J3 Heat
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.
Carnot-Cyclic-Process 
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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