Professor Ramamurti
Shankar: Alright class,
welcome back.
This is our last two weeks.
We're going to have a slightly
different schedule for the
problem sets.
I'm going to assign something
today which is due next
Wednesday.
I'm giving enough time so you
can plan your moves.
Then, I will probably give you
one last problem set with two or
three problems on whatever I do
near the end.
We'll have to play it by ear.
Okay, so this is another new
topic on thermodynamics,
a fresh beginning for those who
want a fresh beginning.
And there's also stuff you
probably have seen in high
school, some of it at least.
So, the whole next four
lectures are devoted to the
study of heat,
temperature,
heat transfer,
things like that.
So, we are going to start with
the intuitive definition of
temperature everybody has.
So, hang on to that;
that's the right intuition.
But as physicists,
of course, we want to be more
precise, more careful.
So, let's say you have the
notion of hot and cold.
Even that requires a little
more precision.
That introduces the notion of
what is called thermodynamic
equilibrium.
Just like mechanical
equilibrium, this is a very
important concept.
So, I'll tell you what
equilibrium is with a concrete
example.
If you take a cup of hot water,
and you take another cup of
cold water, each cup,
if you waited sufficiently
long, is said to be in a state
of equilibrium as long as the
cups were isolated from the
outside world and not allowed to
cool down or heat up.
We think they maintain a
certain temperature.
We say it's in a state of
thermal equilibrium because this
temperature does not seem to
change.
Now, we have not defined what
temperature it is precisely,
but we can talk about whether
whatever it is has changed or
not changed.
So, it will settle down to some
temperature and it will maintain
the temperature.
Be very careful.
If you leave a cup of coffee in
this room, it will cool down
because the room has got a
different temperature.
But I'm talking about a cup of
coffee that's been isolated from
everything;
it maintains the temperature.
Here's another cup of cold
drink at what we feel is a lower
temperature.
They are both in a state of
equilibrium.
Equilibrium is when the
macroscopic properties of the
system have stopped changing.
If you now pour one of these
cups into the other one,
that's going to be a period
when the system is not in
equilibrium in the sense that it
doesn't have a well-defined
temperature.
For example,
if you just poured it from the
top, the hot stuff is on the
top,
the cold stuff is on the
bottom, there's a period of
transition when you really
cannot even say what the
temperature of the mixture is.
Some parts are hot,
some parts are cold;
that system doesn't have a
temperature.
But if you wait long enough
until the two parts have gotten
to know each other,
they will turn into some
undrinkable mess,
but the nice thing is it will
have a well-defined temperature.
That's, again,
a system in equilibrium.
So, you've got to understand
that temperature and thermal
equilibrium represent gross
macroscopic properties and
they're not always defined.
At the microscopic level --
it's no secret -- we all know
everything is made of atoms and
molecules.
The atoms and molecules that
form the liquid or the gas
always have well-defined states.
Each molecule has a certain
location, certain velocity.
But at a macroscopic level,
when you don't look into the
fine details,
focus on a few things like
temperature,
they don't always have a
well-defined value;
that's what you've got to
understand.
Things have a well-defined
value when they have settled
down.
How long does it take to settle
down?
That's a matter of what system
you're studying.
But generally,
you can all tell when it has
settled down.
Here's another example.
Suppose you take a gas and you
put it inside this piston here,
put some gas inside,
you put some weights,
and everything is in
equilibrium.
We say that it's in equilibrium
because the macroscopic things,
things you can see with your
naked eye, nothing is changing.
It's going to just sit there.
But if you suddenly now remove,
say, a third of the weights,
the piston's going to rise up,
shake around a little bit,
maybe settle down in a new
location.
If you wait a few seconds,
then the new location will
again settle down,
and you won't see anything with
the naked eye that looks like
anything is happening.
In between you will see the
pistons moving,
the gas is turbulent,
the pressure is high in some
regions, low in some regions,
then it settles down.
This is the notion of systems
in equilibrium,
and in between,
there are states of the system
which are not in equilibrium.
Now, whenever a system is in
such equilibrium,
we can assign to it a
temperature that we call
T.
Right now, we don't know
anything about this temperature,
so we're going to build it up
from scratch--other than your
instinctive feeling for what
temperature is.
One of the laws of
thermodynamics is called a
zeroth law --zeroth law because
they wrote down the first law,
then they went back and had an
idea which was even more
profound, and they said,
"We'll call it zeroth law."
Zeroth law says,
"if a and b are
at the same temperature,
and b and c are
at the same temperature,
then a and c are
at the same temperature."
Now, I see disbelief in the
audience today.
Why do you call this a law?
Look, I think that is the key
to our being able to speak about
temperature globally,
is the assumption that if I
take a thermometer and measure
something there,
and I come back and dip the
thermometer here,
and it reads the same number,
then I may conclude these two
entities, which never met each
other directly,
are also the same temperature.
That's not--That seems pretty
obvious to you,
but the whole notion of
temperature is predicated on the
fact that you can define an
attribute called temperature
that can be globally compared
between two systems that never
met directly,
but met a third system.
Okay.
So, once we have some idea of
hot and cold,
let us decide now to be more
quantitative.
It's like saying,
you know, somebody's tall and
short is not enough.
We go into how tall,
how many feet,
how many inches,
how many millimeters.
So, we want to get quantitative.
All we have right now is a
notion of hot and cold.
So, what we try to do is to
find some way to be more precise
about how hot and how cold.
So, what people said is,
"Let's look at some things in
the world that seem to depend on
temperature."
One thing that seems to depend
on temperature is the following.
You take this meter stick in
the National Bureau Standards,
kept in some glass case,
at some temperature.
You pull it out--or make a
duplicate of it,
you pull it outside and leave
it in the room.
What you may find is that if
the room was hotter than the
glass case, this rod then
expands to a new length.
So, one rod is outside the
case, one rod is inside the case
so the comparison is meaningful.
Nothing has been done to this
guy in the air-conditioned glass
case, but this one is expanding.
So, one way to define
temperature is to simply ask how
long is this rod,
and somehow correlate the
length of the rod with
temperature by some fashion.
So, you can do that.
So, what you need to do
that--what you need to do first
is to define,
put some markings on it so that
for each extra something it
grows,
we can say the temperature has
gone up by some amount.
So, there we need units for
temperature, that's completely
arbitrary.
And you need some standards,
just like this meter stick,
you know, it's not--nothing
intrinsic in nature about a
meter,
we just made it up and said
"Let's call that a meter."
In the case of the meter,
the zeroth law is if you bring
a meter stick next to mine and
we agree,
you can take the meter stick
somewhere else and define that
to be the meter because if this
stick is as long as that one and
as long as that one,
then those two are equal in
length.
But temperature--You are
similarly going to use this rod
and say, "This rod is a certain
length when kept on top of this
bucket of some fluid and the
same length when I keep it on
that bucket,
then the two buckets are the
same temperature."
So, we can use markings on this
rod compared to the unexpanded
length as a measure of
temperature.
So, what people do is to pick
something a little easier than
this rod.
They notice the liquids expand
when you heat them.
That's why in a summer day if
you fill your gas tank,
you have to leave some room at
the top so the overflow can come
out of the top;
or you shouldn't fill it
completely, otherwise it'll bust
the tank.
So, liquids expand.
So, one way to measure
temperature may be take some
liquid, put it there,
and then put it in hot rooms
and maybe watch the liquid
expand to the new height.
And then draw some markings,
and each marking can be a
certain temperature.
But people had a better idea
than this one.
They had the following idea of
a thermometer,
where you have a lot of fluid
in a reservoir,
a very thin tube evacuated at
the top, and the fluid,
then, is here.
So, what's clever about this is
that if this expands by one
percent, your eyes should be
good enough to see one percent
increase in height.
If this fluid expands by one
percent in volume,
that one percent in volume and
it climbs up this narrow tube
can climb to quite a bit
[pointing to picture],
because the extra volume you
get by expansion will be the
area of this tube times the
extra ∆x by which it expands.
So, you're magnifying the
expansion by making all the
expander fluid climb up this
extremely narrow tube.
In fact, the tube is so narrow,
you cannot probably even see it
well, which is why they have a
little prism that magnifies the
mercury or alcohol in the
thermometer.
Okay, so we have some way of
following temperature now.
We can draw some lines,
arbitrary lines,
it doesn't matter.
That can be zero,
that can be five,
that can be 19;
you've just got to make sure
that it's monotonic.
Then whenever it's on 21,
we may argue that 21 is now
hotter than 19.
But you want a better scale
than that.
Even though that's
mathematically adequate in
practice, what people decide is
to do it as follows.
They said, "We want to set up
thermometers so that people all
over the world,
in different parts of the
world, different countries,
different labs can all agree.
So, we will make it possible
for everyone to make their own
thermometer by the following
recipe."
We will dip this guy in a
bucket which has got some ice
and some water.
That's called the melting point
of water, so that--or the
freezing point of water;
melting point of ice,
or freezing point of water,
it doesn't matter.
We notice that as water cools
down, in the world around us
suddenly ice cubes begin to
form.
We go to the temperature at
which that happens for the first
time and we dip the thermometer
there,
and whatever reading we get we
will postulate to be zero
degrees centigrade.
That is just a definition.
We believe that's a good
definition because people all
over the world can do that.
Of course, if you live in
Kuwait, that's not going to work
for you;
there's no ice.
But they figured out in parts
of the world where you have ice,
this is a very good definition.
You get ice,
you got zero degrees.
Then they said,
"Let's find another universally
accessible thing," which,
as you all know,
is the boiling point of water.
If you put water on the stove
it heats up and heats up and
heats up and suddenly it begins
to bubble and boil and
evaporate.
That temperature is going to be
called 100 degrees,
100 degrees centigrade.
Then, you take this column
between zero and 100,
and you divide it into 100
equal parts.
And that is postulated to be
the temperature anywhere between
zero and 100.
If you have gone 79 percent of
the way to the top,
from here to here,
the temperature is 79 degrees.
That's how the degrees were
introduced, and that's a
centigrade scale,
and you guys know there are
different scales.
You can have the Fahrenheit
scale, you can have any other
scale in which what you want to
call the freezing point is
different.
Somebody thinks it's zero,
somebody thinks it's 32.
And you can again call this
something else,
and you can divide this
interval into 100 parts,
180 parts, whatever you like.
But the philosophy is the same.
You have to find two points,
which are reproducible,
conveniently,
and divide the region between
them into some number of equal
steps.
If there's 100 equal steps,
you say it's a centigrade
scale, provided the lowest one
is called zero.
This is how you have
thermometers.
Now, there are some problems
with this.
One problem is that the boiling
point of water does not seem to
be very reliable.
Because if you boil water in
Aspen, for example,
you know it doesn't seem to
boil--it seems to boil more
readily than in the plains.
You can ask,
"How do you know that?"
maybe it is still doing the
same thing.
I know that because I tried to
cook something,
cook some rice and vegetables,
I find, they don't cook at all.
In Denver, it boils before it
cooks;
that way we know it's probably
boiling earlier in the mountains
than in the plains.
So, who's going to decide what
the real temperature is?
So, you have to be more careful
when you say boiling point and
freezing point,
because things don't seem to
boil at a certain,
predictable and fixed
temperature.
This is a very deep argument I
have never appreciated fully
when I was learning the subject,
is that it's all cyclic
definition.
Because you may not know that
the temperature is changing,
because this thermometer by
postulate, it's going to be the
temperature by definition.
How can it be wrong?
What's wrong is that you know
it's not a reliable method
because physical phenomena,
like when your rice will cook,
are not reproduced by the
boiling point of water.
It cooks in the plains,
it doesn't cook in the
mountains, so we know the
boiling point is to blame.
Rice is the rice.
That's how we know that that's
not a good measure.
So nowadays,
people have much fancier
measures, and I will tell you a
little bit about that.
But for a long time,
this was a very good start.
Don't worry about the fact that
water boils differently at
different altitudes;
you could go to sea level and
that's a good enough definition.
Sea level is pretty much
constant all over the world,
and you can say the pressure of
sea level is the pressure at sea
level;
just the ρgh of the
atmosphere.
Okay, so that's the usual
definition of temperature.
Now, the trouble started when
people realized that if you make
a thermometer with your favorite
fluid,
maybe mercury,
and I make one with alcohol,
they will agree at zero and
they will agree at 100 because
that's how you fixed it.
You rigged it so at zero
everyone says zero;
100 everyone says 100.
But how about 74 degrees,
or 75 degrees?
I say it's 75 if my fluid has
climbed three-fourths of the way
to the top.
At that point,
yours may not have climbed
three-fourths of the way.
In other words,
you've got two things,
two graphs, which have zero and
100 degrees;
one graph may be like this,
one may be like that.
So that when I think it is 75,
you may think it is 72.
At 100, we will agree because
we have cooked it up that way.
In other words,
it's not true that all liquids
expand at the same rate.
So, you will have to then pick
one liquid and say,
"We swear by that liquid,
and when that liquid's gone
halfway, we'll say it's 50
degrees."
So, you will have to pick a
liquid, you'll have to have an
international convention,
you know, there's the alcohol
lobby and there's the no alcohol
lobby;
they argue.
Finally, they found out a much
better solution than these
liquids.
They found out that if you use
a gas--You can define
temperature using gasses,
which have some very,
very nice properties.
And this is the gas thermometer
that I'm going to tell you now.
So, here is how you build a gas
thermometer.
You take some gas in a
container.
A typical container for me in
all--whenever I draw anything
thermodynamics,
it's going to be gas inside
some cylinder with some weights
on it,
and that defines the pressure
of the gas.
Of course, the pressure will be
the mg of these weights
divided by the area of the
cylinder.
That's the pressure,
plus atmospheric pressure.
And the volume is this,
whatever the volume is,
base times height.
Here's what we ask you to do.
Take the product of pressure
times volume for any sample of
gas.
Take some gas,
put it in this tank,
and now put it on different
surfaces, like a hot plate,
like a stove,
like a tub of water,
and measure the temperature
using some standard method up to
that point like a mercury
thermometer.
What you notice is that the
temperature measured by some
reasonable scheme shows that the
product of P times
V lies on a straight line
[drawing diagonal line on
board].
If you connect the dots,
you find the product PV
is linear in this temperature
variable.
And this is zero degrees,
and this is 100 degrees.
Now, here is the beauty of the
gas thermometer.
If you take a different gas and
you put a different amount of
different gas in a different
cylinder, you will get some
other graph;
it may look like this [drawing
another straight line].
For you, that is zero and
that's 100.
But the most important thing is
that's also a straight line.
That it's also a straight line,
has the following implication
you guys can prove at your own
leisure,
which is that,
if I think that my gas has
climbed 56 percent of the way of
this height to the top,
so the temperature is 56
degrees, I ask what's your gas
done, you will find yours also
climbed 56 percent of the way.
It's the property of straight
lines.
You can show that if you took
two straight lines,
whatever be their slope,
if they agree,
if this is zero and this is
100, it has got a different
slope, when you have climbed to
the halfway point,
draw a line at 50 degrees and
ask what has any gas done,
they will all have climbed to
the halfway point from the zero
point to the 100 point.
In other words,
gas thermometers will not only
agree at the end points where
they must, by construction,
they seem to agree all the way
in between.
But there is one requirement.
This gas has to be very dilute.
The more diluted it is,
the better it comes out.
So, take neon or Freon or
whatever you like.
Don't pump it up with a lot of
gas;
put the least amount of gas you
can get away with.
Then, you find all gasses have
the property that if you
calibrate them at zero and 100,
they agree in between.
Is that clear to you?
Take the product P times
V of your gas by putting
it on different surfaces,
measure the product,
plot this graph.
Whenever you're on ice
[freezing point of water]
you call it zero;
whenever you're on boiling
water you call it 100.
You find they're connected by a
straight line,
then every point in between,
you've divided equally,
leads to equal increase in the
product P times V.
P times V for a
gas is better than the volume of
mercury or volume of water
because it doesn't depend on the
gas.
So, everybody can use the gas
thermometer.
That's why we prefer the gas
thermometer.
So, this is the interesting
issue about measurement or
definitions and cyclic
definitions--you've got to be
careful.
The laws of nature allow you to
pick anything you like that
varies with temperature and use
that as a definition of
temperature, as a thermometer.
So, why are some thermometers
preferred over the others?
They're preferred over the
others if the laws of nature
take the simplest form when
described in terms of those
thermometers.
In other words,
take a meter stick.
What makes a good meter stick
for a standard?
You say the one that doesn't
expand, but we don't know what
that means.
That meter stick is the
standard;
by definition it's right.
But then, you will soon find
out that it's not really that
simple, because there are good
and bad meter sticks.
For example,
the same meter stick at one
time out of the year doesn't
match its own length at a
different time of the year;
then we know that it's not a
good meter stick.
Similarly, there are good and
bad thermometers,
and people arrive on the gas
thermometer this way.
If you have a gas thermometer,
something very interesting came
out of the gas thermometer.
If you cool it below zero and
you ask which way is it going,
I don't know how low you could
go.
In the old days,
people couldn't go far below
zero, but now we can go to
one-billionth of a degree above
a certain point.
I'll tell you now,
these thermometers indicate
somehow the product PV
vanishes at a temperature which
is minus 273.16,
suggesting that there is
something very special about
that temperature.
Because if you took another
gas--well, I'm going to do a
little cheating here--that also
extrapolates that same
temperature.
So, all gasses,
all gas thermometers say there
is something very special about
this temperature because that's
when our pressures all vanish.
So, as you cool a given amount
of gas, even at a given volume,
if you keep the volume constant
and ask what pressure do I need,
how many weights do I have to
put on;
that decreases and vanishes at
this temperature.
And this is called the absolute
zero of temperature.
It's called absolute zero for
many reasons.
One is that unlike the zero of
the centigrade,
which is by no means the
absolute lowest possible
temperature,
the absolute zero is the lowest
possible temperature.
Why?
Because the gas pressure can be
reduced and reduced and reduced,
but the worst that can happen
is that it can go to zero.
That's it.
It cannot go below having no
pressure.
We'll find in other ways,
also, this is the temperature
at which you will see
conceptually no further cooling
is possible.
That will require you to
understand what hot and cold
mean.
But right now,
this says all gas thermometers
point at this temperature.
So, people decided,
"You know what,
calling this zero is kind of
artificial."
That's based on human obsession
with water.
But if you think laws of
science describe the whole
universe, what about planets
where there's no water?
Right?
You cannot describe--Suppose
you're talking to a different
civilization;
Planet of the Apes.
You want to tell those guys,
"We're going to set up our
temperatures;
zero is when water freezes,"
and they say,
"What is this thing called
water?"
"You know, the stuff you drink."
You don't know what these apes
are drinking.
Maybe they're drinking methane
or liquid hydrogen.
We don't know.
On the other hand,
you say, "Take any vapor and
wait until the product of the
pressure and volume go to zero,
let's call that zero," that's
the universal standard.
It's not tied to something
called water.
It was fine for a while,
but it is not fine as a
universal aspiration for
thermometers.
So, zero of temperatures can be
set from here.
Once they did that,
they called that zero,
they needed one other
temperature.
And they decided that if you're
starting the new temperature
scale, you will put the zero not
at the centigrade,
but this is now called Kelvin.
And everything will follow a
straight line,
but to define what one degree
means, you've got to define one
other temperature.
That's how we define the
straight line;
that temperature would be
called 273.16.
But this point is called the
triple point of water.
What's the triple point of
water?
You know water and ice can
coexist, and you know that water
and steam can coexist at 100
degrees.
But by varying the pressure and
temperature and volume,
you can actually find a certain
magical point in which both ice,
water and steam can coexist,
simultaneously.
It cannot pick between those
three options.
Ice floating on water is when
water has not decided whether to
be ice or to be water.
That's the coexistence point of
two things.
And when the water starts
boiling on your stove,
that's when water and steam
coexist.
But I'm saying that certain
conditions of pressure and
temperature and volume so that
water, ice and steam will
coexist.
Now, that is a unique situation;
you cannot get to that by any
other means.
And that temperature we will
call plus 273.16 in these
absolute units.
So, basically,
what you have done by going to
the absolute units is you've
shifted the zero to a more
natural point where all graphs
meet;
then, you define one degree
Kelvin to be so that 273.16 of
that Kelvin brings you to the
triple point of water.
So, if you found that
confusing, I'm just saying the
boiling point of water is not a
fixed number.
You go to the mountains,
it changes.
But only under one condition
can water and ice and steam
coexist.
You cannot get that any other
way.
So, everybody will agree on
that particular situation,
that will be called 273.16
Kelvin.
Now there is a rule, apparently.
You can say,
"degree centigrade," you're not
supposed to say,
"degree Kelvin."
There was a big deal made in a
lot of books.
I keep forgetting--In fact,
I forgot again,
and nothing terrible has
happened to me.
So, I don't think you should
pay too much attention to
whether you can call something
"degree Kelvin" or simply
"Kelvin."
I think the purpose of language
is to have no ambiguities.
But when they say,
"degree Kelvin" and I find that
you guys don't get confused,
I don't think that's a big
deal.
But you'll find if you're a
very erudite person,
you will never write "degree
Kelvin."
But having said that,
don't hold me to those
standards--I just don't feel any
affiliation to this particular,
completely artificial and empty
convention.
But you are supposed to
remember, if you take the GRE or
something, it's not called
"degree Kelvin."
Okay so, as far as we are
concerned, the Kelvin scale is
like the centigrade scale,
except the zero has shifted to
here.
That's it.
That's the temperature scale
you will use.
That's the absolute temperature.
Whenever I write T from
now on, I'm talking about
Kelvin, not centigrade.
Now, that's all about heat--I
mean, all about temperature.
Now, I'm going to talk about
heat.
So, heat is denoted by the
symbol Q,
and you've got to ask yourself,
"What are we talking about when
we talk about heat?"
Again, let's use your intuitive
sense of what heat is.
Say I have a bucket of water;
I want to heat it up.
And how do you do that?
You put the bucket on top of
something else which you think
is hotter, and when the two are
brought together,
somehow the water begins to
feel hotter and hotter.
So, we say we've heated the
water, and we say we have
transferred heat.
Now, people were not sure what
really was being transferred.
What is it that's going from
the stove to the water?
Why is it that the stove,
if it's not plugged in,
is getting cooler and the water
is getting hotter?
They just decided to call it
the caloric fluid.
They imagined there was a
certain fluid which is abundant
in hot things,
and not so abundant in cold
things.
When you put hot and cold
together, this magical fluid
flows from hot to cold,
and in the process heats the
cold thing.
And they decided to measure it
in calories.
And so, you have to define what
a calorie is.
In other words,
you want to ask,
"How much heat does it take to
heat this bucket of water?"
And the rule they made up was,
we're going to define something
called a calorie where the
number of calories you need is
equal to the mass of water times
the change in temperature.
That's going to be calories.
In other words,
if I had a container with 10
grams of water,
and the temperature went
up--I'm sorry,
this is mass of water in grams.
If you have 1 gram of water,
and you did something to it and
the temperature went up by seven
degrees, you have,
by definition,
pumped in 7 calories.
If this was a kilogram of
water, this would be called a
kilocalorie.
Sometimes they use grams and
calories;
sometimes they use kilograms
and kilocalories.
But the definitions are
consistent;
if you put a kilo in the gram,
put a kilo in the calories.
Okay.
Now, suppose you say,
"I don't want to just talk
about water, I want to talk
about heating something else.
Maybe I want to heat a gram of
copper."
So, then you write down the
following rule.
The amount of heat it takes to
heat up anything--pick your own
favorite material--gold.
Then, the amount of heat,
I think we can all appreciate,
must be proportionate to the
amount of stuff you're trying to
heat up.
That's our intuitive notion.
If you've got one chunk of gold
that takes some number of
calories, you have a second
identical chunk;
by definition,
that should take the same
number of calories.
You put them together,
it is clear that whatever this
caloric fluid is,
you need double that.
So, it's got to be proportional
to the mass of the substance.
And it's got to be proportional
to what you're aiming for,
namely, increase in
temperature.
But this is true for any
substance, whether you're
heating copper or wood or gold;
no matter what you're heating,
it is true the heat need is
proportional to mass and to the
[change in]
temperature.
So, what is it that
distinguishes one material from
another?
We put a number here,
and that number is called the
specific heat.
The specific heat is the
property of that material.
You've got to understand
certain formulas will depend on
certain parameters in a genetic
way, and some things that depend
on the actual material.
In fact, there's a similar
quantity.
I mean, maybe I'll take a
second to tell you.
If you go to liquids that I
said were expanding,
you can do the same thing.
Take a rod and start heating it
and ask, "How much will it
expand if I heat it by some
amount ∆T?"
What will it be proportional to?
Can anybody think of what it
may be proportional to?
Yes?
Student: Original length?
Professor Ramamurti
Shankar: Depends on the
original length of the rod.
Now, why is that?
Why do we think it's got to be
proportional to the length of
the rod?
Student: Because it
expanded based on what it had
before.
Professor Ramamurti
Shankar: Yeah,
it's based on what it had
before.
Yes?
Student: Well,
each cycle the rod will expand
by some amount,
so [inaudible]
Professor Ramamurti
Shankar: That's correct.
I think one way to say that is
take a meter stick,
it expands to some amount,
put another meter stick next to
it, that expands to the same
amount by definition of
identical things.
For the two-meter stick it will
expand by twice as much.
So, we put the length of that.
So, no matter what you're
heating -- a block of wood,
block of steel -- this is true.
But then, the fact that heat
has different effects on copper
versus wood, is indicated by
putting a number here.
That α is called the
coefficient of linear expansion,
and that depends on the
material.
These are true no matter what
you are heating.
So, these specific numbers,
these coefficients,
these αs that come in
are going to come in all the
time, so you should get used to
them.
Here's another one.
Let's play this game one more
time.
We can ask how much does the
volume of a body change when I
heat it.
Well, the change in the volume,
again, would be proportional to
the starting volume times the
increase in temperature.
Then you put another number;
that's called the coefficient
of volume expansion.
And that depends on the
material.
So, if you take copper,
copper will have a certain
α;
iron will have a different
α;
wood will have a different
α.
Each material will have a
different α.
This is the property of the
material.
If you say, "Well,
I had something and when I
heated it up by one degree,
it increased by nine inches;
another one increased by two
inches."
Is it clear that the first one
expands more readily?
It's not, because the first one
could have been a mile long,
second one could have been a
foot long.
So, you have to take out
certain factors that are
universal, and the rest of it
you put into a property of the
material.
Similarly, when you come to
specific heat,
you ask how much heat does it
take to heat some object,
it depends on the mass.
It doesn't matter what you're
heating.
Depends on the increase in
temperature, because that's the
whole purpose of adding heat;
it's always going to be linear
in the ∆T.
This one is the property of the
material, and by definition,
c equal one calorie per
gram, or one kilocalorie per
kilogram for water.
Once you've got--So remember,
one calorie per gram for water
is the definition.
Once you define water to have a
specific heat of one calorie per
gram, you can define specific
heat for other materials by the
following process.
So, what do you do?
You take a container with some
water in it.
Let's assume the container has
zero mass, so I don't have to
worry about it.
It's an approximation.
If you are worried about that,
you know, take a huge container
so that the volume of water
dominates the surface area of
the container.
Anyway, container's neglected;
you've got some water.
This water is of some initial
temperature
T_1,
and I have some new material,
lead, and I want to find its
specific heat.
So, I take the lead in the form
of pellets and I heat the lead
pellets to some temperature
T_2,
and I drop these guys into this
water.
That'd be an example where
initially, the lead is in
equilibrium, maybe on a furnace,
at temperature
T_2;
water's in equilibrium,
maybe in the room,
at temperature
T_1.
Then, I put the pellets into
the water, and there will be a
period when the temperature is
not defined.
Then, soon they'll settle down
to some common temperature
called T_f.
We will now postulate--this is
a postulate, or a law.
The total change in Q is
zero.
In other words,
if Q is lost by one body
and gained by another body;
the loss and the gain must
equal.
It's a new law.
You can make up all the new
laws you want.
You don't know if they're
right, but this is the law you
first make up.
In that case,
what can you say in this
particular problem?
In any of these heat problems,
I urge you to draw the
following picture.
Here is one temperature,
here is another temperature,
here is the final one,
which we don't know,
but we can measure with a
thermometer and measure it.
Then, you say the mass of the
water, and specific heat of
water, which is 1 times
∆T,
which is the final temperature
minus initial temperature.
Ditto for the lead pellet;
mass of the lead,
lead has got a symbol Pb,
times specific heat which I
don't know,
times a change in temperature
which is T_f-
T_2 = 0.
The sum of all the mc
∆Ts is zero.
This is the gain of heat,
of the water.
This, if you work it out,
will be a negative number,
because you can see
T_f is below
initial T.
This will turn out to be
negative, and the positive and
negative will add up to zero.
So, what is it you don't know?
Well, you know the mass of the
water.
Specifically,
the water is 1 by definition;
T_f and
T_1 are
measured by thermometers.
Mass of lead is for you to
measure;
these are known;
you can find c.
So, this is a birthday present
for you guys.
If you ever see this in an
exam, jump on this first because
you've been doing this in high
school,
and I know kids love this kind
of calorimeter problems.
Yes?
Student: Looking at the
volume in that equation it
expands linearly but wasn't the
problem with the liquid,
measuring liquid,
changing volume,
but it didn't expand
[inaudible]
Professor Ramamurti
Shankar: Yes.
That's correct.
So, the real point is,
if everything expanded
linearly, we wouldn't have the
disagreement between different
thermometers.
So, it turns out to an
excellent approximation,
the change of length is
proportional to the length,
but it's not exactly
proportional to the length.
There will be terms involving
higher powers of length.
Not only that,
specific heated materials is
also not a constant.
We said specific heated water
is 1.
Turns out at a certain
temperature range it'll be 1;
at a different range in fact,
it's not quite 1.
I told you long back.
Everything I tell you is wrong.
The question is,
"How many decimal places do you
have to go to before you honor
my fallacies?"
Specific heat of materials is
not a constant,
with the big industry
calculating the specific
materials starting from atoms
and quantum mechanics.
So, none of the things treated
as constants are ever constant,
including those alphas and
betas.
I can always fudge it by saying
α itself may depend on
the temperature,
and also the dependence on
L may not be linear.
But you should also look at
dimensional considerations and
say if it's not L,
if you want to put an
L^(2) as a correction to
the formula to match the units,
L^(2) has to be divided
by another length to keep the
units.
What other length do we have?
It may turn out to be the
inter-atomic spacing.
So, once the atomic properties
come into play,
then you can find ways to
calculate corrections.
So, all these laws are,
in fact, very tentative and
approximate.
These are pretty ancient
physics.
I think the way I do the
physics course here,
sometimes I'm in the 1600s,
sometimes in the 1400s,
sometimes in the year 2000,
but going back and forth.
This is way back when people
did not even know about atoms.
So, they were trying to do the
best they can,
and what you found empirically
is that once you found a
specific heat for lead,
right, you solve for it,
then you can do another
experiment using that value and
you find if you use the right
values,
∆Q does add up to zero.
Again, when it adds up to zero,
it adds up to zero to a very
good approximation,
during the epoch.
Another epoch when people do
more and more accurate
experiments, everything is shot
down.
In fact, specific heats of all
materials seem to go to zero
when you approach absolute
temperature.
But you have to understand the
laws of quantum physics to know
why that happens.
So, this is in a period when
people are probing temperature
ranges which are around room
temperature,
or boiling or freezing point of
water, which is a very narrow
window in temperature.
If you look at the history of
the universe,
you've got incredibly high
temperatures near the Big Bang,
and even now the rest of the
universe is bathed at some
temperature that happens to be
very,
very low, which is near three
degrees;
it's called a blackbody
radiation from the Big Bang.
So, the temperature of the
universe goes through huge
ranges, and only when you probe
different ranges you see
different physics.
If you come to Sloan Lab,
you can go to temperatures way
below 1 degree Kelvin or
hundredth of a Kelvin,
and we heard a talk last year,
physics at one billionth of a
Kelvin.
If you want to cool them and
cool them and cool them,
by zero degree Kelvin,
see, there I go.
Zero Kelvin is a barrier we're
not able to cross,
just like the velocity of light
is something we're not able to
cross.
These are all big surprises.
The fact that velocity has an
upper limit, not obvious even to
Newton.
Why not?
Why not put rockets on top of
rockets?
Likewise, why not build better
and better refrigerators?
The reason you cannot go below
zero is when you go to zero,
all the mechanical attributes
of pressure simply vanish,
and they cannot have negative
values.
You will see more about this
when you understand heat in
greater depth.
Anyway, right now,
∆Q = 0 is the rule you
use.
I'm sure you guys know how to
do these problems.
Now, there's a little twist
that comes in,
I just want to mention that to
you.
The twist is the following.
So, I take some ice--ice,
by the way, is not always at
zero.
You know, you can go below zero.
Your refrigerator is several
degrees below several tens below
zero.
So, let's take ice,
and let me measure--I take this
container, I put some ice at,
say, minus 30 degrees.
I've gone to centigrade now so
we can relate to ice.
And I put it on some source of
heat, and I watch how many
calories are coming in.
Let me arrange a device that
will pump in a fixed number of
calories every second.
So, as a function of time,
I'm expecting the temperature
of this to go up.
Do you understand that?
In every second,
I get some number of calories,
and those number of calories
are going to produce for me
mc ∆T,
m and c are
constants, so ∆Q is
proportional to ∆T.
But if you divide both by the
time elapsed,
then the rate at which the
temperature rises will be the
rate at which the heat flows
into the system.
If heat is flowing at a steady
rate, temperature should rise,
and indeed it does.
Temperature of the ice goes
from minus 30 to minus 20 to
minus 10 and so on.
But once it hits zero,
it gets stuck.
I know heat is coming in,
but it's not getting hotter.
But I notice that the ice is
beginning to melt.
There will be a period between
here and here when I pump in
calories, I don't get any
increase in temperature but I
get conversion of ice into
water.
And there will be a period when
this guy looks like some water
with some chunks of ice floating
on it.
And until all the ice is
converted to water,
the whole system is stuck at
that temperature.
That's a very interesting
property.
Now, if you really took a real
pot and you put a chunk of ice
on it, you know what will
happen, right?
The bottom of the ice will melt;
it may even evaporate.
That's not what I'm talking
about, because that's not a
system where there's a globally
defined temperature.
I want you to heat the ice so
slowly, the minute you put a
little bit of calories,
give it enough time for all
these guys to share that heat,
so that the whole system has
one single common temperature.
Let's watch the temperature
rise.
I'm saying it gets stuck at
zero, but your calories are
getting you something;
they're converting ice into
water.
Then you can ask,
okay, what penalty do I have to
pay, that's called a latent heat
of melting,
and again, I know only in
calories per gram,
it's 80 calories per gram for
water.
Some of your ∆Q now
goes not to raise the
temperature, but to melt that
amount of stuff at the latent
heat of melting.
That's how much Q you
need to melt that amount of
stuff and the L varies
from substance to substance,
but water is 80 calories per
gram.
If you want to melt mercury
from solid mercury to liquid
mercury, it will have a
different number.
Then, once everybody has become
water, then that uniform system
of water starts growing.
And this is called a phase
change.
A phase change is when it
changes its atomic arrangement
from a regular array;
for example,
that forms a solid into a
liquid.
In a solid, everybody has its
place;
you can shake around where you
are, but liquid you can run
around.
The specific heat of ice is not
the same as the specific heat of
water, so you've got to be
careful.
Even though it's still made up
of water molecules,
the calories needed to heat one
gram of ice is roughly half what
it takes to heat one gram of
water.
So, in these problems,
don't make the mistake.
Okay then, you go along and I
guess you know what the next
stopping point is.
When you come to 100 degrees,
again, it gets stuck until
everybody vaporizes,
and then you get steam.
Then, you can have super-heated
steam, which is at even higher
than 100 degrees.
So, that's the latent heat of
vaporization.
I really don't know what--you
want to write something,
I think it's 500 and something
calories per gram.
That's information I don't
carry in my head.
So, if I tell you I took some
ice at minus 30 and I dumped in
5,000 calories,
where will it end up?
You've got to first spend a few
calories going from here to
here, you got some more money
left you can start melting this,
maybe you'll run out of stuff
there, and that's what you will
have.
Some amount of water and some
amount of ice.
If you have even more calories
at your disposal,
you can melt it all and start
heating it.
You may come this way and you
may be running out of calories;
if not, keep going here and
there and there,
and you may end up there if you
got enough calories.
Or one can ask a question,
"How many calories does it take
to convert ice at minus 30 to,
say, water at 100?"
You'll have to do the mc
∆T for that,
m times latent heat for
this,
mc ∆T for that,
and m times latent heat
of vaporization for that.
So, the kind of problems you
can get are fairly simple most
of the time.
Only kind of problem where you
can really get in trouble is the
following.
I will mention that to you.
Suppose I take some water and
some ice, so this is zero.
The ice is at,
say, minus 40,
the water is at plus 80.
In fact, let me make that water
plus 40.
I bring them together and I ask
you what will happen.
Now, this is a subtle problem.
If you had two--If you had
water at 40 and you had water at
20, you can easily guess that
it'll end up somewhere in
between;
you can calculate it.
Now it's more subtle.
You've got water at 40,
you've got ice at minus 40,
you bring them together and ask
what happens.
Well, the answer will depend on
how much of the stuff you have.
If by water at 40 you mean the
Atlantic Ocean,
and by ice you mean a couple of
ice cubes, we know what's going
to happen.
These guys are going to get
clobbered;
they're going to melt;
you will end up somewhere here.
Then, you can easily calculate
the final temperature by saying
mc times this ∆T
for water,
in magnitude,
is going to be the heat given
to this.
Heat given to this is the mc
∆T to come here;
then, the heat to melt this
amount of ice,
then the heat to raise this
amount of water to that final
temperature.
Then, you can solve for the
final temperature.
So, if you want to solve this
problem, and I give you some
mass for this ice,
of water, and I give you some
mass for the ice,
you can first make the
optimistic assumption that you
will end up as water,
but at an unknown temperature.
We call the unknown temperature
T;
this is the
T_1,
this is the
T_2.
Write your equations,
except you'll have one more
term there.
That's the heat it takes to
melt the ice.
You solve for T.
If you get a positive answer
you can use it,
because the assumption that you
ended up on water meant you
heated up the ice,
you melted the ice into water,
then heated up the water from
zero to the final water.
But if you did the calculation
and got a negative value of
T, that answer cannot be
blindly used,
because the assumption that you
are on the other side of ice is
wrong.
Then, you can try something
else;
you can assume you're down here.
If you think you're down here,
then you've simply heated the
ice from here to here.
This water you brought down to
zero, sucked out mc ∆T
from that, then you've taken out
now the latent heat of melting.
You take out heat when you
freeze, and then you've taken
even more to come down here.
Then, all those losses of the
original water is equal to the
gain of this ice.
You can assume it here,
you can solve for this
T.
When you solve for this
T, if you've got a
negative number,
then you're okay.
That will be a good assumption
if I say I sprinkled two drops
of water on a big iceberg;
we know it's going to end up as
ice and that's a good starting
point.
But if I give you numbers which
are kind of wishy-washy,
where I don't know whether this
will win or that will win,
there's a third possibility.
The third possibility is at the
end of the day,
you end up here with some
amount of water and some amount
of ice at zero degrees.
So, that's a third option you
may have to consider,
if neither of them works.
Then, the question is not what
is the final temperature.
But what's the question then?
What do you want to know in
that case?
How much is ice and how much is
water?
That's the question.
And there are several ways to
figure that out.
Let me just say in words,
I don't want to do this algebra
because for you guys it would be
fairly easy.
If it's a question of--Suppose
both of the things I try fail.
I took a positive T,
assumed I'm up here,
and I assume the ice melted,
and I get a negative answer;
that's shot down.
I take a negative T and
assume everybody froze and that
doesn't work.
Then, I'm down to this option,
which is some amount of water
and some amount of ice.
And the question is,
"How much is left?"
You solve that by doing the
following.
You say all this ice went from
here to there.
It does that by absorbing that
mc ∆T;
mass of the ice times specific
heat of ice times ∆T.
Maybe it was minus 40,
the ∆T is plus 40.
You give that heat to this guy;
that heat you suck,
out of this guy.
When you suck that out of this
guy, first you bring this to
zero, then you still have some
more heat you can extract from
him,
you will use that to convert
water into ice at the price of
80 calories per gram.
Maybe you can freeze 5 grams or
5 kilograms of water;
that will be the extra ice,
the rest will be the water you
started with.
The total mass will be the
same, but if you got 60 grams of
water, you bring the 60 grams to
zero and you still have some
more heat to be extracted;
maybe you'll convert 10 grams
to ice and 50 will remain as
water.
So, the final answer will be 50
grams of water,
10 grams of ice plus whatever
grams of ice you started with.
That's about the most complex
heat-exchange problem.
If you guys want me to tell you
some more I will,
or I can move on.
I don't know what your view on
this is.
Do you understand what you have
to do in each problem?
Okay.
So, it's the conservation of
heat that's applied.
So, the most tricky part is
phase change,
when you've got a phase change,
you've got to remember that the
formula mc
∆T--∆Q has one
more term,
the one more term is this.
Okay, so next question we ask
is, "What's the manner in which
heat manages to flow?"
We say you got these calories,
I mean, how does it flow,
what's the rate at--what makes
it flow.
So, it turns out there are
three popular ways of heat
transfer;
one is called radiation.
Radiation is when the heat
energy leaves some hot body and
comes to you without the benefit
of any medium,
like heat from the Sun.
So, that's really
electromagnetic radiation that
comes from hot,
glowing objects,
and directly comes to you.
Electromagnetic radiation
doesn't need air,
doesn't need anything.
In fact, if it needed air,
we would not get any heat from
the Sun because there is no
medium between the Earth and the
Sun.
Most of it is just vacuum.
So, if you took one of these
space heaters,
you know, with glowing red
coils, and you feel warm.
If I start pumping the air out
of that room,
of course, you will be dying
very rapidly,
but your last thoughts will be,
"I am still warm" [laughter]
because the radiation will keep
coming to you.
Okay?
That's radiation heat.
There are lots of laws for
radiation;
I don't want to give them to
you because there are formulas
you memorize,
and you don't understand too
much of the physics right now.
Other than to say it's
electromagnetic radiation,
whatever that means--we haven't
gotten to that yet.
That's what comes from there to
here and can come in vacuum.
It doesn't need a medium,
is the key.
Then, the second way of heat
transfer is called convection.
So, convection is explained by
the following example.
You've got water;
you put it on a hot plate.
Then, in the lower part of it,
the water gets hot.
When it gets hot it expands,
and when it expands the density
goes down;
therefore, by loss of buoyancy
it will start raising up.
Remember, a chunk of water
belongs in water.
A chunk of something else with
lower density will float to the
top.
But the point is,
water doesn't have a fixed
density.
If you heat it up,
the density goes down,
so the water guys downstairs
have lower density-- they're
like a piece of cork,
they will rise to the top.
When they rise to the top,
the cold water with the higher
density will fall down.
So, you set up a current.
Hot rises to the top and cold
comes down.
And this also happens in the
atmosphere.
On a hot day,
the air next to the ground gets
really heated up and it rises,
and the cold air comes down and
you set up these thermal
currents.
So, here you're trying to
equalize the temperature between
a region which is cold and a
region which is hot by the
actual motion of some material.
In radiation,
you don't have the medium
transferring heat because a
medium is not even present in
radiation.
In convection,
the medium actually moves.
The hot guys physically move to
the other place and the cold
guys come here,
and by that process,
the heat is transferred.
The heat transfer I want to
focus on a little more
quantitatively,
is conduction.
So, heat conduction is
something you've all
experienced.
I mean, if you have a skillet,
why does it have a wooden
handle?
Simple reason;
if you had a steel handle,
you put it on a hot stove and
you put your hand here,
the fact that your body is at
whatever, 98 degrees,
and this one is God knows,
200 degrees,
you're going to have heat flow
from here to here.
So, we want to understand
what's the rate at which heat
flows from the hot end to the
cold end.
So, you can imagine a rod of
some cross-section A,
one end of the rod is in some
reservoir at some temperature
T_1,
other end is at temperature
T_2.
By the way, I'm now introducing
a new term called reservoir.
Reservoir is another body like
you and me, except it's not like
you and me.
It's enormous.
It is so big that its
temperature cannot be changed.
You can sit on it,
you will fry and you'll
evaporate, but its temperature
will not change.
No body is really a reservoir.
If you drop an ice cube in the
Atlantic, you'll lower the
temperature of the Atlantic but
by a negligible amount.
So, take the limit of Atlantic
goes to infinity,
then you have a reservoir.
Reservoirs have one label,
namely, what's our temperature.
So, something big enough can
be--this room is like a
reservoir.
You put a cup of coffee here,
you say it will come to room
temperature.
Actually, the room temperature
meets the coffee,
not halfway but slightly up.
But the room is large enough so
that we can attribute to the
room temperature quite
independent of bodies that go in
and out of it.
So, this is connected on the
left to an enormous tank of
maybe a water-ice mixture at
zero degrees;
this is a water-steam mixture
at maybe 100 degrees.
You put a rod there.
We know heat is going to flow
from the hot body,
from the hot end to the cold
end.
And we want to write a formula
for how much heat flows per
second.
Again, I'm going to write these
formulas over and over again.
So, you've got to ask yourself,
what will it depend on?
What are the properties it will
depend on, in general,
independent of what the rod is
made of?
Can you think of one?
Yes?
Student: [inaudible]
Professor Ramamurti
Shankar: You said the
cross-section.
Now, why do we say--what reason
can you give for cross--
Student: If you just want
to consider a rod with twice the
cross-section area,
you're going to come up with
[inaudible]
two rods and twice [inaudible]
Professor Ramamurti
Shankar: Yes,
okay let me look at this
argument.
You take one rod,
and for convenience let's just
take it to be a rectangular rod.
Take another rod,
rectangular rod;
they will both transfer the
same amount of heat for a given
amount of time.
Just glue them together and say
here is my new rod.
We know it's going to transmit
twice the amount of heat.
So, it's going to be
proportional to the area.
And why is the heat flowing?
It's flowing because of a
temperature difference.
So, that's always there;
that's the underlying force for
heat transfer.
That's the dynamics in
thermodynamics;
that's what makes the heat flow.
But then, we find as an
empirical fact,
that if these two reservoirs
are separated by that distance,
then the heat flow is a lot
less than when they are closer.
It seems to depend on how much
temperature difference is packed
in spatially.
So, you want to divide by a
∆x is not
infinitesimal;
it's the length of the rod
separating the hot and cold
ends.
In other words,
if you dilute the temperature
difference over one mile,
the heat flow will be
correspondingly reduced,
whereas if there's huge
temperature difference between a
very small spatial separation,
there will be very robust flow
of heat;
that's what we're saying.
These happen to be true,
you realize,
independent of what material
I'm talking about.
When I said one rod plus one
rod is two rods,
it doesn't matter what it's
made of.
Again, having put all these
factors which you can argue on
general grounds,
you have to now ask,
"What happens when this is a
copper rod versus silver rod
versus wooden rod?"
So, you've got to put one more
number which is kappa
[κ]
here;
not k you guys,
it's κ,
and it's called the thermal
conductivity of that material.
Sometimes you put a minus sign;
minus sign just means it flows
from hot to cold.
I don't care whether you put
the plus sign or don't put the
minus sign;
anybody knows that the heat is
going to flow from hot to cold.
So, just remember that
direction of flow,
and that's all I care about,
this sign here.
This κ is the property
of the material.
Once again, let me tell
you--You can say,
"Well, I have two reservoirs,
hot and cold.
I connected them with two
different rods.
This rod carried twice the
amount of heat per second as the
other rod.
Is it necessarily a better
conductor?"
No.
Maybe it had 10,000 times the
cross-section.
So, what you want to do is to
make the playing field level,
and compare rods of the same
cross-section,
same temperature difference,
same length,
then ask who conducts more
heat.
That depends on the material
and that's the thing you pulled
out specific to the material.
That is the property of wood or
copper of steel;
that's the heat conductivity.
Okay.
Now, the final topic is just
going to be more hand-waving
now.
I don't want to get into too
many details.
It really has to do with what
is heat.
In the old days,
people just said that it was a
fluid, and they postulated the
conservation law for the fluid.
You can postulate what you
want, you've got to make sure it
works, and it seems to work,
in the sense that all the
∆Qs in any reaction add
up to zero.
But then, people are getting
hints that maybe this thing that
we call heat is not entirely
independent of other things we
have learned.
So, where do you get the clue?
One clue is,
long back when we studied
mechanics, we talked about two
cars that come and collide;
they slam into one big lump.
Now, you've got no kinetic
energy, no potential energy.
Potential energy is always
zero, they're moving on the same
height, kinetic energy was ½
mv^(2) for this,
½ mv^(2) for that;
at the end there's nothing.
No kinetic, no potential,
we just gave up and said,
"Look, conservation of energy
does not apply to this problem."
We just say it's inelastic.
On the other hand,
we find whenever that happens,
we find the bodies become hot.
Here's another thing you can
do, you can take a cannonball,
drop it from a big tower.
This is how some people in the
French army, I think,
first detected this feature;
you dropped cannonballs from a
big height.
When they hit the sand,
they start heating up.
Or you drill a hole in a
cannon, that's what
Count-somebody did,
and he also noticed that you
need to constantly pour water to
keep the drill bit from heating
up.
You'll find very often,
mechanical energy is lost and
things heat up.
So, you get a suspicion
whatever the underlying
mechanism, maybe there's a rule
that says if you lose so much
mechanical energy that you
cannot account for,
then it translates into a fixed
number of calories.
If that is the case,
then we at least get a
dictionary on--between calories
and joules.
So, joules is energy you can
see, calories is energy you
cannot see.
That was going to be the
premise.
But first, you've got to prove
that every time you lose some
number of joules,
you get a fixed amount of
calories.
And that experiment is due to
Joule.
Here is the Joule experiment.
It's very, very simple and
tells you the whole story.
You have a little container in
which there is a paddle.
This is a shaft with a pulley,
and there is a weight here.
So look, try to imagine this
guys.
You got rope wrapped around the
top pulley, and when you let
this weight go down,
it's going to go down like
this;
it's going to spin the shaft.
And put some water here,
and I have some fins that are
sticking out,
so they churn up the water.
So, it's like this thing,
the egg-beater,
right?
In fact, I tried to do the
experiment with an egg-beater
this summer to a bunch of high
school kids,
and I got thoroughly humiliated
because nothing happened as
planned.
But the idea is the same.
You agitate the water in some
fashion.
But this guy did it in a
particularly simple way.
My egg beating was not good
enough;
you will see maybe in a while
why that's not good.
What he did was to put these
paddles, let the weight go down
from there to here.
Now, we can keep track of how
much mechanical energy is lost,
right?
Because if this mass was at
rest, and a drop to height
mg drop to height
h, it's supposed to have
mgh kinetic energy.
Let's say it's got some kinetic
energy, which is not equal to
mgh.
So, mgh minus kinetic
energy is missing.
So, some number of joules are
gone.
So, the water gets hot.
When the water gets hot,
you can immediately ask how
many calories were supplied to
the water.
Because that water heats up the
same way whether or not you put
it on a hotplate,
or whether or not you churn it.
It doesn't seem to depend on
how it got hot.
This has the same effect.
This water is hot in every real
sense.
So, you must have put some
calories.
You can find out how many
calories you put in by looking
at the mass of the water;
specific heat of the water is 1;
looking at the increase in
temperature.
So, some joules are missing,
some calories have been pumped
into the water.
Then you ask,
"Is there a proportionality
between joules and calories?"
And you find that it is.
And that happens to be 4.2
joules per calorie.
In other words,
if you can expend 4.2 joules of
mechanical energy,
you got yourself one calorie to
be used for whatever heating
purposes.
So, in the example of the
colliding cars,
this had some energy,
that had some energy,
all measured in joules;
they slammed together,
they come to rest.
That means you can take those
many joules, divide it by 4.2
and get some number of calories.
Imagine the whole car is made
out of copper.
Then those calories will
produce an increase in
temperature, right,
equal to ∆Q is mc
∆T.
That will be the rise in
temperature of the car.
In practice,
there will be other losses,
because you heard the sound,
well, that's some energy gone;
you won't get it back.
Maybe some sparks are flying,
that's light energy;
that's gone.
You subtract all that out,
you find that in the end,
the calories explain the
missing joules.
So, that made people think that
this is just another form of
energy.
Because if you add this to your
energy balance,
there is no reason to go on
apologizing for the Law of
Conservation of Energy.
Law of Conservation of Energy
is not in fact violated,
even at the inelastic
collision, if you include heat
as a form of energy.
And the conversion factor is
4.2 joules per calorie.
But the question is,
"What right do you have to call
it energy?"
Energy, we think--primarily,
when you say somebody's
energetic, you mean that
someone's running around
mindlessly, back and forth.
Energy is associated with
motion.
These two cars were moving,
and we have every right to say
they have energy.
How about potential energy?
Well, if the car starts
climbing up a hill and slows
down, we think it's got
potential.
If you let it go,
it'll come back and give you
the kinetic energy.
So, most people's idea of
energy is just kinetic energy.
That is lost.
And yet, you get calories in
return, so you ask yourself,
"What can it be?"
Well, the correct answer to
that came only when we
understood that everything is
made up of atoms.
Once you grant that everything
is made up of atoms,
then it turns out that the
kinetic energy of atoms is what
we call heat.
But you've got to be very
careful.
Take a tank full of gas.
I throw it at you.
That whole tank is moving,
that's not what I call heat.
Okay?
That motion you can see.
I'm talking about a tank of gas
that doesn't seem to be going
anywhere;
yet, it got motional energy
because the little guys are
going back and forth.
So, what we will find is what
I'm going to show you next time,
is that if you kept track of
the kinetic energy of every
single molecule in this car,
every single molecule in that
car, before and after,
and you added them up,
you would get exactly the same
number.
The only difference will be
originally the car has got
global common velocity;
macroscopic velocity you can
see.
On top of it,
it's got random motion of the
molecules that make up the car.
So does the other car.
When they slam together,
the macroscopic motion is
completely gone,
and all the motion is thermal
motion.
But it's still kinetic energy,
and that's what we will see the
next time.
 
