The second main theorem. Impossibility of the perpetuum mobile of the second kind.
According to the first main theorem, all processes in Nature can only involve conversions of energy (never its generation or destruction). In particular, the second main theorem deals with the direction, in which conversions occur, that is, with the question whether under given conditions work is converted into heat or heat into work, and whether the conversion is complete or not. Prior to discussing this theorem, we will discuss the conversion of heat and mechanical work during a readily understood process (Fig. 361).
The cylinder A, a good heat conductor, contains a gas, for example, air. It is separated from the outside by the air tight piston B and exchanges heat through the wall from an unbounded container of (warm) water. The water temperature remains constant, whatever heat enters or leaves. The weight of the piston and the masses M, m balance the pressure of the gas, which has the temperature T of the container. If you remove m, the gas expands and raises the piston and the mass M. The gas obtains the amount of heat corresponding to this work (according to the first main theorem) out of its own heat content. It would cool during this process, if it did not exchange heat through the heat conducting wall with the warm water container. It thus maintains its temperature T (isothermal expansion). The amount of heat, which corresponds to the work done, is taken from the warm water container. Hence (1) the working substance, the gas, has changed in a certain sense, it has increased its volume. (The fact that the working substance changes is fundamentally important for an understanding of what follows!). If the weight M is subdivided, so that you can remove it in several steps, the gas can gradually perform more work, while the warm water container always provides an equivalent amount of heat. Simultaneously, the state of the gas changes more and more, its volume increases, its density drops.
If you now replace the weights on the piston one by one, they compress the gas as the piston sinks. They perform work on it and there arises in it a corresponding amount of heat. It would raise its temperature, if there were not the exchange of heat with the warm water container; it thus flows into the water container and the temperature T of the gas remains constant (isothermal compression). If finally the piston carries the same load as at the start, the gas has returned to its initial state: It has passed through a cyclic process.
Did the second part, the reversion of the process, use the same amount of work (for the compression of the gas) as the gas performed during the first part of the process (during its expansion)? Moreover: Does the warm water container have at the end of the cyclic process the same amount of heat as at the start? Experience tells us: In general, the return process demands more work than the first part has yielded, whence the performance of the cyclic process demands work input and the quantity of heat, corresponding to this effort, enters the water container, that is, we have converted work into heat.
According to the theory, you can imagine a limiting case, during which the gain in work equals the supply of work and the water container has at the end of the cyclic process the same amount of heat as at the start. This limiting case occurs when the piston moves without any friction and when the gas expands so slowly and is compressed so slowly that there do not arise in it pressure and temperature differences.
Taking into consideration the executable, in face of the ideally imaginable, conditions, we may conclude that there can never be gained work in the experiment of Fig. 361 at the expense of heat. Hence work can only then be obtained at the expense of heat, when the working system undergoes a change of state. Generalizing this insight, we say: "It is impossible to gain work through a cyclic process or through a sequence of several cyclic processes (or, what is the same thing: Through a periodic process); one only withdraws heat from the container of water, which is at a given temperature". (At the expense of the heat of a container, you can only gain work when there remains a change of state in the work performing body or in its environment.)
This theorem - note that it relies on experience! - Planck places at the front of his reflections regarding the second main theorem; he derives from it all the facts, which are contained in it. It follows immediately from it, that it is impossible to build an engine, which does nothing else than all along withdraw heat from the sea or the atmosphere or Earth and convert it into work. Such a hypothetical machine would not at all contradict the first main theorem and would without cost perform arbitrary amounts of work; hence it would be effectively a perpetuum mobile, whence it is referred to as a perpetuum mobile of the second kind, following Ostwald. The second main theorem is therefore called : The theorem of the impossibility of the perpetuum mobile of the second kind.
It is impossible to convert heat into work continuously by the above cyclic process. However, we know of other cyclic processes, which make it possible. The practically most important example is the steam engine, and especially clear is the condenser engine (Fig. 362). Water is converted in the boiler A into steam, due to its expansion the steam performs work on the piston B of the engine and eventually is reconverted into water in the condenser C. The fundamental difference from the just described cyclic process is: There are two heat containers with which the working substance of the steam engine (water, steam) exchange heat - two heat containers, indeed at different temperatures - a steam boiler A at the temperature T1 and cooling water in the condenser C at the temperature T2.
For the sake of clarity, we idealize the process to the cyclic process, named after Carnot 1824. (It avoids generation of frictional heat and the loss of heat to deeper temperatures without performance of work.) We employ again a cylinder with an air tight piston and a gas as working substance, similar to what was shown in Fig. 361, but use now (Fig. 363 below) two (unlimited) heat containers with the (therefore) unchangeable temperatures T1 and T2. The heat exchange between the working substance, the gas, and the heat containers occurs through two valves at the bottom of the cylinder; they can connect or disconnect each of the containers to the inside of the cylinder. We will call the heat containers according to their temperatures T1 and T2, assume that T1 > T2 and at times refer to T1 as heat source and to T2 as cooler - corresponding to the steam boiler A and the condenser C in Fig. 362. It is unimportant how the valves are closed and opened, you can imagine with each of them a Maxwell demon, which undertakes the task.
Before we describe the process, we make a comment of fundamental importance: We let the process progress quite slowly! We let the working substance, the gas, expand so slowly and let it be compressed by the sinking piston so slowly, that there arise only infinitesimal pressure differences between it and the external pressure. For this purpose, we need only make the external pressure by a very little smaller or larger than the pressure of the gas and in doing so commit only an unimportant error, when we (in computations) set the pressure of the gas equal to the external pressure. The smallness of the pressure difference, which occurs through this slow process, offers the advantage , that we can also revert the process; for this purpose, we need only invert the pressure difference which by itself is unimportant.
Similar arguments apply to the heat transfer: If we let the process take place so slowly, then the working substance need only have with respect to the heat container, connected to it, a quite small temperature difference, in order to pick up heat from it or to return heat to it. Also here, the error is negligible, when we invert the temperature difference which by itself is unimportant. Thus, we can imagine that every sufficiently slow (you say: infinitesimally slow) process is also executable in the opposite direction.
We will now present a description of the Carnot-Process. It has the following four stages:
Stage 1. Fig. 363 A (1): Let the gas have at the start the volume and pressure, which in the p,v coordinate system (Fig. 363 C) has been denoted by the point a, and the temperature T1. We now reduce the weight on the piston. Then the gas expands and lifts the piston. Let it expand until it reaches the p,v point b isothermally, that is, by accepting a certain amount Q1 of heat from the heat containerT1. The work, done by the gas (lifting of the piston) is given by the area in Fig. 363 B(1); let it be positive.
Stage 2. Fig. 363 A(2): We remove the remainder of the weight on the piston. Then the gas expands more and lifts the piston further. Let it expand adiabatically from the p,v point b to the p,v point c in Fig. 363 C. The gas must take the work required for the expansion from b to c out of its own heat stock. In this process, it cools and its temperature sinks to T2. The work, which it performs between the p,v points b and c (lifting the piston) is given by the area of Fig. 363 B (2); let it also be positive.
Stage 3. Fig. 363 A(3): We load the piston with some of the former weight. Then the piston sinks, compresses the gas, indeed isothermally, until its reaches the p,v point d in Fig. 363 C. The heat Q2, which thus arises, is taken to the container T2. The work, which is performed on the gas between the p,v points c and d, is given by the area in Fig. 363 B(3). We must make it negative - in contrast to the work it performed.
Stage 4. Fig. 363 A(4): Load the piston as it was loaded at the start. It then compresses the gas more. Let this compression be adiabatic until the gas returns to the p,v point a (Fig. 363 C ). During the compression along da, let the temperature of the gas rise from T2 to T1. The cyclic process is now finished and the working substance, the gas, is in the same state as at the start. The work, which is performed on the gas between the p,v points d and a, is given a negative sign, Fig. 363 B (4).
Here is now the result: The warmer container T1 has lost the heat Q1, the colder container T2 has taken in the heat Q2, the working substance has performed the work A, which graphically is represented by the sum of the four areas, which represented the work done during the four stages. The combination of the two positive areas with the two negative areas yields the area of Fig. 363 c, enclosed between the two isotherms and the two adiabates, the area abcd; it must correspond to the work A. According to the first main theorem, one must have A = Q1 - Q2 and, if A is to be non-zero, Q1> Q2.
Useful effect of the reversible process
We now invert each of the partial processes, in order to let the cyclic process take place in the opposite direction. If we let the process proceed slowly enough, the temperature differences can be contained between arbitrarily small bounds, so that the Carnot-Cyclic-Process can be considered to be strictly invertible. The reverse process withdraws from the colder body T2 a quantity of heat Q2 and feeds to the warmer body T1 a quantity of heat Q1, which is larger, and indeed at the expense of work, the size of which is given by the area abcd, which therefore also assesses the quantity of heat, into which the work during the process has been converted. The inversion of the Carnot cyclic process shows therefore: It is possible, to transfer heat from a colder (!) body to a warmer body (!), but only at the expense of mechanical work of a certain magnitude.
The reversible cyclic process yields new, physical insight of great importance, namely the conclusion: If a given, reversible cyclic process occurs between the upper temperature T1 and the lower temperature T2 while it takes in the heat Q1 at the upper temperature and performs mechanical work A, then no machine of whatever construction can perform more work with the same quantity of heat and between the same temperatures. In other words: The useful effect of a reversible machine is a maximum, which cannot be reached with a given temperature difference - by useful effect is understood the fraction A/Q1(both measured mechanically). The reason follows: Let us assume that there exists a machine M, which works between the temperatures T1 and T2 and has a greater useful effect than a reversible machine N. Link M and N in such a manner that M drives by its direct action the machine N in the opposite direction. At each stroke of this combination machine - its new name - N receives from the colder body T2 the heat Q2 and passes on the heat Q1 by its work A to the warmer body T1. The machine M will accept this amount of heat and, according to our assumption, perform more work as it passes it on to T2 than is required for driving the machine N. Hence the combination machine performs at each stroke useful work,
This result does not at all violate the law of the conservation of energy, because, if M yields more work than N would perform, it also converts at each stroke more heat into work, whence M returns to the cold body a smaller quantity of heat than N withdraws from it. Thus, our assumption forces us to conclude: The combination machine converts the heat of the colder body into work and will eventually convert all its heat into work. However, this contradicts obviously experience, whence we must conclude: Our assumption is wrong in that between given temperatures the reversible machine has the largest useful effect.
The question regarding the computation of the useful effect of the Carnot process out of its dependence on the temperature bounds leads to a new temperature scale, which enables us to survey better than the scales introduced so far the balance of the process and formulate it mathematically.
Thermodynamical temperature scale (Lord Kelvin)*
The temperature scales of Celsius et al. depend on the character of the selected material (mercury, hydrogen, platinum) and are therefore limited below and above (by freezing, combustion, condensation of the thermometer substance). However, strict thermometry demands a temperature scale, which does not depend on the accidental property of a substance. The second main theorem yields such a scale, called the thermodynamical temperature scale. At first, it is only defined theoretically. It is employed in practice by converting to it the results of the gas thermometer scale. For this purpose, you can employ any equation between measurable quantities, which arises out of the second main theorem, for example, the equation of Clapeyron-Clausius , the Joule- Thomson-Effect, the Carnot-Cyclic- Process. We will use the last to demonstrated the basic idea of the thermodynamic scale:
In Fig. 364, let JJ be for a given substance the isotherm which belongs to the (Celsius) temperature J. Let it expand from the state, to which its volume and pressure correspond at the point a, while it is taking in heat at the same temperature J until it has received the heat quantity Q, when its state corresponds to the point b. Then let it expand, at the temperature J, until it has received an equal quantity Q of heat to the point c, etc. along the isotherm JJ; let two neighbouring points be determined by the expansion of the substance due to input of the quantity of heat Q. We now draw through a, b, c, ··· the adiabates aa, bb, cg, ···, which establish the relationship between the pressure and the volume as the substance expands from a, b, c, ··· without addition of external heat; let J1J1 and J2J2 be the isotherms, corresponding to J1and J2.
*This presentation follows that of Maxwell in his book "Theory of heat", published in 1871.
During a Carnot-Cyclic-process, which occurs between the temperatures J and J1, the quantity Q of heat from any source is converted into the work A, the magnitude of which only depends on J and J1. However, the segments ab and bc now correspond to equally large quantities of heat Q, whence the areas 1 and I, which represent the corresponding work, must be equally large. The same applies to the areas, which the adiabates bound between another pair of isotherms. If you now draw, as in Fig. 364, a number of adiabates in such a way that the points, at which they intersect an isotherm, correspond to a sequence of successive equally large increases in temperature, then these adiabates cut off a sequence of equally large areas from any strip, bounded by any two isotherms.
Kelvin arrives now at a temperature scale in the following manner: He chooses the points a, a1, a2, from which he draws several isotherms, in such a manner that the area 1 between two neighbouring isotherms JJ and J1J1 becomes equal to the areas 2, 3, ··· between every other pair of neighbouring isotherms J1J1 and J2J2. It ends up by him calculating the number of degrees between the temperatures J and J2 proportional to the area abb2a2. Only the zero point and the degree unit of the new scale remain arbitrary. He chooses them so that the new scale at two fixed points coincides with any of the ordinary scales (gas thermometer), whence then also every other temperature is determined - independently of the character of the substance and by a method, which yields with each substance the same result.
We will now draw isotherms and adiabates in the following manner: We let the isotherm of a temperature J be intersected by the adiabates at such points, where the segment between two neighbouring adiabates corresponds always to the expansion of the substance on intake of always the same quantity of heat Q, whence the sequence of adiabates is fixed.
We draw the isotherms so that neighbours cut off from the strip between the pair of adiabates aa and bb equally large areas 1, 2 etc. The thus determined isotherms cut off also with each other pair of adiabates equal areas. (Therefore the two systems of lines form a net, all the meshes of which have equally large areas.) Hence the isotherms in Fig. 364 present now a subdivision into degrees on the basis of a method, which rests solely on a general thermodynamical principle and is independent of the peculiarity of a working substance. Possibly required changes of the distances between the isotherms and of the selected zero line obviously allow us the obtain the degree subdivision with the two fixed points of an ordinary scale .
Next, consider the useful effect of the cyclic process, the ratio of the heat input to the work performed, measured by means of the new temperature scale. For the sake of brevity, call the warmer heat container heat source (W) and the cooler heat container condenser (K). If J is the (Celsius) temperature of (W) and the working substance takes in at this temperature the heat Q, then the magnitude of the work, performed by Q, only depends on the temperature of (K). Let it be denoted by J2. The work performed by Q is then represented by the area abb2a2. Since the expansion from a to b is linked to the heat input Q, the work is represented by the area between the adiabates aa and bb and the isotherms J J and J 2J 2. The difference (J -J2) is the number of unit areas between them along the adiabates aa and bb . The quadrangles between neighbouring isotherms and neighbouring adiabates are equally large. If we denote the area of a unit quadrangle by Q·C, then the work performed by Q is Q·C(J -J2). - The product Q·C, introduced here, is a constant, but only the product is constant, not the factors; the quantity of heat Q differs depending on the isotherm, along which the process takes place, and C depends on the temperature. For example, the areas 1 and 2 are equally large, but Q, as a quantity of heat, enters the magnitude of the area 1, Q2 the magnitude of the area 2.
If the temperature of (W) is not J, but J 1, then the working substance changes its state along the isotherm J 1J 1. Let it take in the quantity of heat Q1 between the points a1 and b1 . Denote the work it performs by A; it is represented by the area 2 and equals, as above, Q·C(J1 -J2). The useful effect of this process is A/Q1. Hence A/Q1 = Q·C(J1 -J2)/Q1. By the first main theorem, Q1= Q - abb1a1, measured mechanically, or Q1 = Q - Q·C(J -J1), whence the useful effect of the Carnot process between J1 and J2 is
A/Q1 = Q·C(J -J1)/Q - Q·C(J -J1) = (J1 -J2)/(1/C + J1 - J).
The useful effect is the larger, the larger is the part of Q1 converted into work. If it were possible for Q1 to convert itself into mechanically equivalent work so that it (mechanically measured) became equal to A, the maximum possible useful effect would be obtained. In this - manipulated - case we would have 1 = (J1 -J2)/(1/C + J1 - J), whence J2 = J - 1/C would be the temperature, that is, the cooler would have to have this temperature, the deepest physically possible, whence we could call it the absolute zero point and set J2 = 0. However, the deepest physically possible temperature is that, at which an ideal gas has volume zero, that is, the temperature -273.2ºC on the Celsius scale. By employing this number, we define: The Celsius temperature zero is absolute 273.2 = T0, and we must set J100 =273.2º + 100º = T100. Hereby the fixed points of the thermodynamic absolute scale has been made to agree with those of the Celsius scale, but this does not tell anything about another agreement between the scales. At the same time, we have 1/C = 273.2 or C = 1/273.2. We have here assumed that the quantity 273.2 is known, because it was derived more accurately from the gas laws than would be possible from the Carnot process. (Note: We had introduced the absolute temperature, defined on the air thermometer, in order to be able to formulate the gas laws comfortably. The temperature, which has now been defined thermodynamically, has no link to any substance. It is accidental that the difference between the two scales is very small.) In order to convert a temperature, referred to the ordinary scale, into the absolute temperature, we must, according to the equation J = J2 + 1/C, add to the ordinary temperature number a constant number of degrees; this constant number is the absolute temperature of the scale zero.
The mathematically simplest form of the second main theorem for invertible cyclic processes
In order to be
able to relate the useful effect A/Q to the new scale,
we must denote the temperature J correspondingly
and set (in the denominator) 0 for 1/c - J. If we
call the absolute temperature of the heat source T1 the
absolute temperature of the cooler T2, then A/Q1 = (T1 - T2)/T1. The
heat Q2, brought to the colder body at the absolute temperature
T2, is Q1 - A. If we replace A by the value,
given by the first equation: Q1(1 - T2/T1), then Q2 = Q1·T2/T1, whence
Q1/T1 =Q2/T2 or
Q1/ Q2=T1/T2, that is, in a reversible machine, the ratio of the intake of heat to that exported to the cooler is equal to the ratio of the absolute temperatures of the heat source and the cooler. If we make the heat Q positive, as it enters the cyclic process, and negative, as it leaves it, we find
Q1/T1 + Q2/T2 = 0 as the mathematically simplest formulation of the second main theorem for invertible cyclic processes.
Clausius has generalized the theorem for cyclic processes, in which heat input and output occur at more than two temperatures. For example, you can decompose a cyclic process with three exchange temperatures into two cyclic processes with two input temperatures each and arrive in the end at the formula Q1/T1 + Q2/T2 + Q3/T3 = 0. A generalization of this method leads to a cyclic process with arbitrarily many exchange temperatures in the form S Q/T = 0.
We have now the two relations, which link the work A and also the quantities of heat Q1 and Q2 to the temperatures T1 and T2:
A = Q1(T1 - T2)/T1 = Q1(1 - T2/T1) and Q1/T1 - Q2/T2 = 0.
Given T1,T2 and Q1, the work A and the heat Q2, given off at the temperature T2, have always the same values irrespectively of whether the working substance of the Carnot process is a gas or a fluid or rigid, whence the two relations are valid for every substance. The work gained increases with the input quantity of heat Q1 and is the larger, the larger is the temperature difference T1 - T2. We see that only a fraction of the heat Q1is converted into work. The generated work A equals Q1 times the real fraction h, the useful effect of the process, whence
h = A/Q1 = (T1 - T2)/T1 = (1 - T2/T1).
For example, if the higher temperature is J1 = 200ºC, the lower temperature is J2 = 50ºC, then T1=200+273=473º and T2=50+273=323º and h =Q 0.32, that is, only 32% of the input heat Q1 is converted into work. The remainder Q1 - A = Q2 = 68% passes on to the container at lower temperature as heat and is lost for the work. The same follows also from the second main equation in the form Q2 = Q1 · T2/T1.
Retrospect. The cyclic process, which uses only a single heat container, consisted of two parts, the second was the exact reverse of the first. It is different in the case of the Carnot process. The working substance passes from a via b to c through other states than on the return path from c via d to a. However, we can attach to this process - we will call it positive - a second exactly oppositely directed, that is, negative one, which returns from a via d, c, b to a. During this negative cyclic process, as a result of the expansion of the working substance from d to c and the associated work, a quantity of heat Q'2 is taken from the heat container at the lower temperature T2 and correspondingly a heat Q'1 onto the heat container at the higher temperature T1. Just as here the quantities of heat have the opposite sign from that of the positive cycle, so also has the work: The negative process does not perform work, in fact, it uses work A'. The theory tells that otherwise the same relations apply as for the positive process, namely that
A' = Q'2(T1 - T2)/T1 and Q'1/T1 - Q'2/T2 = 0.
Moreover, the theory tells: A' and A, Q'1 and Q1 as well as Q'2 and Q2 are equal, but opposite, so that the negative Carnot process is the exact reverse of the positive one. While the positive process transmits heat from a higher temperature to a lower one and performs work, the negative process transmits heat from a lower temperature to a higher one and utilizes work.
Only one part of the available heat can be converted into work by a positive Carnot process, the remainder drops to a lower temperature and remains heat. Heat can go from a lower to a higher temperature level by a negative Carnot process, however only by input of work (refrigerating machine).
Idealized and real processes (reversible and irreversible)
All processes discussed so far have been idealized. How do real processes differ from idealized ones? Answer: For example, the vibration of the physical pendulum is real, that of the mathematical pendulum is idealized; sliding with friction of a piston along the cylinder wall is real, it is idealized, if it is frictionless; loss of heat by conduction and radiation during the operation of a steam engine is real, without such losses, it is dealized, etc. If the process described in Fig. 361 is a real one, the gas employs a part of its work to overcome the friction of the piston along the wall of the cylinder, whence less useful work is performed (it lifts the piston less high) than corresponds to the consumed heat energy. In order to compress it, we must for the same reason perform more work than corresponds to the quantity of heat, which it has passed on to the container, whence the process yields during the expansion of the gas less work, when friction is present, than the effort applied to the compaction of the gas. - It is similar during every real working process: Friction consumes a part of the useful work, whence the final result is that one part of the work has become friction heat.
A second unavoidable source of loss for all real processes, in which temperatures change, is heat conduction. It changes the heat without performance of work from a higher to a lower temperature level; it is just as when falling water misses the water wheel. While friction destroys useable work, heat conduction destroys the possibility of doing work; every real process yields therefore less gain in work than an ideal one.
You have to consider idealized processes as limiting cases of the real ones; they perform, ceteris paribus, more work than the real processes. In order to reverse real processes completely , we must perform a certain amount of excess work, because they are always accompanied by friction and heat conduction. - The principle of the impossibility of the perpetuum mobile of the second kind allows to prove that it is impossible without expenditure of work to raise heat from a lower to a higher temperature (refrigerator) and that heat of friction cannot be converted into work without some change of state of the participating bodies. Thus, while during friction, so to say, work automatically transits into heat, that is, without that any body undergoes simultaneously a change of state, an inversion of this process is always connected with a change of state. Thus, friction is a process which cannot be reversed without any change left behind. Such a process is called irreversible. Also the spreading of heat by conduction or radiation is irreversible: This spreading occurs without performance of work, but an accumulation of heat at a higher temperature demands work. Moreover: If gas flows into a vacuum, it expands without performing work, but you apply work to recompact it. And: Two gases mix and a substance spreads in its solvent without doing work, but work is required to separate again the gases and to recover the solved substance from the solution.
Summary: Without leaving behind other changes in Nature, work can be converted by friction into heat and, without application of work, heat can be transmitted from higher to lower temperatures (conduction, radiation) or mass can be transmitted from a smaller into a larger volume (diffusion), but the processes in the opposite directions leave behind changes in state or demand work. Frictional heat arises at every opportunity by itself outside the system, which is involved in a process, just as heat and mass tend to spread without external action. Nature prefers this direction of the processes and resists invincibly their complete reversion. Friction, heat conduction, diffusion are processes, the traces of which are never completely erased. Hence every process in Nature occurs always in the sense that more work is converted into heat than conversely, that more heat is transported to lower temperatures than to higher temperatures and that mass is to a higher degree subject to spreading than contraction.
Irreversible processes are confronted by reversible ones, processes which for their inversion do not demand additional work. For example, this is so during the to and fro motion of the mathematical pendulum, the amplitudes of which retain their magnitude during rising and falling. All completely reversible processes are idealized, all real ones are linked in some manner or other to irreversible ones (also pendulum motion), whence all are more or less irreversible and differ from each other only through their degree of irreversibility.
Entropy (Clausius 1865)
The degree of irreversibility of a process can be assessed with the aid of a mathematical function, the entropy (Greek: troph = conversion. Entropy = conversion-content). During every irreversible process, the entropy of a system increases. It can never decrease, at the best, it can remain constant, but only during an ideal, reversible process. The entropy of a single body or a group of bodies, which only form a part of a system, can well decrease; however, then the entropy of other parts of the system must simultaneously increase the more strongly, so that the sum of all changes of the entropy in the system increases. We have called entropy a mathematical function - what is it physically? Every body has in its present state a certain entropy, just as it has a certain temperature, a certain volume, a certain pressure. Maxwell conceives therefore the entropy of a body to be among its physical properties. We understand by a physical property of a body its capacity of being able to display under given conditions certain actions. What kind of conditions are here involved? And what kind of actions characterize entropy? The condition is: You take the body from its instantaneous state, as far as its pressure p and its temperature J characterize it, to a normal state (p0J0), that is, adiabatically to the normal temperature J0, next isothermally to the normal pressure p0. Hence its heat either increases or decreases. If it loses the heat quantity Q, its entropy S in the initial state was larger by Q/J0 than in its normal state. The entropy in the normal state is employed as the conventional zero of entropy, whence its entropy S in the initial state is Q/J0.
If a body takes in the quantity of heat Q, in order to come into the normal state, its initial entropy was -Q/J0. The entropy of a body in a given state is the larger, the larger is its mass. Therefore entropy is referred to unit mass, that is, it is measured as the quantity of heat per degree/mass, whence entropy is dimensionless. The entropy of a system of bodies equals the sum of the entropies of the individual bodies.
It can be shown that there exists for arbitrary bodies always a mathematical function with the properties of the entropy. For an arbitrary body, the expression for the entropy cannot be written down, because, in general, the equation of state is not known. But it can be written down for an ideal gas, because this equation is known. Related to unit mass, its entropy is
s = cvlogT + (R log v)/m + const,
where m is its molecular weight.
Two states (1) and (2) of a body may differ by their energy as well as their entropy, for example, by the energy and entropy of a perfect gas. If the temperature rises from T1 to T2, related to unit mass, the energy increase is u2 - u1 = cv(T2 - T1), the entropy increase
s2 - s1 = cvlog T2/T1 + (R/m) log v2/v1.
The energy increase can be computed from the fact that the energy of the ideal gas only depends on the temperature.
The mathematical expression for the change of entropy has an especially simple form in the case that possible volume changes are invertible, that is, do not involve diffusion. The increase in entropy during a time interval, during which the body has the absolute temperature T and has the heat input Q, is given by Q/T. If the temperature changes, the entropy is equal to the sum of the quotients Q/T, which correspond to the individual time intervals.
During a positive Carnot cyclic process, the working body takes in at the temperature T1 the heat Q1 and loses at the temperature T2 the heat Q2 , or, in other words: It takes in the heat -Q2, whence its entropy increase is Q1/T1 - Q2/T2. However, since this expression is zero, its overall entropy has not changed, but indeed the entropies of the two heat containers have changed. The entropy of the container at the higher temperature T1, which has given the heat Q1 to the working body, has now decreased by Q1/T1, the entropy of the container at the lower temperature T2, which has received the heat increase Q2, has increased by Q2/T2. The two quotients have the same magnitude, but opposite signs, whence they meet the condition: During reversible processes, the entropy in the complete system (the working body and the two heat containers) remains constant.
If a cyclic process is irreversible, the conversions of work into heat and of heat at higher temperature into heat at lower temperature are in excess, whence you have for every cyclic process the equation S dQ/T >= 0. The symbol d represents the corresponding, very small changes of entropy, the inequality the presence of irreversible processes. And all real processes are irreversible! Note: The Carnot cyclic process is idealized. The really invertible process demands, that all bodies involved should return in the end to their initial states. For this to be really the case, heat must flow from somewhere into the heat container T1, which loses the heat Q1, and the container T2, which accepts Q2, must give it away to somewhere; moreover, we cannot set the pressure and counter pressure at the piston completely equal to each other. If we take this into consideration, also this process is irreversible.
Boltzmann has improved the understanding of the concept of entropy by molecular theoretical considerations: If the heat energy of a body is the sum of the kinetic energies of all its molecules, which during their enduring, mutual impacts equalize their energy , the energy exchange must last until all molecules have the same average kinetic energy, that is, until the temperature in the body is the same everywhere. If you were to cause in an initially uniformly tempered body differences in temperature, you would have to remove from certain groups of molecules some of their energy and give it to other molecular groups. This cannot occur on its own, because molecular motion acts in the sense of an energy equalization and in that of a local energy accumulation. This is the start of reflections, which are always made in probability theory: Molecules can be distributed with different densities in their space and can possess differently large energies, whence you speak of the probability of a certain grouping and of the probability of a certain state.The most probable state is always that in which equalization is realized as far as possible. If a system changes its state on its own, it transits from a less probable state to a more probable state, that is, it increases the probability of the state: The probability of a state becomes the measure of entropy; the larger the probability, the larger the entropy. It has reached its maximum when complete equalization has been reached. (Apart from an additive constant, the entropy is equal to the natural logarithm of the probability of the state.) This concept becomes clear, when you replace the word probable by the word stable or permanent. Just as the impossibility of the perpetuum mobile of the first kind has led to the first main theorem, the principle of the conservation of energy, the impossibility of the perpetuum mobile of the second kind has led to the second main theorem, which, following Planck, we will call the Principle of the Increase of Entropy.
Technical significance of entropy
At all times, even in the ideal case of the Carnot process, during conversion of heat into mechanical work, the quantity of heat Q2 = (Q1/T1)·T2 leaves the cyclic process and becomes useless. As a consequence, although heat is physically equivalent to mechanical work, it is technically, that is, economically of less value. (In this sense, you speak of lowly valued energy; mechanical energy and electrical energy are highly valued energies, heat is lowly valued.) A mental experiment shows: The owner of a steam engine installation has a partner using the installation in return per second of 427 mkg*. His steam boiler is under 2 atm, that is, contains water at 120ºC. He has raised the temperature of the water from room temperature (20º) to 120º, that is, he has put in 100 kcal per litre and delivers to his partner the agreed 42 700 mkg* per 100 seconds in the form of heat in every 1 l of water at 120ºC. Theoretically, they are equivalent to 427 mkg*/sec, but not technically, for, as the recipient converts the heat into work, he obtains even in the (unrealizable) ideal case at the most 25%, that is, 75% of the heat have no value for him. If he has received every 100 seconds 1 l water at 150ºC, about 69 would have been useless for him. The lower the level of the temperature of the heat delivered to him, the less is its value, when converted into mechanical energy. In this sense, we ascribe to each quantity of heat a certain temperature, the temperature of its container. During its conversion into work, the heat drops from the higher temperature level T 1 to the lower level T2. The higher the level from which it drops, and the deeper the level to which it drops, the larger is the fraction which is converted in the process into work ( and the less useless heat is left at the lower temperature level.)
If we let a heat carrier (steam, gas, heating oil) in a suitably designed installation - we mean here the entire system of components forming it, that is, boiler, machines, conducts and all other parts - increase its entropy (by reduction of its pressure and temperature), it will convert one part of its heat content, and indeed the substantially smaller one, into mechanical work. (One says: Into the work of the machine, but it is its work: It drives the machine, whence you use the term propellant.) However, the larger part remains heat at a lower temperature. It pays Nature a high tax for the mechanical work, which it performs, at the expense of its heat content . This is a fact of experience. For the magnitude of the as useless eliminated quantity of heat Q2 = (Q1/T1)·T2, the factor Q1/T1, the entropy, is decisive. You cannot change T2 very much; it is the room temperature or that of the condenser of a steam engine or such like. The larger the entropy, the larger is also the quantity of lost heat, which we must write off as debit work, whence the worse is the useful effect of the work performing system. In order to improve it, you must obviously make T1 as large as possible (T2 as small as possible, the ideal being the absolute zero; however, we must be satisfied with a around 300ºC higher temperature) and indeed the design of steam engines aims in this direction since a long time.
A knowledge of the entropy is therefore of fundamental importance for insight into technical processes. The entropy of steam - it is the soul of the steam engine as heat carrier; the useful effect with which it is employed depends on its energy content and its entropy - is calculated with the aid of an empirical equation of state and has been collected in tables and diagrams for all pressures and temperatures and their relationships, which characterize vaporization heat, saturation temperature, specific weight, etc. Just as one can represent performed work in a pressure-volume-diagram (pv-diagram) by an area, one can represent the heat used in a temperature-entropy-diagram (TS-diagram) by an area. The mechanical energy performed during the volume change dv at the pressure p by 1 kg of a body is dL = p·dv mkg*. The corresponding change of heat energy dQ at the temperature T is dQ = TdS kcal. Isothermal changes of state in the TS-diagram are parallel to the entropy axis, adiabatic changes are parallel to the temperature axis. Fig. 365 below presents a simple sample of a TS-diagram.
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