An Introduction to Thermodynamics phần 1 pdf

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 mv2 1 The

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 theth

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 newth 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 byth 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 thatth 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

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 mv2 (1) 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 twoo Dewar flasks (Figure 1) To the first flask we add 50 g of water initially at 20 C, and stir until1 o

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, ato

20 C, and find the final temperature to be about 82.2 C We musto 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)

Sample Tfinal ∆Ts ∆Tw -∆T /w∆T -s ∆T M /w w∆T Ms s

* Temperatures in C o ∆ T is the temperature change of the sample and

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 theo o second a block of aluminum, at 20 C, with a mass of 25 g (Figure 2) After a few seconds weo 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

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 theo 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

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.

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 energy3

transfer is as work,

W = ∆E (3)

Second, and equally important, all investigations thus far of transfer of energy as work

∑ = ∑ ∫ ⋅

=

i i i

W

W f dx (4)











ds f

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 = If ·ds, where p = mv, has been called the

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.

indicate that the amount of work done is equal to a product of the force exerted (on the object)

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 = net (5)

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 = (mv) 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

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

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

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, the2

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 change2 1

in V, the volume Then the law of conservation of volume would be written

(∆V )system + surroundings = 0 (1)

The best-known of the three thermodynamic postulates is known as the first law of