Download PH 213 ENERGY CONSERVATION The Fisrt Law of

A. La Rosa
Lecture Notes
PH 213
________________________________________________________________________
ENERGY CONSERVATION
The Fisrt Law of Thermodynamics and the Classical Work
Kinetic-Energy (CWE) Theorem
I. CONSERVATION of ENERGY
Ref: This section taken from The Feynman Lectures Vol. 1, Ch. 4]
I.1. What is energy?
Illustration of conservation of energy with an analogy: Conservation of blocks
I.2. Reversible and non-reversible machines in the context of the gravitational
potential energy
I.2.A Weight lifting machines and the non-perpetual motion principle
I.2.B Ideal Machines and Reversible process
I.2.C Comparing reversible and non-reversible machines
I.2.D Universal behavior of reversible machines
I.2.E Figuring out the actual lift capability of the reversible machine
I.2.F The Potential Energy
II. CONSERVATION of ENERGY in MULTI-PARTICLES SYSTEMS
Ref: This section taken from University Physics, by Ronald Lane Reese, Vol. 1, Ch. 13]
II.1 The Classical Work Energy (CWE) Theorem: Case of a single particle
II.2 The Classical Work Energy (CWE) Theorem: Case of a multiple particle rigid
body
II.3 Limitation of the CWE theorem
II.4 Generalization of the Classical Work Energy (CWE) theorem: its extension to a
thermodynamic system
II.4.A Macroscopic and microscopic contributions to the total energy
II.4.B The transfer of energy
II.4.B1 Macroscopic external work W ’ done on a system
II.4.B2 Heat-transfer Q
II.4.C Fundamental Energy Conservation Law
Generalization of the classic work/kinetic-energy theorem
III CONSERVATION of ENERGY. Case: Pure Thermodynamics
The First Law of Thermodynamics
1
I. CONSERVATION of ENERGY
[This section taken from The Feynman Lectures Vol. 1 Ch. 4]
I.1. What is energy ?
Illustration of conservation of energy with an analogy: Conservation of blocks

There is a fact or law governing all natural phenomena that are known to date,
called the conservation of energy. It states that:
There is a certain quantity, which we call energy, that does not
change in the manifold changes which nature undergoes
It is not a description of a mechanism.
It is a mathematical principle
It says that it is a numerical quantity that does not change
when something happens.
A strange fact that we can calculate some numbers and
when we finish watching nature go through her tricks and
calculate the number again, it is the same.

Illustration of conservation of energy with an analogy: Conservation of blocks
Little John plays with 28 indestructible blocks. At the beginning of the day, his
mother puts him, with his blocks, into a room.
At the end of every day, she counts the blocks very carefully, and discovers a law:
No matter what little John does with the blocks, there are
always 28 remaining.
One day she finds only 26. There is consternation. However, when she looks through
the window, she finds that 2 blocks lies on the garden.
Another day she finds 30! … Later she finds out that a friend came over to visit John
and left 2 blocks.
She starts enjoying the counting blocks game. But from now on she will close the
windows and will not allow extra blocks entering the room.
Everything goes all right …until one time she counts only 25. At another occasion,
she counts 20. Next day 22, … . What is going on?
The mother discovers there is another box where apparently little John place them,
but the kid does not allow her to open.
Being curious, she invents a scheme: Knowing that each block weights 20 grams, she
first weights the full closed box, then subtract the weight of the box (she figures the
weight of the box on a day when she sees the 28 blocks), and divides the result by 20
grams. This way she encounters:
number of
(weight of the box) - 50 grams

 constant = 28
blocks seen
20 grams
2
But one day, checking again and again the formula throughout the day it appears
that it does not to work; the total does not add up to 28.
But she notice the dirty water in the bathtub is changing its original height level of
40 cm. It may be that the kid is throwing some blocks at the bath tube (but can not
be seen, because the water is dirty. She check that each block raises the level by ½
cm, so she modifies her formula:
number of
blocks seen
(weight of the box) - 50 grams

20 grams
(1)
height of the water - 40 cm

 constant
(1/2) cm
And so on, …. In the complexity of her world she finds a whole series of terms
representing ways of calculating how many blocks are in places where she is not
allowed to look. As a result she finds,

A complex formula ( a quantitity that has to be computed) which always
renders the same value.
What is the analogy of this to the conservation of energy?
The most remarkable aspect that must be abstracted from this picture is
that there are no blocks
Take away the the first term in expression (1) and we find ourselves
calculating more or less abstract things:
(weight of the box) - 50 grams

20 grams
height of the water - 40 cm

 constant
(1/2) cm
(2)
The analogy has the following points:
When we are calculating the energy, sometimes it leaves the system and
goes away, or sometimes some comes in. To verify the conservation, we
have to track that we have not put any in nor taking out.
The energy has a large number of different forms, and there is a formula for
each: gravitational, electrical, radiant energy, …
Quoting Feynman,“In physics, today, we have no knowledge of what energy is… however
there are formulas for calculating some numerical quantities that when added together
gives ’28’__always the same number. It is an abstract thing and does not tell us the
mechanism or the reasons for the various formulas ”
3
4
What other conservation laws are there in Physics
 Based on symmetry
 Based on counting
5
I.2. Reversible and non-reversible machines in the context of gravitational
potential energy
Conservation of energy can be understood if we have a formula for all its terms.
Here Feynman discusses the formula for the gravitational energy near the surface of the
Earth. In Feynman’s description,
the point of interest is not necessarily the formula mgh (which we may
already know); instead the focus is on the line of reasoning to arrive to
such a formula.
The line of reasoning is patterned after Mr. Carnot’s arguments on the efficiency of
steam engines. We include it in this course also for its direct introduction to the concept
of reversible process.
I.2.A Weight lifting machines and the non-perpetual motion principle
A weight lifting machine:
It lifts one weight by lowering another
Fig.1 Simple weight-lifting machine.
Let’s state the following principle:
There is no such a thing as perpetual motion
(3)
with these weight lifting machines
(Actually, the no perpetual motion is a general statement of the law of conservation of
energy).
For the case of lifting-weight machines:
If, i) when we have lifted and lowered a lot of weights and ii) restored the machine to
the original condition, we find that the net result is to have lifted a weight,
then we would have a perpetual motion machine,
(because we can use that lifted weight to run something else, which can be repeated
again and again.)
Accordingly, for the case of weight-lifting machines, in the
absence of perpetual motion, after bringing the machine to
(4)
its initial state, the net lifted weight should be zero.
I.2.B Ideal machines and reversible processes
Consider an ideal frictionless lever.
6
m’
Y
3
1 meter
g
1
Figure 2. An ideal machine. A massless, friction free,
weight lifting machine.

We wish to lift a block of mass 3m.
For that purpose, the block of mass 3m is placed at the bottom on the right side of
the lever, and a mass m’ is placed at the left of the lever at a height h= 1 m.
Here m’ is slightly smaller than m. [ m’ = m -  with   0 ] .
Since m’ < m the system in Fig. 3 will remain stationary.
Y
m’
g
3
1 meter
1
3m
Figure 3. Lever in its initial position.
Let’s try to design a reversible machine out of this weight-lifting machine prototype.

Notice in Fig. 3 that the lever will be able to lift the block of mass 3m if some
additional energy K were added to the mass m’.
K
Y
K
m’
g
g
3
1 meter
1
1
meter
3m
1
m'
3
3m
X
Figure 4. Lifting a mass 3m. Left: A little bit of kinetic energy is given to m’ just to make
m’ to go down to the bottom level while lifting the 3m block. Right: Once the mass m’
is at the bottom, the mass 3m has been lifted a distance X.
Let’s choose the magnitude of K such that, afterwards, m’ lowers down to the
ground level height=0 , and the lever becomes stationary. Thus, the lever machine is
able to lift the block 3m to a height X (see right diagram in figure 4).
At this stage, we clamp the lever to, momentarily, avoid further motion.
7

The motion could continue if we just unclamp the lever (this occur because m’ is
slightly less than m, so a small torque acts on the lever).
When m’ reaches a height of 1 meter, it would have a kinetic energy K.
We can place a spring at the proper position such that , when the mass m’ reaches
the height 1 m, the energy K gets stored into the spring. Right after this occurs,
K
we clamp the lever again.
K
Y
m’
g
1 meter
3
g
3
3m
1
m
’
Spring compressed a bit
1 meter
1
X
3m
X
Figure 5. Reverse process of Fig.4 to bring down the mass 3m. Left: The lever is
unclamped. Right: Masses m’ and 3m are back to their initial positions (as in Figure 3),
after which the lever is clamped again.

If we wanted to lift the 3m block again, all we need to do is to unclamp the lever. We
notice we can repeat this process, back and forth, again and again.
What we have, then, is a reversible process:
The work (energy) that the lever invested in lifting the 3m
block by a distance X (Fig. 4) is at the expense of lowering
the mass m’ by 1 meter. By operating the lever in reverse,
mass m’ is lifted at the expense of lowering the 3m block
(Fig. 5).
(5)
Notice that for operating this reversible process,
we (the external agent) just need to clamp and (! )
unclamp the lever during the process; i.e. we do
not input into, nor extract from, the machine any
energy.
K
K
Y
m’
g
3
1 meter
g
1 meter
1
3m
m
’
X
3
1
3m
X
Figure 6. Reversible machine. Left: Releasing a clamp on the top left (no shown in the
figure) allows the machine to lift the mass 3m a distance X. Right: Releasing a clamp on
the bottom left (no shown in the figure) allows the machine to lower the mass 3m to
the bottom.
8
I.2.C Reversible and non-reversible machines
The machine in Fig 6, described above, is called a reversible machine.
It is an ideal machine (where friction is absent), which is in fact unattainable no
matter how careful our design. Its concept is however useful, for comparing it with
other non-reversible machines.
Reversibility has the connotation that the machine returns 100% of the energy
invested on it.
A non-reversible machine includes all real machines. They are subjected to friction
and other adverse factors that detriment their motion.
Comparing reversible and non-reversible machines.
Consider the reversible machine shown in Figures 6; let’s call it “machine-A”.
This machine lowers the mass m’ by 1 meter and lifts a 3m weight to a maximum
height X; it can also run in reverse (as described above in Figure 5.)
Consider another machine-B, which is not necessarily reversible. This machine lifts
a 3m weight to a maximum distance Y, while lowering a mass m’ by one meter (m’ 
m) to its initial position (Fig. 7.) The mechanism of how machine-B operates is
unknown. We now prove that Y cannot be greater than X.
More general, we state that it is impossible to build a
weight-lifting machine that, by lowering a mass m’ by
one meter, it will lift a weight any higher than it will
be lifted by a reversible machine.
(6)
Proof: Suppose Y >X.
That is, the special (unknown) design of machine-B allows lifting the 3m block to a
height Y while bringing the mass m’ one meter down. The design (its inventor claims)
is so good that Y>X.
K
Y
STARTING STATE
m’
Machine-B
3
1 meter
0
Unknown
arbitrary
design
Machine-B
g
1 meter
Y
3m
3m
3
m’
X
Unknown
arbitrary
design
Y
1
X
Figure 7. A non-reversible machine. Left: Initial state. Right: After one step operation.
Once the 3m block is at Y, we can let it free-fall to a height X and thus obtain free
energy.
9
g
Machine-B
Free
energy
3m
3
Unknown
arbitrary
design
1 meter
m’
Y
1
X
Figure 8. The mass 3m is brought from the
height Y to a height X, thus releasing energy.
We then transport the masses to the machine-A. The mass m’ (being already at the
lower position) is placed at the left side of the lever; and the block of mass 3m
(being already at the height X) is placed at the right side of the lever (see figure 9.)
K
Machine-B
Machine-A
3
Unknown
arbitrary
design
1 meter
m’
Y
1
1 meter
1
3
X
X
K
Machine-B
Machine-A
3
Unknown
arbitrary
design
1 meter
Y
1
1 meter
X
3m
1
m’
3
X
Figure 9. Masses from machine-B transported to machine-A.
We now run the the machine-A K
backwards (raising the mass m’).
K
Y
m’
g
1 meter
1
m’
3
3
3m
1 meter
1
X
3m
X
Figure 10. Machine-A operated in reverse.
If we now proceed to transport the masses to machine-B, we would have brought all
the masses (m’ and 3m) to their initial height (shown in Figure 7).
10
K
Y
Machine-B
m’
3
3
Unknown
arbitrary
design
1 meter
1 meter
1
K
3m
X
K
Y
X
Y
Y
FINAL STATE
m’
Machine-B
g
3
3
1 meter
1 meter
1
X
0
Unknown
arbitrary
design
Y
1
3m
X
Figure 11. Transfer of masses from machine-A to machine-B. The latter one ends up at its
initial state (as in Fig. 7).
Not only both machines would have been brought to their initial states. In addition,
we would have had extra energy (when the 3m block was thrown from Y to X, as
shown in Fig. 8). But this would constitute perpetual motion, which we postulate is
not possible. Therefore Y cannot be greater than X.
Accordingly, among all the machines that can be
designed, the reversible machines are the best.
They lift the 3m block to the highest height.
(7)
I.2.D Universal behavior of reversible machines
All reversible machines must lift the 3m block to exactly the same height.
Notice the proof of this statement is similar.
If one reversible machine-C were to lift the 3m weight to a height Z >X, we could free
fall that block to a height and then operate machine-A in reverse. This would
constitute perpetual generation of free energy, which is not possible. Therefore, Z
cannot be higher than X.
But X cannot be smaller than Z either. (The same argument used above with Z and X
interchanged).
Therefore, all reversible machines lift the 3m block up to
the same height.
(8)
This is a remarkable observation because it permits us to analyze the height at which
different machines are going to lift something without looking at the interior
mechanism.
If somebody makes an enormous elaborated series of lever that lift a 3m block a
certain distance X by lowering a mass m by one unit distance, and we compare it
with a simple lever that does the same thing and is fundamentally reversible, his
11
machine will lift it no higher, but perhaps less height. If his machine is reversible, we
also know exactly how high it will lift.
In summary, we have a universal law:
Every reversible machine, no matter how it operates,
which drops one kilogram by one meter and lift a 3-Kg
weight, always lifts it the same distance X.
(9)
The next question is: what is the value of X?
I.2.E Figuring out the actual lift capability of the reversible machine
3
m’
Y
Feynman now introduces an ingenious procedure to figure out the value of X.
alluded in the machines above.
Fig. 12 shows a reversible machine that lifts three blocks a distance X by lowering
one block a distance 1 meter. (By comparison with Fig. 6, the spring is omitted in
Fig.12 for simplicity). The blocks on the right side are located on weightlesscompartments separated,3by design, a vertical distance X.
For convenience, a fixed rack (with shelve compartments also separated a distance
K located near the machine, as shown on the right side of figure 12.
X) is conveniently
Y
m’
1 meter
0
1 meter
0
X
X
Figure 12. Left: Reversible machine (with slight modification relative to the machines
described in the paragraphs above). Right: An additional rack of shelves has been
placed for convenience at the right side.
The left side of Fig. 13 shows the position of the blocks after operating the machine,
lowering one mass on the left by 1 meter, and lifting each of the three masses on
the right by a distance X each.
Subsequently, the masses on the right are displaced laterally and deposited into the
rack (diagram at the right in Fig. 13). Since the latter involves just lateral motion, we
assume there is no need to invest energy to undertake this step (recall we are
assuming there is no friction).
12
3
3
Y
m’
m’
1 meter
1 meter
1 meter
X
0
X
Figure 13. After operating the machine (left), the masses are emptied from the machine
and loaded into the rack (right).
Comparing in Fig. 13 the position of the masses on the right side, before and after
the machine operation, we notice that,
the net effect of bringing one block down one meter (on the
left side) is to lift just one block from the bottom to the top
(10)
by a distance 3X.
3
m’
The reason is that shelves #2 and #3 effectively remain filled in both cases (before
and after the machine operation). The only difference is that shelve #1 was filled at
the beginning and empty at the end, while shelve #4 was initially empty but is filled
at the end. Therefore, energy wise the block in shelve #1 “jumped’ to shelve #4; i.e.
on the right side one block was raised by a distance 3X.
Next, let’s put the machine back to its initial state. Notice in Fig. 14 that this can be
done without investing energy.3 First, shelve #4 is emptied by moving (laterally) the
mass to the left side. Second, the mass on the bottom left is laterally moved to the
K
shelve #1 on the right.
m’
1 meter
1 meter
0
1 meter
0
1 meter
3X
X
X
Figure 14. Lateral motion of the masses to restore the machine to its initial state (as in Fig.
12).
Notice, the machine is now back to its initial state (compare Fig. 12 to Fig. 14).
We claim that 3X has to be equal to 1 meter.
So X= 1/3 meter.
(11)
Indeed, since by lowering one mass by a meter on the left side produces a lift of one
by a distance 3X on the right side, then 3X has to be equal to 1m. Otherwise,
i) if 3X were, for example, higher than 1 meter, we could throw the masses on the right
by a distance such that the top mass aligns with the 1 meter level; we would get that
13
free energy available for other useful work. Then we would proceed to put the
machine to its original state by translating the masses laterally (the top to the right,
the one at the bottom to the right). That way we would have obtained free energy in
the process and placed the machine back to its initial state. However, this would
constitute perpetual motion.
ii) Similarly, if 3X were smaller than 1m, we would start with the masses on the right at
their high level. Operating the machine in reverse, we would bring the three masses
down, while lifting the mass at the lift by 1 meter. We could then throw the left mass
to a level equal to 3X, thus obtaining free energy (useful for other work). Then we
move the masses laterally (the top-left mass to the right side, and the one at the
bottom of the shelf to the left side). We would have restored then the machine to its
initial stat and had obtained free energy in the process; i.e. ready for perpetual
motion.
In short, if the lifting weight machine is reversible then 3X has to be equal to 1 meter.
I.2.F The Potential Energy
We can generalize then expression (11):
One block falls a distance “h” in operating a reversible machine;
and the machine can lift “p” blocks by a distance “h/p”.
Or, if we use more than one block on the left side:
(Weight) x (distance) on the left side equals
(Weight) x (distance) on the right side.
We call the weight times the height gravitational potential energy.
Gravitational
potential energy
of an object
= weight x height
We state that
The potential energy, when operating the machines in a cycle,
conserves
(12)
(13)
Quoting Feynman: This is a very beautiful line of reasoning. The only problem is that
perhaps it is not true. (After all, nature does not have to go along with our reasoning.)
For example, perhaps perpetual motion is, in fact, possible. Some of the assumptions
may be wrong, … It turns out, it is found experimentally that expression (13) is, in fact,
true.
14
II. CONSERVATION of ENERGY in MULTI-PARTICLES SYSTEMS
Ref: This section taken from University Physics, by Ronald Lane Reese,
Vol. 1, Ch. 13]
II.1 The Classical Work Energy (CWE) Theorem: Case of a single particle
First, let’s consider the CWE theorem for the case of a single particle. Consider a
particle of mass m, which moves from the initial position indicated by “i” to the final
position “j” along the dashed curve shown in the figure. During its motion, a net force

F
acts upon the particle.
total
f
m dr
Ftotal
i
f
r
Wtotal ( i f ) =  Ftotal. dr
i
r is the vector
position of the particle
Figure 15. Force acting on a particle. As the particle moves from the
initial position I to the final position, the force does work on the particle.
f
 Ftotal. dr
f
=  m a. dr
i
i
But notice,
a. dr = d v . dr = dv . d r = dv . v
dt
dt
= v. dv = 1 d(v . v ) = 1 d(v2)
2
2
Accordingly,
f
 Ftotal. dr
i
f
=
1
2
m d(v2)
i
= 1 m [ (vf) 2 - (vi) 2 ] = Kf - Ki
2
That is,
Wtotal (i  f ) = Kf - Ki
(14)
15
The work done on the particle by all the forces is
converted into kinetic energy of the particle.
Equivalently,
Wtotal = K
(14)’
The contribution to the total work may come from conservative forces and other nonconservative forces;
Wtotal = Wconsrv + Wother
The nature of conservative forces is that they can be derived from a potential energy
function PE, such that,
Wconsrv = - PE)
Using the last two expressions in (14), one obtains Wother = K + PE) ; or,
Wother = K + PE )
(15)
The sum of potential energy and kinetic energy of the particle at any point in space is
called the total mechanical energy E.
Wother = E
The work done on the particle by the “other” (non-conservative)
forces acting the particle is converted into mechanical energy of
the particle.
(16)
II.2 The Classical Work Energy (CWE) Theorem: Case of a multiple particle rigid body
The result in (16) is extended to a multiple particle rigid body system. In such a case,
the internal forces between any pair of particles in the system are assumed to point
along the direction that joins the pair; the result is then that the work done by all the
internal forces is zero. The result in (16) then holds, although the calculation of the total
mechanical energy becomes more involved.
Wother = E
The work done on the particle by the “other” (non-conservative)
forces acting the particle is converted into mechanical energy of
the system of particles in the rigid body.
(17)
Indeed equation (17) allows us to calculate the velocity at which a cylinder, rolling down
through an inclined plane, will arrive to the bottom after being release from a given
height.
16
II.3 Limitation of the CWE theorem
We are already familiar with the CWE theorem, which establishes the source (work)
that causes a change in the kinetic energy of a system. Fig. 15 illustrates this theorem
for the case of an individual particle. There exist examples also shown its validity when
describing the dynamics of system of particles constituting a rigid body. The later
allowed us to solve, in a very straightforward manner, problems involving bodies rolling
down an inclined plane, for example. But, until now, cases involving the work done by
internal forces in non-rigid bodies have not been considered. We will consider one of
such a cases in the next example.
Case: Inelastic collision
Fig. 17 considers the inelastic collision between two objects. This example illustrates,
as we will explain below, the need for generalizing the CWE theorem, in order to include
cases in which disordered (microscopic) energy is involved.
Before the collision
Both particles are initially at the same
temperature and in thermal equilibrium
v
m
At rest
Kinetic energy:
Kbefore = ½ m v2
frictionless
M
X
Figure 16. Two blocks before undergoing a collision.
After the collision
V’
(m+M)
Kinetic energy:
Kafter = ½ (m +M) V’ 2
frictionless
X
Figure 17. Two blocks after an inelastic collision.
Since the linear momentum is conserved mv = ( m + M ) V’. Thus, V’ = mv / ( m + M ),
which gives,
Kafter = ½ (m +M) [ ( m /(m+M) ) v ]2
= ½ [ ( m2 / (m + M) ] v2
= ½ m v2 [ ( m / (m + M) ]
= Kbefore [ ( m / (m + M) ]
The change in kinetic energy is given by,
K = Kafter - Kbefore =
[ ( m / (m + M) - 1 ] Kbefore
17
= - [ ( M / (m + M) ] Kbefore
that is, the kinetic energy is less after the collision than before.
According to the CWE theorem this change should have resulted from the work done by
the forces acting on the system. But notice, all the external forces acting on the system
(normal forces
and weight)
perpendicular to the displacement of the particles,
Both particles
initially atare
the same
rest(WN = 0, WW = 0.) That is Wother in expression
temperature
and
in
thermal
equilibrium
hence, their work on the system isAt
zero
(17) is zero.
Nm
NM
frictionless
i
(m+M)
X
W1
W2
Figure 18. External forces acting on the system are perpendicular to
the displacement. Hence Wext =0.
Apparently, then, the CWE theorem, as written in expression (17) is not valid to describe
this inelastic interaction. However, the experiment is valid; that is, it is observed a
decrease in the total mechanical energy after the collision.
Thus, this example reveals the need to modify the CWE theorem. As described in the
next section, we will have to include the possibility that the total mechanical energy can
be transformed into kinetic energy taken away by the microscopic constituents (atoms,
molecules) of the two macroscopic bodies. Such an increase of the kinetic energy of the
microscopic components is then at the expense of the macroscopic mechanical energy.
As a consequence, the temperature (a concept ascribed to the motion of microscopic
particles) of the object will increase. In short, in an inelastic collision, the change in
mechanical energy of the system (change in macroscopic kinetic energy in this case) is
converted into microscopic energy U
II.4 Generalization of the Classical Work Energy (CWE) theorem: its extension to a
thermodynamic system
In a rigid body, we can predict the motion of every particle of a rolling cylinder. In a
more general case, a real body is not rigid; it deforms. Some regions may acquire higher
kinetic energy than other region, so an additional channel of energy distribution may
surge. It is clear then that expression (17) will have shortcomings to predict the motion
of this more general type of system. Accordingly, the purpose of this section is to
generalize the result (17). Such a modification will lead us to state the first law of
thermodynamics.
18
Expression to be modified:
Wother
E
=
In the CWE, the term on the left side
consideres WORK as the only mechanism
to change the kinetic energy of the system.
We will extend it to include heat-transfer,
driven by temperature difference, as
another mechanisms that causes the
energy of a system to change.
In the CWE the term E includes only
the (macroscopic) mechanical energy.
We will extend this right term to
include also, in addition to the
mechnical enery E, all types of
microscopic interactions processes
that lead to corresponding potential
and kinetic energies.
II.4.A Modification of the right side of Eq. (17)
Macroscopic and microscopic contributions to the total energy
In the right side of expression (17), E stands for the changes in the mechanical energy
of the system. The modified version will replace the change in the mechanical energy E
with one expression that includes, in addition, the change of the total energy of a
system.
The total energy of a system has two distinct contributions:


A macroscopic energy, the total mechanical energy E of the system,
associated with the macroscopic position and motion of the system
as a whole.
A vast collection of microscopic energies, known collectively as the
internal energy U of the system.
II.4.A1 The macroscopic mechanical energy E is the sum of:
- Kinetic energy of translation of the center of mass and
the rotational kinetic energy about the center of mass.
- Potential energies associated with the position of the center of mass of the
system on space, be they gravitational, elastic, or other potential energies
associated to other interactions (i.e. electrical forces).
Since the zero reference of these potential energies is arbitrary, only the changes
in the total mechanical energy are physically significant when the system
changes its state of mechanical equilibrium. That is, E is physically important,
not E itself.
II.4.A2 The microscopic internal energy U of the system is the sum of:
- the individual kinetic and potential energies associated with the motions and
interactions of all the individual particles (atoms and/or molecules) of the
systems.
- Many of these interactions involve rather complicated potential energy
functions on a microscopic distance scale.
19
Since the zeros of these microscopic potential energy functions, we can talk
about a definite value for the internal energy of the system when it is in a state
of thermodynamic equilibrium, but this is a rather complicated endeavor.
When the system changes its state of thermodynamic equilibrium, it is only
changes in its internal energy U that are physically significant in
thermodynamics.
Uinternal=  12mi ui 2   12 Pij
i
RCM
“Disordered” energy
(gas molecules inside the
container.)
ui = velocities
Pij = potential energies
i, j
VCM
Emacro = (1/2) MvCM2 +
+ (1/2) I 2 + Mgz
Ordered energy
(of the can cylinder)
Fig. 19 Identifying the macroscopic and microscopic contributions to the total energy of a
container (that has microscopic material components inside) thrown out in the air.
2
If the container had only
monoatomic components
Uinternal =  1mi ui 2
2
1
i
i
If the container had di-atomic components,
we have to consider rotational motion
Uinternal =
[
i
i
1
(mCM )i (uCM )i
2
2

1
I
2 i i
2
]
… and if the temperature were high enough
we have to also consider vibrational motion
Uinternal =
1
[
(mCM )i (uCM )i
2
i
20
2

1
I
2 i i
2

1
2
k x  ... ]
2 i i
Fig. 20 Simple examples of internal energy. The container has a great number of molecules
inside (of the order of the Avogadro’s number). Only one representative molecules is shown
in the figures.
II.4.A3 Ordered and disordered energies
The mechanical energy E of a system is also referred to as ordered energy. This
may responds to the fact that, because of its macroscopic character, one can
track its location (i.e. because one is able to track the location of the center of
mass).
The internal energy U of a system is called disordered energy due to the
microscopic un-tracked motion of the system’s many constituent particles.
II.4.B Modification of the left side of Eq. (17)
The transfer of Energy
In the left side of expression (17), Wother stands for the macroscopic mechanical work
done on the system (see figure 15 above). This mechanism is the way to provide kinetic
energy to the system (the system in that case is a particle, or a rigid body).We are going
to generalize this concept.
Energy can be transferred to a system (from other systems) by two mechanisms or
processes:


via work, exerted by macroscopic objects (i.e. driven by external macroscopic
forces). We will see that Work done on a system is a macroscopic mechanism
for transferring energy to a system.
via heat transfer, caused by temperature difference between the system and the
surrounding medium. We will see that Heat-transfer to a system is
fundamentally a microscopic mechanism for transferring energy to a system.
Example: Distinguishing work from heat-transfer
Fig. 21 Catching a cold soda can.
W’ext: In this case W’ext is numerically equal to (1/2)mv2. It is the macroscopic work
you do on the can to stop it (at the moment you catch the can).
21
Heat exchange
Interchange of random motion energy
between the can and your hand
Heat-transfer Q: The Interchange of random motion energy will not stop but until
the temperature of the can soda and your hand are equal
II.4.B1 Macroscopic external work W ’ext done on a system
The macroscopic external work W ’ext done on a system can cause a change in either
- the internal energy U of the system (example given below), or
- the total mechanical energy E of the system
Example where the external work causes a change of purely internal energy:
Insulation
(no heat transfer
Fext
Gas
External
non conservative force)
Q=0)
Fig. 22 Work done by an external force while compressing a gas
towards a smaller volume. That energy associated to the external
ends up causing an increase the internal energy U of the gas.
Figure 22 show a gas enclosed in a heat-insulated container. The insulated walls
ensure an absence of heat transfer (i.e. interchange of kinetic energy) from or
toward the system (the gas.)
A movable piston allows an external agent to compress the gas (by pushing the
piston), thus doing work on the system.
The work on the gas by the external agent results in an increase of the gas
temperature (indicative of an increase in the internal energy U.)
Example where the external work causes a change of purely mechanical energy:
- Using the same system of the example in Fig. 22, simply lifting the gas container
would be an example of increasing the mechanical energy of a system, without
changing the internal energy.
- Another example constitutes stopping the flight of a soda can (see Fig. 21 above)
II.4.B2 Heat-transfer Q
Mechanisms for heat-transfer are fundamentally microscopic (at the atomic and
molecular level.) Heat-transfer occurs by random molecular collisions and other
essentially molecular interactions.
Heat transfer is that energy transferred between two systems simply because they are
at different temperatures. The process of heat transfer continues until both systems
have the same temperature.
22
Heat transfer is essentially a microscopic atom-to-atom transfer of energy.
Heat-transfer occurs mainly by three mechanisms: conduction (two objects, each at
different temperature, placed in intimate contact), convection (via mass movement;
a hot fluid moves towards a region of lower temperature), and radiation (emission
of electromagnetic radiation due to the agitation of charges in a body).
In heat-transfer via conduction, the direction of heat transfer is from the higher to
the lower temperature (an aspect to be explained later with more detail in the
context of the second law of thermodynamics.)
Caution: Do not confuse heat-transfer Q with the internal energy U.
Just because a system has received a given amount of heat-transfer one cannot say
“a system has a given amount of heat.”
Heat transfer is not a property of the state of a system.
There is not a mathematical “heat function” associated with a system in thermal
equilibrium by virtue of its position in space or speed of its components.
In other words, (a system in thermal equilibrium does not have an amount of heat or
heat-transfer.)
On the other hand, a system in thermal equilibrium does have (in principle) a
specific internal energy U.
That is,
Q is not a state variable
U is a state variable
In the case of the gas system shown in Fig. 22 above, the state of the system is specified
by the value of its pressure, volume and temperature. A given state will have a specific
value of internal energy U.
II.4.B3 Thermal Equilibrium
Two systems are said to be in thermal contact if they are able to interchange heattransfer. If the heat transfer between two systems in thermal contact is zero, the
systems are said to be in thermal equilibrium; that is, they are at the same temperature.
Thermal equilibrium is the analogous to static equilibrium in mechanics.
The Zeroth Law of Thermodynamics
If systems A and B each are in equilibrium with a third system C,
then A and B are in thermal equilibrium with each other.
II.4.B5 Monitoring graphically the work W’ done on a gas system by a external force
Example: The work done on a system depends on the particular PV process.
A gas inside a container is subjected to compression by a external force. The state of the
gas is specified by its pressure and volume (P, V).
23
The transition from a state “i “ to a state “f “ can occur through different P-V processes
(depending on the external conditions to which the gas is subjected).
Fext
Gas
Pressure
External
non conservative
force)
Pressure
PV process
(I)
f
f
PV process
(II)
i
i
W’ ( I )ext >
0
W’ ( II )ext > 0
Volume
Volume
Fig. 23 The work done on the gas by the external force, when taking it from a state
“i “ to a state “f “ depends on the particular P-V process.
Left: The gas is taken from a state i to a state f following a particular P-V process (I).
In the process the external force makes a positive work W ( I )ext.
Right: The gas is taken from state i to a state f but following a different P-V
(II)
process (II). In this process the external force makes a positive work W ext.
Notice W(II )ext < W (I )ext.
Example: Net work done in a cycle PV process
Pressure
f
(I)
i
Pressure
Pressure
f
f
(II)(III )
W’ (I )ext >
0
W’ (II )ext < 0
Volume
Volume
i
i
Net work done during a
cycle process
W’ ext > 0
Volume
Fig. 24 The net work done by an external force in a PV-cycle process on the gas is positive
in this case.
24
Example: Net work done in a cycle PV process could be positive or negative
Pressure
Pressure
f
f
(II)
(II)
i
(I)
(I)
i
Net work done during a
cycle process
Wext < 0
Net work done during
this cycle process is
W’ ext > 0
Volume
Volume
External
External
Fig. 25 Left:External
The net external work in a cycle
process is positive.
non conservative force)Right:
nonThe
conservative
force)
non
conservative
net external work
done
by an
external forceforce)
on the gas in a
PV-cycle process is negative.
Fext
F
Fext
ext PV cycle process
Example: A different
Pressure
Pressure
Pressure
f
f
f
i
i
i
W’ ext < 0
W’ext > 0
W’ ext > 0
Volume
Volume
Volume
Fig. 26 Illustration of net external work in a cycle process.
II.4.C Fundamental Energy Conservation Law
Generalization of the classic work/kinetic-energy theorem
The ideas expressed in II.4.A and II.4.B about a) the total energy of a system and b)
the energy transfer into a system are consolidated here into a fundamental energy
conservation law.
We express the generalized classical work energy (CWE) theorem as follows,
Q + Winternal + W ’ext-non-conserv + Wext-conserv =
= KCM + Kmicroscopic
In expression (18),
25
(18)



Q is the heat-transfer into the system caused by temperature difference
between the system and the external world.
It explicitly separates out the macroscopic work (done by external macroscopic
forces, whether conservative or not) from the work done by microscopic forces.
W’ext-non-conserv refers to the work done by forces that cannot be classified as
conservative, like, for example, the force pushing the piston in Fig. 17 above.
Wext-conserv stands for the work done by a macroscopic object. It could refer,
for example, to the the gravitational force.
It is assumed that the kinetic energy changes in both forms: macroscopic (the
CM kinetic energy) and microscopic (the chaotic motion of the microscopic
atoms or molecules).
Let’s modify expression (18) a bit, in order to express it in terms of more frequently used
quantities.
First, for external conservative forces, the work can be derived from a potential energy
function Ep,
Wext-conserv = - Ep
(19)
which gives in (18),
Q + Winternal + W ’ext-non-conserv + (-Ep ) = KCM + Kmicroscopic
Q + Winternal + W ’ext-non-conserv
=
( KCM + Ep ) + Kmicroscopic
(20)
By calling
KCM + Ep ≡ Emacro
(21)
the macroscopic mechanical energy,
expression (20) can be written as,
Q + Winternal + W ’ext-non-conserv = Emacro + Kmicroscopic
(22)
Second, we can envision that, ultimately, Winternal causes a change in the microscopic
potential energies of the interacting microscopic particles that constitute the system.
That is, Winternal =  -Pij. Replacing this expression in (22) we obtain,
i j
Q +
Q
+

i j
-Pij
+ W ’ext-non-conserv = Emacro + Kmicrosc
W ’ext-non-conserv
= Emacro + Kmicrosc +
26

i j
Pij
The last two terms in the right side of the expression above constitute what we called at
the beginning of this section the disordered internal energy U of the system.
Q
+
W ’ext-non-conserv = Emacro + Kmicrosc
+

i j
Pij
(23)
Change in the
internal energy U
Through the derivation process given above, we notice that the work energy is
deposited (transformed) into the system as either,
macroscopic mechanical energy, or
internal energy.
The work Wext-conserv done by conservative macroscopic external forces has been
assimilated into the mechanical energy, while the work Wint done by microscopic forces
ended up being grouped into the internal energy term.
The expression above also shows that the work energy W’ext-non-conserv done by external
non-conservative forces could end up either as macroscopic mechanical energy or
internal energy (the latter case is illustrated in Fig. 17 above where a gas was
compressed by a piston; the force acting on the piston is the non-conservative force.)
In summary
Q
Heat-transfer
into the system
caused by
temperature
difference
+ W’ext-non-conserv
Work done on the
system by a nonconservative
macroscopic external
force
Emacro
=
Change in
macroscopic
mechanical
energy of the
system
+
U
Change in the
internal energy U
of the system
which constitutes our Fundamental Energy Conservation Law.
Energy is transferred to a system by heat-transfer Q into the system and by work W’
done on the system
Inelastic collision (revisited)
Before the collision
27
(24)
Both particles are initially at the same
temperature and in thermal equilibrium
At rest
v
m
frictionless
Kinetic energy:
Kbefore = ½ m v2
i
M
X
After the collision
V’
(m+M)
frictionless
Kinetic energy:
Kafter = ½ (m +M) V’ 2
i
X

Here Q is the flow of energy by heat transfer, caused by temperature differences. In
our case it is zero.

W’ is the work done by external forces. In our case it is zero.

U change in the internal energy

E change in the mechanical energy
In this case E = - [( M / (m + M) ] ½ m v2
Accordingly, in expression (24) we have,
0 + 0 = - [( M / (m + M) ] ½ m v2 ) + U
which gives,
U = [( M / (m + M) ] ½ m v2 )
That is, the missing (ordered) macroscopic kinetic energy appears as an increase in the
internal (disordered and essentially microscopic in character) energy U of the system.
The increase in the internal energy of the system typically manifest itself in an increase in
the temperature of the system. As the temperature of the system increases above the
ambient environment because of the increase in the internal energy, heat-transfer
subsequently occurs from the system to the environment until the system-ambient reach a
common temperature.
III CONSERVATION of ENERGY. Case: Pure Thermodynamics
In pure thermodynamics, one typically considers only systems whose total
mechanical energy does not change, Emacro = 0. The general statement of the energy
conservation becomes,
Q + W’ext non-conserv = U
(25)
28
nsulation
no heat transfer
Q=0)
nsulation
no heat transfer
Q=0)
where W’ext non-conserv is the work done by external nonconservative force on the system. Or, more generally,
external work on the system by the surroundings.
Notice
Before
After
F
F
Work done by the external force F > 0
W’ext non-conserv > 0
Before
After
F
F
Work done by the external force F < 0
W’ext non-conserv < 0
It is typical to consider the work done by the system (no the work done on the system
by the external non-conservative forces.) Since, according to the Newton’s third law, the
force exerted by the system is equal in magnitude but opposite in direction, then
W’ext non-conservative
= - W done-by-the-system
=
Work done on the system
by the external surroundings
(26)
Work done by the system
on the surroundings
Thus, for pure thermodynamic systems
QHeat-transfer-into-the-system
- Wdone-by-the-system = U
(27)
When all the terminology is understood, the subscripts are omitted and one simply
writes
Q - W=
U
The First Law of Thermodynamics
29
(27)’
nsulation
no heat transfer
Q=0)
nsulation
no heat transfer
Q=0)
Notice
Before
After
F
F
Work done by the gas < 0
W <0
Before
After
F
F
Work done by the gas > 0
W >0
Question:
wedge
Before
After
W=?
30
Question: Consider two cases
1) a gas with initial volume Vi is free expanded to a volume Vf and then adiabatically
compressed back to its volume Vi quasi-statically, after which the temperature of the
gas has increased by T.
2) a gas with initial volume Vi is heated up causing its temperature to increase by T.
What ius the entropy change in each case.
31