2010 On Line Technocracy Study Course project
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Mechanical
Energy.
In the preceding lessons we have already
learned that matter on the earth is not destroyed, and that movements and
changes of matter involve work or energy. We further learned that there is an
exact relation between work and heat; namely, that when a given quantity of
work is converted into heat the same amount of heat is always produced...
Lesson 4
...It was also pointed out in discussing the weight-and-flywheel experiment that if no friction were involved, and hence no heat produced, the loss of potential energy by the falling weight would be completely compensated by the gain in kinetic energy of the flywheel. After the falling weight had reached its lowest point it would be relifted by the flywheel which would slow down and lose kinetic energy as the lifted weight gained potential energy. Furthermore, the gain in potential energy would be exactly equal to the loss in kinetic energy and vice versa.
Hence we arrive at the conclusion that in any purely mechanical system involving no friction, and hence no heat loss, the sum obtained by adding all the potential energies and all the kinetic energies existing simultaneously is a constant.
When there is friction (which in reality involves all cases) heat is produced, and the amount of heat produced is proportional to the loss of kinetic and potential energy by the system. Since heat is a form of energy and I gram calorie of heat is equivalent to 4.18 joules of work, if the heat loss be stated in terms of joules instead of calories it will be found that the energy appearing as heat is exactly equal to the loss of mechanical energy---potential and kinetic---by the system.
Energy of
Evaporation.
When water boils at I atmosphere pressure
the temperature remains constant at 100 degree C. If heat is added at a faster
rate the water boils more vigorously but the tem-
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temperature still remains constant. If this be continued long enough all the water will finally disappear as steam or water vapor. Here we have a case where energy in the form of heat is being added to a system without any increase in temperature of either the water or the vapor, but in which there is a progressive change of water from its liquid to its gaseous state. It follows, therefore, that the energy must be required to effect this change. By careful measurement of the amount of heat required to vaporize a known quantity of water it has been determined that 539.1 gram calories of heat are required to vaporize I gram of water at 1 atmosphere pressure and 100 degrees C.
At first thought it might appear that this energy has been lost. If the steam is made to condense back to water again, however, while at 1 atmosphere pressure and 100 degrees C., it has been found that 539.1 gram calories of heat must be extracted. Thus the heat of evaporation has not been lost but stored in the vapor.
This energy required to produce evaporation serves two purposes: (1) Part of it is required to pull the water molecules apart against their own mutual attractive forces and hence becomes stored as potential energy. (2) Part of it is required to perform the work of vaporization against the atmospheric pressure when 1 gram, of liquid water expands into 1 gram of steam. Thus we may say that the heat of vaporization is employed to perform two kinds of work, an internal work against cohesive forces, and an external work against atmospheric pressure. When the reverse process occurs this energy is again released in the form of heat.
Chemical Energy.
If 2 grams of the gas hydrogen are mixed
with 16 grams of the gas oxygen and the mixture ignited by an electric spark
while being maintained at a constant pressure of 1 atmosphere, there will be a
mild explosion and 18 grams of water vapor at a greatly elevated temperature will
result. If this water vapor be cooled down to the temperature of the original
mixture (room- temperature) it will become 18 grams of liquid water, but to
produce this result it will be necessary to subtract 68,300 gram calories of
heat. Thus we may write:
2H2 + O2 = 2H2O + 2 x 68,300 cals.
4 grams + 32 grams = 36 grams
Hydrogen + Oxygen =
Liquid Water
It is also possible by means of an electric current to separate liquid water back into its components, hydrogen and oxygen, at room temperature and I atmosphere pressure. When this is done we find the electrical energy required to decompose 18 grams of water is equivalent to 68,300 calories.
While this is only an isolated instance, the same kind of thing is true for all chemical reactions. Some release free energy and
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are said to be exothermic; others require the addition of energy and are endothermic. In all cases, however, if a chemical change when proceeding in one direction releases energy, then an exactly equal amount of energy would have to be supplied if the constituents of the system are ever to be restored to their initial state.
Thus a storage battery releases energy upon being discharged, but the same amount of energy must be supplied if the battery is to be recharged. Coal and wood release energy in the form. of heat upon being burned (reacting with oxygen) but this energy was originally supplied by the sun when the components of these fuels were originally combined.
Still other forms of energy are those of light, electricity, magnetism and sound. Space here does not permit a detailed discussion of all of these forms. Enough has already been said to lead one to suspect that energy is interchangeable among all of these various forms. This is indeed the case.
If we generalize the facts already noted we arrive at one of the most important conclusions of all science. Let us take any system of matter, and let us cause this to change from some initial state, A, to some final state, B. In this process a definite amount, E, of energy will be released in the process of transition. (If energy is absorbed E will be negative.) Now by any method whatsoever, let us restore the system to its initial state, A. In this case the same amount, E, of energy will have to be restored to the system as was originally released by it. Were this not so it would be possible to obtain more energy, E1, in changing the system from state A to state B than the amount E2 required to restore the system from state B to state A. In this manner a complete cycle would leave us with a surplus of energy which could be used in lifting a weight or in otherwise performing work. This would enable us to build a self-contained, self-acting machine that would operate continuously and perform work, a form of perpetual motion.
On the basis of our experience, however, we have never found it possible to build such a machine, and so we conclude that to do so is impossible. If this be so, then we must also conclude that it is impossible to obtain more energy when any system goes from an initial state, A, to a final state, B, than must be restored to the system in order to change it back from state B to state A.
Consequently, if this be true, it follows that either to create or to destroy energy is impossible. Thus in processes occurring on the earth when a given amount of energy in one form disappears an equal amount always appears in some other form. Energy may change successively from radiant energy to chemical energy to electrical energy to mechanical work and finally to heat, but in none of these processes is any of it lost or destroyed.
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It is this indestructibility and non-creatibility of energy that constitutes the First Law of Thermodynamics.
Direction of Energy
Transformations.
It is not enough, however, to know that in
processes occurring on the earth, energy is neither created nor destroyed, or
that when an engine performs external work such as lifting a weight, an
equivalent amount of energy must have disappeared somewhere else. We must
inquire whether energy transformations occur with equal facility in opposite
directions, or whether there is a favored direction in which energy
transformations tend to occur.
To do this we may begin with simple instances of our everyday experience. If we could build a flywheel that was perfectly frictionless, once started it would turn indefinitely at constant angular velocity. Similarly a frictionless pendulum would swing with undiminished amplitude. In each of these instances the mechanical energy originally supplied would be retained in undiminished amount. In actual practice, however, we have never been able to completely eliminate friction so the flywheel gradually slows down, and the pendulum swings with steadily diminishing amplitude of swing, both finally coming to rest. In each case the initial energy has been gradually dissipated by the friction into waste low temperature heat.
Had we tried the reverse process, however, of supplying energy in the form of heat to the bearings of the wheel or pendulum while initially at rest, this energy would never have resulted in the wheel's turning or the pendulum's beginning to swing. Thus we observe that while there is a spontaneous tendency for mechanical energy to be converted into low temperature heat, the process does not appear to be reversible.
In a more complicated case we might consider a waterfall such as Niagara. Here the water falls from a height of 167 feet. In falling, the potential energy due to height is converted into heat, and the water at the foot of Niagara is about one-eighth of a degree Centigrade warmer than it was at the top. Thus, the energy of Niagara is being continuously converted into waste heat.
Suppose, however, that a part of this water is made to go through a hydro-turbine. Then over 90 percent of this energy is captured by the turbine, which, in turn, converts it into electrical energy. This electrical energy is then used to drive electric motors and run machinery, to produce light, to heat electric furnaces, or to produce chemical reactions such as charging storage batteries or producing calcium carbide. If it drives an electric motor, friction exists in the motor, and in the machines which it drives, and the energy is lost as waste heat of the bearings and the air, plus
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the heat losses in the windings of the motors due to electrical resistance. If it is used for lighting or for an electric furnace, again it produces heat. Light is absorbed and becomes heat. If the energy is used to produce a chemical reaction, such as making calcium carbide, this, when placed in water, reacts to release acetylene gas, which when burned in air, produces heat.
Now if we add to this apparently exceptionless tendency for all other forms of energy to be transformed spontaneously into heat, the further fact that heat always tends spontaneously to flow from regions of higher to those of lower temperature, we obtain the remarkable result that all other forms of energy tend finally to be degraded into heat at the lowest available temperature of the surroundings.
Absolute Scale of
Temperature.
What we wish to know is to what extent it
is possible to make the opposite transformation from heat to work and to other
forms of available energy.
In order to throw more light upon this subject we require some sort of a standard or unit of measurement by which we can judge the direction of various energy transformations. But prior to that we need to know what the absolute zero of temperature is. We note, for example, that a gas like hydrogen, when kept under a constant pressure of 1 atmosphere, expands as the temperature is raised and contracts when it is lowered. In particular, when its temperature is raised from 0' C. to 1' C. its volume increases by 1/273 of its volume at 0' C., and for each additional degree increase in temperature the volume increases 1/273 of its amount at 0' C.
Likewise for each degree the temperature is lowered the volume decreases by 1/273 of the volume at 0 degree C. This is approximately true for other gases also.
Obviously, if this relationship held, the volume would become zero when the temperature became 273 degrees below 0 degree C., or minus 273 degrees C. This we take to be the coldest temperature possible, or the absolute zero of temperature. Temperatures within a fraction of a degree of this have actually been attained.
Now if we employ the Centigrade scale but set the zero point at minus 273 degrees C., we would get an Absolute scale of temperature. 0 degree A. would be minus 273 degrees C.; plus 273 degrees A. would be 0 degree C.; 373 degrees A. would be the same as 100 degrees C., etc.
Entropy.
Now we can introduce another type of
quantity we have not dealt with heretofore. When a quantity of heat, Q,
flows into a body at the absolute temperature T, let us agree to call
the quantity Q/T the increase in the entropy of the body.
If the heat flows out of the body the entropy of the body will, of course,
decrease. If a body were heated from a lower temperature, T2, to a
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higher temperature, T1, its entropy would increase, but to obtain the amount we would have to add up all the separate entropies step by step from the lower to the higher temperature. Thus for water, since 1 calorie raises the temperature of 1 gram approximately I degree C. or 1 degree A., the entropy-increases, would be, when the temperature is raised from 273 degrees A. to 278 degrees A., approximately:
Delta S= 1/274 + 1/275 + 1/276 + 1/277 + 1/278
where Delta S (read delta S) is the increase in the entropy of 1 gram of water.
Now let us consider the entropy changes that occur in various energy transformations of the kind we have already considered. If we take any frictionless mechanical system such as a pendulum or flywheel at constant temperature no heat would be produced and no heat conduction would occur, consequently the entropy change would be zero.
Delta S = 0
for all such systems, and they are said to be isentropic or constant entropy systems.
If, however, friction exists, heat is produced and the entropy increases by an amount
Delta S = Q/T
where Delta S is the increase of the entropy of the system, Q the amount of heat generated and T the absolute temperature.
Now let us consider two adjacent bodies, one at an absolute temperature T1, and the other at T2, T1 being higher than T2. The heat will flow by conduction from the hotter of the two bodies to the colder. Let a small quantity of heat, dQ, flow in this manner from the body at temperature T1 to that at temperature T2.
The entropy lost by the hotter body is dQ/T1; that gained by the colder body is dQ/T2. The total entropy increase will be the difference of these two entropies.
Delta S = dQ/T2 minus dQ/T1
Now dQ is the same in both cases, but T2 is less than T1. Therefore dQ/T2 is greater than dQ/T1. Hence the total entropy change, Delta S, consists in an increase in the entropy of the two bodies taken together.
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Thus we see that an idealized frictionless mechanical system involves a zero change of entropy, while any process involving friction, or heat conduction, results in an increase of entropy.
Now let us see if we can find a process that results in a decrease of entropy. A direct conversion of heat to work would be such a process. Suppose we could construct an engine which was self-contained and operated cyclically, that is, one that repeated the same cyclical operation over and over, which did nothing but take heat from a heat reservoir and lift a weight. This is manifestly no contradiction to the first law of thermodynamics, because we are not proposing to create energy but merely to transform already existing energy from heat to work.
If T be the temperature of the engine and the heat reservoir, and if Q be the heat taken in at each complete cycle, then since the engine returns at the end of each cycle to its initial state, its entropy remains unchanged. The lifting of a weight is an isentropic process. Consequently the only entropy change of the system is manifested by the disappearance of an amount of heat Q at temperature T per cycle. This would correspond to a decrease in the entropy per each cycle:
Delta S = minus Q/T
But no such engine has ever been built. If one could be built it could be made to run on the heat from the ocean or from, the ground or the air. It would act both as a refrigerator and as an engine for doing work. If such a machine could be built it would not violate the principle of the conservation of energy, but it would still constitute a sort of perpetual motion machine in that it could operate from the heat of, say, the ocean and perform work, which could be transformed by friction back to heat, thereby maintaining the initial supply. This has been called perpetual motion of the second kind.
Our failure to build such an engine leads to the conclusion that to do so is impossible. This conclusion is based entirely upon negative experience and can only be upset by actually producing this kind of perpetual motion.
Another instance of a decrease of entropy would be given if heat flowed from a colder to a hotter body. By reasoning exactly analogous to that employed for heat conduction from a hotter to a colder body, we arrive at the fact that if heat ever flowed from a colder to a hotter body the entropy of the system would decrease, or, the entropy change would be negative. But such a heat flow is contrary to all of our experience. All of our experiences thus far may be summed up by saying that in all processes of whatever sort so far observed, the changes in the entropy involved
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are such that the total entropy of the whole system either remains constant or increases.
Conversion of
Heat to Work.
Now if we have a difference of temperature
between two heat reservoirs, the higher temperature being T1 and the
lower T2, the entropy would increase if heat were allowed to flow
directly from the one to the other by conduction. On the other hand, we know it
is possible to operate a steam engine between these two different temperatures,
using one for the boiler temperature and the other for the condenser.
In this case if an amount of heat Q1 be taken by the engine per cycle from the temperature T1, and Q2 be the heat discharged into the condenser at T2 then Q1 minus Q2 is equal to the work, W, done by the engine per complete cycle. The maximum possible value of the work, W, is obtained when we consider that the limiting case of the operation---the limit that the engine can approach but never exceed---is given for the case when the entropy change is zero.
For each cycle the entropy lost by the heat reservoir at temperature T1 is Q1/T1, while that gained by the condenser is Q2/T2, the entropy of the engine itself being the same at the completion of each cycle. Then if the total entropy change is to be zero,
Q1/T1=Q2/T2
or
Q2=Q1 times T2/T1
Now, since the work, W, done by the engine is equal to the loss of heat, Q1 minus Q2,
W = Q1 minus Q2 = Q1 minus Q1 times T2/T1,
or
W = Q1 times (T1 minus T2) over T1
Thus the maximum possible fraction of the heat, Q1 taken from the higher temperature reservoir that can be converted into work is given by the fraction (T1 minus T2)/T1, which is said to be the efficiency of the engine.
The nearer the two temperatures are together, the smaller the value of this fraction, becoming zero when the two tempera-
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tures become the same. Hence it is impossible to operate any heat engine except when a difference of temperature exists. Under no circumstances can the work produced or the efficiency be greater than that given above.
Reversible and
Irreversible Processes.
Now we come to the concept of reversible
and irreversible processes. A reversible process is in reality an
idealization and occurs only in those cases for which the entropy change is
zero. All actual cases involve friction or its equivalent and therefore result
in an increase of the entropy of the system. Such systems are said to be
irreversible and the entropy increase is a measure of their degree of
irreversibility.
An irreversible process is characterized by the fact that when once it has occurred, by no process whatsoever can it be undone. For example, if a book is pushed off the desk and falls to the floor its potential energy is changed into heat and the entropy increases. It is physically impossible ever to put the book back on the desk and at the same time to restore everything else to the state it was in before the book originally fell. The book can be lifted back by hand but that degrades chemically the energy inside the body. It could be hoisted by an electric motor, but that would discharge a battery. So with every other process of replacing the book. It is impossible to put everything involved back to its initial configuration. In consequence of this fact the universe has experienced a new event and has made a stride forward.
Transformations
in an Isolated System.
Now let us imagine a system completely
isolated from all outside energy transfers, that is to say, that no energy is
allowed to enter or escape. For such a system we may imagine a large
heat-proof, light-proof, sound-proof room. Let it be stocked with all sorts of
physical and chemical apparatus and supplies such as storage batteries,
gasoline, oxygen, food supplies, water, electric and gasoline motors, electric
or fuel lights, etc. Into this room we will also place a physicist and then
seal the door to isolate the system.
Now this isolated universe, as it were, is all equipped to run. Our physicist can have light and food, oxygen to breathe and water to drink. In addition to this he has engines and motors and an energy supply to drive them. To make the problem even more interesting we might even allow him soil and plant seeds so he could grow his own food supply.
What would be the future of this isolated universe? Merely from our everyday experience we would know that the food supply, the free oxygen, and the fuel would all diminish with time. The storage batteries would become discharged; the water would become contaminated; and ultimately, if not rescued, our phys-
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icist would die from lack of food, oxygen, or water, and chemically disintegrate thereafter.
Now it is instructive to analyze the problem thermodynamically. The room, by hypothesis, consists of a isolated system. The matter in the system is constant; the energy is constant; but both the matter and the energy are undergoing continuous transformations. If the matter is initially at state A it successively occupies states B, C, D, etc. At successive intervals of time. Since, from what we have seen, all actual transformations of matter from any given state to the next successive state involved an increase of entropy, we may say that the entropy of the system is continuously increasing. Thus the entropy of state B is greater than that of state A; that of state C is greater than that of state B, etc. This being so, if the room were ever to regain any earlier state such as going from state D to state B, a decrease in entropy would occur. But this, we have seen, is impossible. Consequently we may say that when any isolated system has once occupied and passed through any given state it is physically impossible, by any method whatsoever, for it ever to regain that state.
Consequently the history of any isolated system may be regarded as the record of the changes of the material configurations of that system. These changes are, however, unidirectional and irreversible. Consequently it is a physical impossibility for the history of the system ever to repeat itself.
Unidirectional
Nature of Terrestrial History.
Now that we have said with regard to the
room is equally valid with respect to the earth if we recognize that although
it is not an isolated system the changes in the configuration of matter on the
earth, such as the erosion of soil, the making of mountains, the burning of
coal and oil, and the mining of metals are all typical and characteristic
examples of irreversible processes, involving in each case an increase of
entropy. Consequently terrestrial history is also unidirectional and
irreversible.
In order to repeat the history since the year 1900, for example, we would have to restore to the world the configuration that it had in the year 1900. We would have to put the organisms back to their 1900 state; we would have to put the coal, the oil, and the metals back into the ground; we would have to restore the eroded soil. But these are things which by no method whatsoever can be done.
It is this unidirectional tendency of energy transformations; this fact that all actual physical processes, at least on a macroscopic scale, are irreversible; the fact that no engine operating cyclically can convert heat into work without a difference in temperature existing and then only incom-
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pletely; the fact that heat flows only from regions of higher to those of lower temperature; the tendency for the entropy, of a system only to increase with time, that comprises the Second Law of Thermodynamics.
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