An Introduction to Thermodynamics phần 1 pdf

7/10/07 Intro- 1
An Introduction to Thermodynamics
Classical thermodynamics deals with the flow of energy under conditions of equilibrium or
near-equilibrium and with the associated properties of the equilibrium states of matter. It is a
macroscopic theory, ignoring completely the details of atomic and molecular structure, though not
the existence of atoms and molecules to the extent required for writing chemical reactions. Time
is not recognized as a variable and cannot appear in thermodynamic equations. For students who
have become familiar with atoms and molecules, it may be surprising to find how far one can go
toward treating chemical and physical equilibria without employing any simplified models or
delving into theories of molecular structure.
The detachment of thermodynamics from molecular theory is an important asset. The
fundamental principles of thermodynamics were developed during the 19 century on the
th
foundation of two principal axioms, supplemented by a small number of definitions, long before
atomic structure was understood. Because of this lack of dependence of theory on models, even
today we need not worry about our vast ignorance at the molecular level, especially in the areas of
liquids and ionic solutions, in applying thermodynamics to real systems. It has been said, with
some justification, that if you can prove something by thermodynamics you need not do the
experiment. Such a strong statement must be handled with care, but it should become clear in the
following pages that common practice is quite consistent with this assumption.
Two developments associated primarily with the 20 century introduced substantial new
th
insights into thermodynamics. Statistical thermodynamics, or statistical physics, originated with
the efforts of Maxwell and of Boltzmann in the late 19 century and grew with additions by
th
Gibbs, Planck, Einstein, and many others into a companion science to thermodynamics. Because
statistical thermodynamics relies on specific models of atomic and molecular structure and
interactions, it provides important tests of those models, at the expense of substantially greater
mathematical complexity than classical thermodynamics. More important for present purposes,
statistical mechanics provides much greater insight into the quantities that appear in
thermodynamic equations, and thus a clearer view of why things happen. Thus we will not

hesitate to introduce some basic principles of statistical mechanics (without the extensive
mathematics) when necessary to explain what is going on.
The other new development, largely responsible for the change in physics from what is
generally considered purely Newtonian to relativistic and quantum physics, arose from the
introduction of operational definitions at the end of the 19 century. This viewpoint requires that
th
any definition (of energy, position, or time, for example) must include a statement of how we can
measure the quantity. Application of this criterion demands clarification of some quantities that
were introduced casually, without a solid foundation, in the early days of thermodynamics. We
will try to be more careful in explaining what is meant by our symbols, and what can or cannot be
measured, than has been customary in thermodynamic textbooks.
One of the characteristics of thermodynamics is that most of the terms are familiar. Everyone
has heard of energy, of heat, and of work. The difficulty is that we must sharpen our definitions
to distinguish between loosely associated ideas. We will therefore be particularly careful to define
these familiar quantities carefully, often emphasizing what our technical meanings do not include
as much as specifying the intended meanings.
These are also called vacuum flasks, because the space between the silvered double walls
1
has been evacuated, a design developed by Sir James Dewar. Another common name for these
and for containers of different design but for the same purpose is “Thermos” bottle, which is the
trade name of the American Thermos Products Co.
7/10/07 Intro- 2
One of the guidelines of early thermodynamics was that all energy transfers (under
equilibrium conditions) can be classified as either “heat” — transfer of energy because of
temperature differences (thermo) — or as “work” — transfer of energy because of forces and
motions (dynamics). It seems appropriate, therefore, to begin with definitions of energy, heat,
and work.
ENERGY. Energy has been a difficult quantity to define because it has so many faces, or forms in
which it may appear. Initially, energy was defined to be the energy of motion, or kinetic energy,
which for most objects under usual conditions is half the mass times the square of the speed,

E = ½ m
v
(1)
2

The most convenient, and generally reliable, definition of energy is that it is kinetic energy or
any of the other forms of energy which can be changed into kinetic energy or obtained from
kinetic energy. These other forms of energy include rotational energy (a spinning ball or
weathervane), vibrational energy (a mass oscillating up and down on a spring), and potential
energy (a skier at the top of a slope), as well as energy within an object, called internal energy.
HEAT. Most of the internal energy is associated with the nuclei or with the chemical state of the
object. We will generally ignore the nuclear energy. For any given sample of matter, the nuclear
energy typically remains unchanged. Changes of chemical energy will be considered when there
are chemical reactions. For now, we are more concerned with the relatively small portion of the
internal energy that changes when the temperature changes; it is most often called “heat”, or more
narrowly defined as thermal energy.
The meaning of the thermodynamic term “heat” can best be explored by consideration of a
few qualitative or semiquantitative experiments. For each of these we will develop a working
hypothesis, select a crucial test, and revise the hypothesis as necessary.
Our understanding of heat is based upon common experiences. When we stand before a fire,
or when we place a pan of water over a gas fire or in contact with an electrically heated coil, our
senses and the change in character of the water tell us that something passes from the fire or hot
coil to nearby objects (specifically to us or to the pan of water). The effect is to “heat” the
objects, by which we mean that there is a sensation of warmth that can be verified by a
thermometer. The thermometer, in some way, measures this “heat”. We seek to find the
relationship between temperature, heating, and heat.

Temperature balance? As an initial hypothesis, assume that a thermometer measures the
amount of heat. If so, we should find that a loss of temperature by one body is compensated by a

gain of temperature by another. To test this we put 200 g of hot water, at 90 C, into each of two
o
Dewar flasks (Figure 1). To the first flask we add 50 g of water initially at 20 C, and stir until
1 o
7/10/07 Intro- 3
the temperature becomes steady. The new temperature is found
to be about 76.0 C. To the second flask we add 25 g of water, at
o
20 C, and find the final temperature to be about 82.2 C. We must
o o
ask now whether the experiment shows the initial hypothesis to be
fully satisfactory or not.
There has indeed been a loss of temperature by the water in
the flask and a gain in temperature by the water added. But there
is clearly no “temperature balance”. The water in the flasks
changed temperature only slightly, whereas the water added
increased in temperature several times as much. Also, the water
in the second flask dropped in temperature less than that in the
first flask, but the water added to the second flask increased in
temperature more than that added to the first flask. Examination
of the results (Table 1) shows that the drop in temperature of the
water originally in the flasks is roughly doubled when twice as
much cool water is added.
The experiment just described suggests how the original
hypothesis might be revised. It appears that a larger amount of
water can absorb more heat for a given temperature increase. The
temperature, therefore, is more nearly a “concentration” of heat.
From this revised hypothesis we predict that the temperature
change times the amount of the substance should be the same for both the added and the original
water (see Table 1).

Table 1 Temperature Measurements*
Sample T
final
∆T ∆T -∆T /∆T -∆T M /∆T M
s w w s w w s s
H O,50g 76.0 56.0 -14.0 0.25 1.0
2
H O,25g 82.2 66.2 -7.8 0.125 1.0
2
Al, 50g 86.35 66.35 -3.65 0.0050 0.22
Al, 25g 88.12 68.12 -1.88 0.0276 0.22
* Temperatures in C. ∆T is the temperature change of the sample and
o
s
∆T is the temperature change of the water originally in the flask.
w
Initial temperature of the water is 90 C and of the sample, 20 C.
o o
M is the mass of the sample added and M the mass of water in the flask.
s w
It is necessary to find out whether the same relationship will hold if we exchange heat
between two different substances. To do this we again prepare two flasks, each containing 200 g
of water at 90 C, then add to one a block of aluminum, at 20 C, with a mass of 50 g and to the
o o
second a block of aluminum, at 20 C, with a mass of 25 g (Figure 2). After a few seconds we
o
may assume that the temperatures of the aluminum blocks are equal to the temperatures of the
surrounding water — about 86.35 C for the larger block and 88.12 C for the smaller block.
o o

7/10/07 Intro- 4
Multiplying temperature change by mass and comparing the result for the aluminum block and the
water shows that the ratio is the same for both parts of the experiment with aluminum, but
appreciably different from the results of the earlier experiment. We conclude, therefore, that
aluminum and water have a different “heat capacity”, so that a given amount of heat added to a
certain mass of one produced a different temperature change than equal heat added to the same
mass of the other.

Volume or mass? We have left unanswered the question whether
the “heat capacity” depends upon the volume or the mass of the
substance that is absorbing the heat. The choice can be easily made by
means of an experiment employing a substance, such as air, that can
readily change volume without changing mass. We fill one flask with
air, evacuate a second identical flask, and immerse both in water, with
a connection provided between the flasks, as shown in Figure 3. The
temperature of the water is measured; then the stopcock is opened,
allowing the air to expand to twice its initial volume, and the
temperature is remeasured. The temperature is found to be
unchanged. From this we conclude that the temperature of the air did
not change with the change in volume, and therefore that it is better to
define the “heat capacity” in terms of mass rather than of volume.
(The result is confirmed by more sensitive tests.)

Is heat gained or lost? In each of the measurements described
thus far it has been possible to follow heat as it flows from one body to
another; the amount lost by one substance has been equal to the
amount gained by the other. It is necessary to determine whether this is always true. (If it is, we
would say that heat is “conserved”, or that the “amount of heat” is constant.) Taking a hint from
the famous observations of Count Rumford, who noted the great quantities of heat evolved during
the boring of cannons, we design our next experiment to include mechanical motion, in which

energy will be added from motion (i.e., by “doing work”). Instead of expanding the air from one
flask into an evacuated flask, we can let it expand against a piston, as shown in Figure 4. This
time the temperature of
the gas drops (about
50 C) during the
o
expansion, even though
we add insulation
around the cylinder to
prevent the flow of
heat outward from the
gas. The change of
temperature cannot be
solely because of the
volume change; the
7/10/07 Intro- 5
previous experiment showed that the change of volume did not cause any change of

A doubling of volume causes a temperature drop
from 25 C to about - 25 C.
o o
temperature. The fact that the gas pushes on the piston, causing it to move, must be the
important difference.
A few additional experiments will provide more information on the relationship between
expansion, with work being done, and temperature effects on gases. (For brevity, only the results
of these experiments will be discussed.) Compression of a gas causes an increase in the
temperature just equal to the decrease of temperature during expansion, if both expansion and
compression processes are slow. It is therefore possible, by repeated expansion and compression,
to cycle the temperature between two values. Any other property of the gas that we might
measure, such as density, volume, or viscosity, will be found to depend only on the temperature as

measured by a thermometer, and not on how that temperature was achieved (for any specified
pressure). In other words, the “heating effect” of a compression seems to be exactly the same as
the heating effect of a flame or other source of heat. Thus it is possible to compress a gas,
thereby raising its temperature; then extract heat from it by removing the insulation until the gas
has returned to room temperature; expand it into an evacuated space without change of
temperature; compress it to again increase its temperature; extract heat; and so forth, as many
times as we wish.
Clearly, heat is not a quantity that retains its identity after it is absorbed by a substance, for
we can add any amount of heat without changing the properties of a gas in any way (provided
only that the proper amount of work is done by the gas). There is no property that will enable us
to determine the amount of heat added to any substance, or the amount of heat removed. The
description of temperature as the “concentration of heat” is therefore untenable, and must be
abandoned.

If not heat, then what? Temperature is related to a “concentration” of something more
fundamental, which can give rise to heat or can cause a gas to do work and which is increased
when the substance absorbs heat or when work is done on the substance. This quantity so
directly related to temperature is called energy.
In classical physics, any measurement of energy is necessarily an energy difference. We often
Be particularly careful to distinguish between Q, which is an amount of thermal energy
2
transferred, and the change in a property, such as thermal energy or E. To write ∆Q displays a
confusion to all who may read your notes.
Be aware that older thermodynamic literature quite generally defined work with opposite
3
sign. Because of an emphasis on steam engines, W was taken to be work done by an object,
rather than work done on an object. Some textbooks have chosen both definitions, changing from
one chapter to another. Again, the amount of energy transferred is W, not ∆W.
7/10/07 Intro- 6
write E for energy, where we mean ∆E, the difference between the current energy value and some

implied reference level of energy. There are several meanings for “energy”. Total energy includes
information on where the sample is located and how it is moving. Measure-ments made on a
sample at rest with respect to our apparatus would measure the internal energy. A small portion
of that internal energy is the thermal energy, which is a collection of several different forms of
energy, or modes of storage of energy, in matter, that can be changed by a change of temperature.
It will be sufficient to let E represent total energy, keeping in mind that the changes we are
primarily concerned with will be changes of internal energy and, most often, changes of thermal
energy.

Transfer of “heat”. Furthermore, in thermodynamics we are primarily concerned with the
transfer of energy to or from a system, including the transfer of thermal energy. This quantity is
nearly always represented by the symbol Q. Thus Q represents a change of energy of the system;
2
Q = ∆E (2)
To ensure that all the energy transfer occurs as transfer of thermal energy, only, we sharpen
the definition of Q:
Q (thermal energy transfer) is the transfer of energy between two objects in
physical contact as a consequence of a difference in temperature between the
objects.
The importance of this refinement will become apparent later.
WORK. We distinguish “work” from “play”; we say that machinery “works” or “doesn’t work”;
and we read that if a force acts on a body and the body moves through some distance, work is
done on the body. But in thermodynamics, work has a special meaning.
First of all, work is a transfer of energy from one object to another by the action of a force. If
work is done on an object, that increases the energy of the object. Thus, if the only energy
3
transfer is as work,
W = ∆E (3)
Second, and equally important, all investigations thus far of transfer of energy as work
∑

∑
∫
⋅==
i i
iii
WW (4) dxf
∫
∑
⋅






dsf
i
i
The product of force and distance is an integral of a scalar product, or dot product.
4
That will be automatically taken care of in most of our applications. In mechanics, we often
measure work done on a “particle” (or, better, a “physical particle”), which can change only its
kinetic energy. For such a particle it is easily shown that work is If·ds. We are concerned here
with more general bodies, subject to rotation and deformation, for which the definition of work
must be more explicit.
The imposter equation, (∆p) /2m =
I
f ·ds, where p = m
v,
has been called the second-

5 2
net
law equation, SLI. It is a valid statement of Newton’s second law, net force is equal to mass
times acceleration, but except for special circumstances it has no connection to W.
7/10/07 Intro- 7
indicate that the amount of work done is equal to a product of the force exerted (on the object)
4
times the displacement of the point of application of the force.
If more than one force is acting, each force must be multiplied by the distance through which
it acts. Then the work terms may be added.

What is the imposter equation? The expression for work done is often abbreviated to “work is
equal to force times distance”, but this is often interpreted to mean a net force times the distance
traveled by the object. Is this imposter equation ever wrong?
Imposter equation W =?If ·ds = (5)
net
Yes, if it is interpreted as giving the work done. We give here just two examples. First, if
you jump, you move upward because the floor exerts a force on you (which is the reaction force
to the force you exert on the floor). You move, and momentum p = (m
v
) is transferred between
you and the floor, but the force exerted on you times the distance you move (while in contact
with the floor) cannot be equal to work done on you because the floor does not transfer any
energy to you. The point of application of the force is at the floor, and that point doesn’t move.
Second, if you drag a box across the floor, it is easy to find the force you exert on the cord
and the distance the cord moves, so work done by you on the cord is known. Similarly, the force
exerted by the cord on the box can be measured, as well as the distance the box moves, so work
done by the cord on the box is known. (Energy transferred, as work, to the cord is equal to
energy transferred, as work, by the cord to the box) But the forces of friction between the box
and the floor are molecular-level forces, not known in detail. And the distances over which these

molecular-level forces act, individually, cannot be known. A simple “force times distance”
calculation gives an answer, but it is demonstrably wrong!
5

Analogues. Fortunately, we can usually avoid ambiguities in the definition of work in our
7/10/07 Intro- 8
analysis of equilibrium thermodynamics. We must, however, be sensitive to the differences
between what is transferred and what is stored.
The description of energy transfers as “heat” (thermal energy transfer) and as “work” may be
compared with deposits and withdrawals from a savings bank. The deposit slip may ask for a
separate listing of bills and coins, and a withdrawal may be in the form of bills or coins. Yet the
account balance itself is neither bills nor coins. In the same manner, energy may be put into a
substance, or withdrawn from it, either as Q, a thermal energy transfer (“heat”) or as W, “work”,
but it exists within the substance only as energy — not as Q or W. (Remember that money may
also be deposited by check or electronic transfer. We must be alert to the possibility of
transferring energy by means other than Q and W.)
If we are being very casual, it may be sufficient to describe thermal energy and thermal
energy transfers as “heat”, without additional labels. That, however, is very much like an
accountant choosing a single label (perhaps “money”) to indicate income and net worth, or trying
to balance your check book without distinguishing between net deposits for the month and
balance at the end of the month. Learning to distinguish between thermal energy and Q, the
transfer of thermal energy, will go a long way toward helping you understand discussions of
thermodynamics.
By going beyond thermodynamics, into statistical mechanics, the internal energy can be
described in terms of thermal energy and other forms of energy (such as chemical), and the
thermal energy may be further broken down into kinetic energy and potential energy of individual
atoms and molecules. But for purposes of thermodynamics we need know nothing more about
energy than that it includes a component related to the temperature. This component, called
thermal energy, can be internally converted to increase or decrease the temperature without
changing the internal energy. (If a mixture of oxygen and hydrogen is ignited, the gases become

substantially hotter, without any transfer of energy to or from the surroundings.) Or the thermal
energy can be transferred to or from the surroundings either as thermal energy transfers or as
work. We can find by experiments, such as the compression-cooling experiment or a variety of
others performed by Joule, how much thermal energy transfer is equivalent to how much work
and, for a given substance, how much energy (by either transfer method) must be put in for a
given temperature rise. The relationships between temperature, energy, thermal energy transfer,
and work will be considered quantitatively in the discussion of the first law of thermodynamics.
7/10/07 1- 9
1 The First Law of Thermodynamics
Thermodynamics is based on a small number of postulates, or assumptions. These are called
the “laws” of thermodynamics because they are suggested by a great amount of accumulated
experimental evidence. In fact it is extremely important to keep in mind that thermodynamics is
important just because there is total agreement between the results of thermodynamics (properly
applied) and all careful experimental results available to us. Because it is not possible to prove
the fundamental assumptions of thermodynamics, both the postulates and the derived results of
thermodynamics have often been challenged. In every showdown thus far, thermodynamics has
been shown to be correct.
Energy
The first step in understanding thermodynamics, and making it serve your purposes, is to
learn how to evaluate changes in energy in any object. As you probably recognize already, that
means understanding how to evaluate Q and W, the transfer of thermal energy to (or from) the
object and the work done on (or by) the object. To do so, we will write what is known as “the
first law equation,” which directly relates Q and W to the change of energy. But before we can
move ahead, we must look at how we define the object or quantity that is to give up or receive
energy; we must consider what is meant by a property; and we must examine the meaning of a
conservation law.

System and Surroundings. When we consider a ball projected through the air, or a block
sliding down a plane, it is quite satisfactory to talk about the ball, or the block, or simply the
object. If we wish to consider a gas or a solution, “object” is no longer a very appropriate

description, but we could still refer to “the gas” or “the solution” or simply call it “it”. A better
method is to call whatever is of primary interest (ball, or block, or gas, or solution, or whatever)
the system. One advantage is that we needn’t change descriptions when we substitute one gas for
another, or a solution for a gas, and so forth. Another advantage is that it allows for a smooth
transition to problems in which we choose an open system, that is, where we allow material, as
well as energy, to pass back and forth, into or out of the system. Unless specifically indicated
otherwise, however, we will assume our systems to be closed.
Whatever else is around, we simply call the surroundings. Then, ideally, we need consider
only system and surroundings. To avoid difficulties that we might encounter in trying to describe
the far reaches of the universe, it is quite sufficient to limit the term surroundings to include all of
the universe that might be affected by whatever change, or process, we are considering.
It is important that we clearly define what we mean by our system. When we have done so,
the surroundings is usually adequately defined.
There will be times when you will find it more convenient to deal with the interactions of two
bodies, or two “systems”, with each other, ignoring anything outside those systems. That is
perfectly legal. Terminology such as “system” and “surroundings” is meant to be an aid, not a
liability. On the other hand, it has been shown many times that disdaining the definitions of
7/10/07 1- 10
system and surroundings provides sufficient ambiguity to allow erroneous conclusions to be
"proved". Also note that system and surroundings are essentially equivalent and arbitrary labels,
so we are free to interchange them if we wish.

Properties, or State Functions. Examine a block of copper and you will find that it has a
certain shape, a certain volume, a certain temperature, a certain density, and various other
properties that characterize the sample. Each of these may be changed, by changing the
temperature of the copper, or to some extent by changing the pressure exerted on the copper.
Neglecting the shape, which depends on the specific sample but not on the nature of the copper
itself, we call each of these measurable quantities a property of the copper. The state of the
copper, or state of the system, is defined by these properties, so they are often called state
functions.

A gas, such as a gram of oxygen, O , is adequately described if we know the temperature, the
2
pressure, and the volume, but two of these are sufficient to determine the third, so only two state
functions are sufficient to define the state of the system. Other substances, such as iron or plastic,
may have additional properties (such as magnetic state or strain) that must be specified.

Conservation Laws. In mechanics we pay special attention to quantities that do not change
during a process we are studying. A “free particle” (not acted on by any net force) will have a
constant velocity — no change in speed or in direction. Such a “constant of the motion” may be
said to be “preserved” under the special conditions of the process. In a frictionless system, energy
will be preserved, or unchanged. When a liquid is poured from one container to another, the
volume of the liquid is preserved. Unfortunately, it has become common practice to refer, in such
instances, to the unchanging quality as being “conserved”, although the term “conservation” has
quite a different, very important meaning.
Volume is something we can see and often measure with a meter stick, whether the sample is
a copper block or the gram of oxygen gas (in a rigid container). For many processes, the volume
of the system (the liquid) is "preserved", but we also are aware that the volume of such a system
can be changed. With a gas, we need only move a piston, or change the pressure or the
temperature. With the block of copper or a free liquid, a change of temperature will change the
volume.
What we cannot do is change the volume of our selected system without causing some
change elsewhere. If the volume of the system increases, then less volume is available outside the
system, so the volume of the surroundings must decrease, by the same amount. Thus we have the
seemingly trivial, but very important, conclusion that the volume of the system plus the volume of
the surroundings is constant. That, of course, is because of the assumed properties of local space.
We state this result as a conservation law. Volume is a conserved quantity. The volume of a
gas, or liquid, or a solid may be changed, but the volume of the system plus the volume of the
surroundings remains unchanged for all changes we make. Let ∆V = V - V represent the change
2 1
in V, the volume. Then the law of conservation of volume would be written

(∆V ) = 0 (1)
system + surroundings
The best-known of the three thermodynamic postulates is known as the first law of
7/10/07 1- 11
thermodynamics. It is just the principle of conservation of energy. It may be stated in the form:

First Law The energy of the universe is constant.

A better form for our purposes is to write
First Law (∆E) = 0 (2)
system + surroundings
Remember that ∆ (Greek capital delta) indicates a change in the quantity that follows, in this
instance a change in the energy, E. Expressed in symbols,
∆E / E - E (3)
final initial
Because we are primarily interested in changes within that part of the universe that we
choose to call the system, which generally is a small, fixed quantity of substance, a symbol such as
∆E (unless specifically labeled otherwise) will always refer to the system under discussion. When
the energy of the system increases, ∆E is positive; when the energy of the system decreases, ∆E is
negative. In any given problem, the system to be considered must be carefully defined. The
surroundings are then adequately defined by what is outside the system. It should be clear that
equation 2 is, indeed, a statement of a conservation law, equivalent to the previous statement of
the first law.
It is often convenient to think in terms of a system that is totally isolated from its
surroundings. Unfortunately, such arrangements are not possible. We may prevent energy from
moving to or from a system, or hold volume of the system constant, or keep the pressure on the
system (exerted by the surroundings) constant, or make other special provisions, but there is no
practical way to define a truly isolated system and, if we had one, we could not make
measurements on the system. We must therefore carefully specify in what sense we want our
system to be isolated. It may be isolated in the sense that no energy is transferred to or from the

system, or it may be isolated in a more narrow sense. If the system is insulated, there is no
thermal energy transfer, Q. If the volume of the system is being held constant, there can be no
transfer of energy as work of expansion or compression. To simply call the system “isolated”
leaves the question indeterminate as to what is being prevented or what is maintained constant.

Heat, Work, and the First-Law Equation. Thus far we have considered energy transfers, to
or from a system, only as Q or as W. These will suffice for the present. Then we can write the
total energy change for the system in the form
∆E = (∆E) + (∆E) (4)
heat work
— but this is awkward. It is better, and conventional, to introduce new symbols for the terms on
the right-hand side, as we have already indicated. We let Q be the amount of thermal energy
transferred to the system from the surroundings, and let W be the amount of work done on the
system. Then, allowing for both modes of energy transfer, at the same time,
“First-Law Equation” ∆E = Q + W (4a)
The notation (V) at the left of the equation indicates that the equation is valid when the
1
volume of the system is held constant. We are also assuming, for the present, that there is no
other form of work, such as electrical work.
Other choices are δQ and ñQ, each of which is often misinterpreted, including confusing
2
the small magnitude with an inexact differential (i.e., change).
7/10/07 1- 12
Note that this “first-law equation,” by itself, tells us nothing about whether the energy for the
entire universe changes or not.
Roughly, the division of energy transfers into “heat” (thermal energy transfer) and “work”
corresponds to a division into random motion and directed motion (or collective motion). At this
point we have no good criterion for deciding whether light, for example, should be called “heat”
or “work”, but for the present it is arbitrary. Later we will find that the second law of
thermodynamics deals with these distinctions.

Insofar as all energy transfers may be measured either as heat (thermal energy transfer) or as
work, or a sum of these, this equation accounts for all possible changes in the energy of the
system. This equation is called the first-law equation. If we combine this equation with the first
law (conservation of energy) we may conclude that Q = - Q and that W = -
system surroundings system
W . It is necessary to add additional terms to the first-law equation when there are other
surroundings
modes of energy transfer (including especially the transfer of matter to or from the system). A
few simple examples of energy changes, in the following sections, will show the meaning of the
terms in equation 4.
TEMPERATURE CHANGES AND HEAT CAPACITY, C . Consider first the process of heating a gas
V
confined in a container of constant volume. Because there is no directed motion, no work is
done.
(V) ∆E = Q (5)
1
We are often concerned with very small (infinitesimal) changes in energy, which would imply
an infinitesimal value for Q. The temptation is to write dQ, but that would be misunderstood.
2
Because Q = ∆E, it follows that ∆Q = ∆(∆E) and dQ = d(∆E), which is not at all what we
intend. We therefore choose the notation
q = dE (5a)
intended to serve as a reminder that we want an infinitesimal amount of thermal energy transfer.
The amount of thermal energy absorbed by the system, Q, divided by the temperature rise is
the quantity commonly called heat capacity, C. C = Q/∆T. However, because C may itself vary
with temperature, it is better to take the limiting value of the ratio for small changes of
(6) LimLim
00
dT
q

T
Q
T
E
C
TT
V
=
∆
=
∆
∆
=
→∆→∆
V
T
E






∂
∂
(6a)
V
V
T
E

C






∂
∂
=
( ) ( )
7 dTCdT
T
E
dT
dT
dE
dEV
V
V
=






∂
∂
==

( ) ( )
( )
∫
∫
−==
=−=∆
2
1
2
1


12


12

8
T
T
VV
T
T
TTCdTC
dEEEEV
See Appendix.
3
For convenience, numbers specified for illustrative purposes, such as amounts of
4
material, temperatures, and pressures, will be treated as integers, known exactly. Also, for

present purposes we approximate C for hydrogen as 5/2 R (see Table I), although the actual
V
value is somewhat less and increasing with temperature.
7/10/07 1- 13
temperature. This is just a derivative.
3
It is common practice to represent a derivative, such as dE/dT, which is evaluated with some
variable (in this case V) held constant, by the special symbol
These are called “partial derivatives”. Employing this notation we arrive at the usual form of
equation 6.
Consider now the change in energy for 2 mol of H warmed at constant volume from 25 C to
2
o
50 C. For a diatomic gas near room temperature, C is constant and is about 5/2 R = 21
o 4
V
J/mol·K. The gas can do no work at constant volume so the total energy change is equal to the
thermal energy transfer (“heat absorbed”).
The total energy change is obtained by integration.
Thermodynamic temperature is expressed on the Kelvin scale, by adding 273.15 to the