution what other
laudable states of the system and I have
day for each quantum state at the
equilibrium I can tell what's the
average number of particles and I can
count how many states and using the
concept of density of states right so
and of course in my star point is really
sum up all the discrete states but this
summation
times I can use the tens of states and
of course the condition for that is that
the separation of energy levels a lot
for our part compared to KBT so at
between two adjacent level I can
actually convert that summation using
integration approximation right
summation is actually accurate
integration is approximation it's a
reverse typically when we do integration
we do discrete salvation right and see
the other comment is on quantum systems
and that summation becomes sometimes
invalid because the separation is large
so I want to show you we discuss on
electrons before and I'm going to show a
few slides on the full on specific heat
if you have nano structures what will
happen okay so this was actually a
student mine and I took two meanwhile
two to actually really appreciate it in
fact in my original concept that was not
quite right so it's a density of states
on machine if you want to get a specific
heat of material you sum up within sub
state so if you have a slab right a wire
a slab and the 3d if the in one
direction the distance is very small
they say you can you can look after your
energy levels and they start with
separate and as you can say they dense
up state alpha 1d and 2d and 3d began as
a 3d for a Debye model that's a omega
square right and 2d
and steps and 1d spikes this is a
symmetry fault for electron their
similar spike 1d and 2d we already say
it's a staircase right here's the steps
for electron and fourfold ah it's pretty
much similar except the e if you average
them out that you can follow this Omega
square controller okay so you feel add
this together you get the specific heat
so this was a measurement that we did on
titanium dioxide nanotubes and you can't
circular modes and the spacing between
the layers and you integrate you get a
which they say specific heat trend for
different say 3 D 2 D 1 D structures ok
and there was a experiment this was a
measurement and specific heat if you
really want to say for phalaris you want
to say quantization in fact you have to
go to very low temperature and you can
say this is the most of the case 3 is a
pretty good go to very low temperatures
say the deviations
okay and let me see whether in fact they
are not any really much measurement on
the electronics specific heat quantum
structures let me see whether there are
any is this know
so the the for electronics structure
actually calculating specific heat is
not easy because remember we see as you
have to calculate the chemical potential
because it's a fermi dirac system right
and chemical potential is calculated by
the number of particles equals to the
modes that's the one and the spin so how
many Oh another one is the Fermi Dirac
distribution because this is a how many
quantum state each quantum state how
many particles you have so and then
combine with the energy both F content
chemical potential and temperature so
when this is a discrete it's not easy
you have two equations you have to solve
and that's why it's a little complex for
electronic specific heat in quantum
systems okay football K is 1 over si
they just a specific increase with T
right okay so what are we going to go
laxed is we know how to calculate
equilibrium quantities and the heat
storage for example and we're like
looking into the transport process and
then we need to find out lung
equilibrium so if I have point one and
point two and either different
temperature or different chemical
potential what are the mass flux what's
the heat flux so I have not discussed we
have not discussed this quantity is say
going from one to two what's the
probability that's a transmission and
also what's the velocity you know in
fact I have shaded away I have not
really talked about that what's the
speed of the energy moving right this is
speed of Nye the speed of sound but the
when we discuss the speed of energy
carrier you have to actually think about
the energy packet and the group velocity
and so in the next chapter next lecture
we're going
to start of philosophies will first
discuss a wave when goes through the
interface there's reflection that's how
we calculate transmission right and then
we'll discuss the wave as a packet the
group velocity and then we go transit
from the wave picture to see you in
surely we neglect the phase carried by
the wave so that's a little extra few
lecture will focus on the waves and
that's pretty much it so far we've been
really building a foundation based on
waves so go from wave you make sure the
forgetting wave because that's where the
diffusion picture start to become valid
you