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this means is ultimately converted into heat. A delicate
thermometer at / gives the temperature with great exact-
ness, and, the usual precautions being taken to eliminate
the effects of radiation and conduction, it is evident that
the amount of heating effect may be accurately measured,
and by knowing the thermal capacity of the box and its con-
tents this may ultimately be expressed in terms of the unit of
heat. Joule also made experiments on the friction of iron.
In these a disc of cast-iron was made to rotate against
another disc of cast-iron pressed against it ; the whole being
immersed in a cast-iron vessel filled with mercury. By all
these experiments it was found that the quantity of heat
produced by the friction of bodies, whether solid or liquid,
is always proportional to the quantity of work expended,
and that the number of units of work in kilogrammetres
necessary to raise by iC the temperature of one kilogramme
of water taken at about ioC was as follows :

423.9 from friction of water mean of 40 experiments.
424-6 ,, mercury 50 ,,

425-2 ,, cast-iron ., 20

332 RELATION BETWEEN HEAT

Quite recently (Phil. Trans. Jan. 1878) Joule has repeated
his experiments on the friction of water. The method he
adopted was to revolve a paddle in a vessel of water, deli-
cately suspended ; to find the heat thereby produced ; measur-
ing the work by the force required to hold the vessel from
turning, and the distance run as referred to the point at
which the force was applied ; in other words making use of
the principle that the work transmitted by a shaft is equal to
the moment of the force multiplied by the angle of rotation
of the shaft.

He obtained from these experiments the value 423*8, which
agrees extremely well with his previous results.

Still more recently an experiment of a similar kind has
been made by Professor Rowland, of the John Hopkins
University, who obtains a result agreeing extremely well with
that of Joule.

328. II. Magneto-electricity. Other methods were
used by Joule ; one of these took advantage of magneto-
electricity, and was essentially the same experiment which
was afterwards put in the following form by Foucault.

If a metallic disk or top in rapid rotation be brought
between the poles of a powerful electro-magnet, induced
currents will be generated in the top in consequence of its
rotation in presence of the magnet, the tendency of which
will be to bring the top to rest. The effect is exceedingly
curious, and if it be asked what becomes of the energy of
the rotation, the answer is that it is converted in the first
place into electricity in motion, but ultimately into heat, and
that in consequence the temperature of the disc will be found
to have increased. If the disk be turned by hand the effect
is very strange; it is found almost impossible to move it
while the electro-magnet is in its neighbourhood, but when
the current is broken it is of course exceedingly easy to do
so. At the expense of much labour the disk, so revolving

AND MECHANICAL EFFECT. 333

between the poles of the magnet, may be heated until it is
too hot to be touched.

Joule's final results coincided in giving 424 kilogrammetres
as the mechanical equivalent of the heat necessary to raise
one kilogramme of water (weighed in vacuo and having the
temperature of about ioC) through iC. Strictly speaking,
this determination is for the value of gravity at Manchester,
and for the specific heat which water has between 8C and
io c C.

329. III. Condensation of gases. Before leaving this
subject it will be desirable to consider the method of deriving
the mechanical equivalent of heat from the condensation
of gases. Many familiar experiments shew that when a gas
is suddenly compressed there is a production of heat, and
that when suddenly expanded there is an absorption of heat.

and vapours for the purpose of determining the mechanical
equivalent of heat ; and air, the substance chosen by Mayer,
was no doubt very good for such a purpose ; nevertheless,
the suggestions of these philosophers do not seem to have
been accompanied with a clear appreciation of all the data
necessary to a complete proof.

330. Joule, however, in his experiments supplied what
was wanting in order to derive a good determination of the
mechanical equivalent of heat from the known gaseous laws.
By compressing air forcibly into a receiver surrounded by
water he found that the water was considerably heated. It
is not, however, correct to infer without further experiment
that the amount of heat produced in this case is the exact
equivalent of the energy expended in compressing the air.
A familiar instance will make this clear. By a blow of a
hammer upon a small quantity of fulminating mercury it is
exploded and produces a considerable amount of heated gas,
but we are not at liberty to suppose that all the heat thus,

334

RELATION BETWEEN HEAT

developed is merely the mechanical equivalent of the energy
of the blow, as will be evident by supposing such an extreme
case as a ton of the fulminating powder.

Evidently the substance is in different molecular condi-
tions at the end of the experiment and at the beginning,
and it may be supposed with much truth that the heat
produced is nearly all due to the conversion into a kinetic
form of a certain potential energy present in the compound.
Now in the experiment above described, in which air is
compressed, the air is evidently in a different molecular con-
dition after compression, for the particles are much nearer
together. The first thing therefore is to determine how
much, if any, of the heat produced may be due to this
change of the molecular condition of the air, and how much
to the work expended in compressing the air.

331. The following very ingenious experiment performed
by Joule is conclusive in shewing that the mere change of

Fig. 73-

distance of the molecules of a permanent gas neither pro-
duces nor absorbs heat to an appreciable extent. In Fig.

AND MECHANICAL EFFECT. 335

73 we have two strong vessels, of which A contains com-
pressed air, say under the pressure of 20 atmospheres ; B,
on the other hand, is a vacuum. The two vessels are con-
nected with each other by a tube having a stop-cock which
we may suppose to be shut. The whole apparatus is plunged
into a vessel of water. After the temperature of the water
has been very accurately ascertained, open the stop-cock, and
thus allow both vessels to have the same pressure.

When the experiment is finished it will be found that
there is no change in the temperature of the water. The
prevalent idea is that when air expands it becomes colder,
and that when condensed it becomes hotter; but Joule by
this experiment has shown that no appreciable change of
temperature occurs when air is allowed to expand in such a
manner as not to develope mechanical power. It follows as
an inference that when air is compressed the rise of tern-,
perature is scarcely at all due to the mere diminution of the
distance between the particles, but almost entirely to the
mechanical effect which must be spent on the air before this
condensation can be produced.

332. Specific heat of gas of constant volume. In
a previous part of this work a distinction was made between
the specific heat of a gas of constant volume and that of
the same gas of constant pressure, and the determinations
therein exhibited were those of Regnault.

His determinations give the specific heat of various gases
under constant pressure; that is to say, when the gas re-
mains at the same pressure during the various temperatures
to which it is exposed.

Experimentally it would be very difficult to find the specific
heat of a gas of constant volume ; nevertheless, the one
specific heat can be obtained from the other without trouble
by means of the knowledge derived from the experiments of
Joule.

336 RELATION BETWEEN' HEAT

Thus let us consider a rectangular prismatic vessel one
square metre in section (Fig. 74), and suppose that we have
a cubic metre of air under the ordinary
pressure of 760 millimetres of mercury re-
duced to oC contained in it, the tempe-
rature of the air being also oC. The whole
pressure on the surface a a, which shuts in
this air, may easily be found ; it will in fact
be that of a column of mercury whose base
is one square metre and whose height is
0-760 metres. Now the specific gravity of
mercury at oC is 13-596 (Art. 75), that is
to say, a cubic decimetre (or .001 of a cubic metre) of mer-
cury at oC will weigh 13.596 kilogrammes, and hence the
above column of mercury will weigh 760 x 13.596 = 10333
kilogrammes nearly. We may suppose, for the sake of sim-
plicity, that there is no atmosphere above this air, and that it
is kept down by a veritable weight of 10333 kilogrammes
above it. The weight of this cubic metre of air will at oC
be (Art. 149) 1.2932 kilogrammes nearly, and if it be raised
in temperature through an interval of 272C its volume under
the same pressure will be exactly doubled ; that is to say, it
will have raised the weight 10333 kilogrammes one metre
high and done work represented by 10333 kilogrammetres.

Now according to Regnault's determination, Art. 299, the
specific heat of air is 0.237 ; tnat i g to sav > ^ w ^ on ty
require 0.237 f tne amount of heat necessary to raise a
kilogramme of water one degree in temperature, in order
to raise a kilogramme of air under constant pressure one
degree in temperature.

Hence the amount of heat necessary to raise this air from
oC to 272C under constant pressure will be

1.2932 x 0.237 x 272 = 83.365 heat units.
In this expression the first factor refers to the weight of the

AND MECHANICAL EFFECT. 337

air, the second to its specific heat, and the third to its
increase of temperature. But in the course of this increase
of temperature work equal to 10333 mechanical units, or

(Art. 328) 222 =24.37 neat units, has been done.
424

Hence, of all the heat expended upon this air, or 83.365
units, 24.37 units have been spent in work. Hence also
83.365 24.37 or 58.995 units denote the amount of heat
consumed in the mere heating of the particles.

But, according to Joule's experiment, if the air had
throughout been confined to one cubic metre, and afterwards
made to occupy two cubic metres without doing any work,
the whole heat of the particles would be the same in the
second case as in the first. But we have seen that the mere
heat consumed in heating the particles occupying the two
cubic metres is 58.995 units; hence this also will represent
the heat required to raise the air remaining at the constant
volume of one cubic metre through 272C. It thus appears
that if 83.365 be used to represent the specific heat of this
air under constant pressure, 58.995 will represent its specific
heat of constant volume.

According, therefore, to our usual method of measuring

specific heat, 0.237 x o ? = - l6 7 w ^l denote the specific

y 3-35

heat of air, the volume of which remains constant during
experiment.

SECOND LAW OF THERMO-DYNAMICS.

333. Reversible engines. Having now described at
some length the first law of thermo-dynamics and the ex-
perimental proofs of the same, let us proceed to consider
the second law, which relates to the conversion of heat into
work.

338 RELA TION BETWEEN HE A T

The following proof of this law is deduced from that
given by Professor Sir W. Thomson.

In establishing this very important principle recourse is
had to a conception of Carnot, to whom this branch of
science is much indebted, although his idea of the nature
of heat was erroneous.

This conception is that of an engine completely reversible
in all its physical and mechanical agencies. Such an engine
must be supposed to have a source of heat and also a re-
frigerator, the temperature of the first being of course higher
than that of the second, and it produces work while it trans-
fers heat from the source to the refrigerator. If worked
forwards, such an engine will produce a certain amount of
work from a certain amount of heat which leaves the source;
but if worked backwards, owing to its perfect reversibility,
it will, at the expense of a similar amount of work, bring
back the same amount of heat into the source.

334. Now it may easily be shewn that a perfectly re-
versible engine will produce as much mechanical effect as
can be produced by any heat engine, with the same tempe-
ratures of source and refrigerator, from a given quantity of
heat. For let there be two heat engines A and B, of which
B is a reversible engine, both working between the same
temperatures, and if possible let A derive more work from a
given quantity of heat than B. Now if A be worked for-
wards, a quantity of work W is produced by it, during the
conveyance of a quantity of heat Q from the source of heat.

If B were worked forwards, a quantity of work w (less
than W by hypothesis) would be derived from the same
quantity of heat Q; but since B is completely reversible if
worked backwards it would restore to the source of heat
a quantity of heat Q by the expenditure of a certain amount
of work w (less than W).

Thus we have

AND MECHANICAL EFFECT. 339

A working forwards and producing an amount of work W
by carrying heat = Q from the source.

B working backwards and spending an amount of work w
(less than W) in order to carry heat = Q to the source of
heat.

But, since the work produced by A is greater than that
spent by B, A may be made to work B, and hence the whole
arrangement becomes self-acting; while, on the whole, the
source neither gains nor loses heat, and work equal to Ww
is produced during each double cycle of operations. Now,
as far as this problem is concerned, we may suppose all the
bodies surrounding the source (with the exception of the
refrigerator) to be of the same temperature as the source,
and therefore, if the hypothesis with which we started is
correct, we may go on continually producing work by the
mere presence of a refrigerator, or body of lower tempera-
ture ; while, at the same time, no heat is conveyed from the
bodies of higher temperature which are supposed to sur-
round this refrigerator.

Since however, consistently with the conservation of
energy, heat must disappear as heat in order to produce
this work, we see that this heat must in this supposed case
really come from the body of low temperature; that is to
say, work is produced by abstracting heat from a body of
already low temperature. A little reflection will shew that
such a process might be carried on for ever, and would
result in a perpetual motion ; but since we cannot admit
the possibility of such a case, we are forced to conclude that
our hypothesis is erroneous.

But our hypothesis was, that of two engines A and B, of
which the latter is reversible, working between the same
source and refrigerator, A could produce more work than
B out of the same quantity of heat. We are thus driven to
the conclusion that under similar circumstances B produces

Z 2

349

RELATION BETWEEN HEAT

as much work as A ; and therefore that the test of maximum
work under given circumstances is reversibility.

335. Reversible engines of infinitely small range.
In the next place let us take a mass of any substance (for
the sake of simplicity we may suppose it to be fluid), and
let each unit of its surface be subjected to the uniform pres-
sure/, also let its volume be v while its temperature is /.

Let us imagine this substance to form our heat engine ;
that is to say, let us imagine certain operations at different
temperatures to be performed on this substance whose result
is that heat is transmuted into work.

And here it is well to observe that we need not trouble
ourselves about the practicability of making such an engine ;
all that we need care about is that our conception is pre-
cise and mechanically conceivable.

Let us reckon pressures and volumes along two axes
at right angles to one another as in Fig. 75, and let us
suppose our substance to have a volume v denoted by
og, its pressure p being represented by ag. Now, in the
first place, let it expand from volume v to v + dv, its

temperature being kept con-
stantly /; at the end of this
expansion its volume may be
supposed to be oi and its
pressure tb, necessarily less
than ga ; let it now, secondly,
be allowed to expand further,
without either emitting or ab-
sorbing heat, till its tempera-
ture goes down through an ex-
ceedingly small range to / r,
(T being very small); c may
now be taken to denote the place of the substance in our
scale of pressures and volumes; thirdly r , let it be com-

ff h

Volumes.

Fig. 75-

t k

AND MECHANICAL EFFECT. 34!

pressed at the constant temperature ir (differing infinitely
little from /), so much that when,/0#rM/j/, the volume is further
diminished to the original volume v without the substance
being allowed either to emit or absorb heat, its temperature
may be /.

336. Here, then, we have first of all two expansions,
and next two similar compressions bringing back the body
to its original state, and the first compression is the reverse
of the first expansion, while the second compression is the
reverse of the second expansion, and all are supposed to be
extremely small.

Evidently, therefore, the line which denotes the first com-
pression will be parallel, but opposite in direction to that
which denotes the first expansion, and the same will hold for
the other two lines ; so that, provided all the movements be
sufficiently small, the four positions of the body will be the
four corners of a parallelogram abed. Now the work done
by the body during the expansion between a and b (equal
to the mean pressure on unit of surface multiplied by the
volume passed over) will be denoted by the area abtg, also
that done by the body between b and c will be denoted by
the area bcki. Hence the whole work done BY the body be-
tween a and c will be denoted by the area abckg.

In like manner the work performed AGAINST the body during
the compression between c and dfwill be the area cdhk, and
that done against the body by the further compression be-
tween d and a will be the area dhga. Hence the whole
area ckgad will denote the whole work done AGAINST the body.

Now the difference between the area abckg, or the work
done by the body, and the area ckgad, or the work done
against the body, is the parallelogram abed.

Hence this parallelogram denotes the whole surplus work
done BY the body in its cycle of operations.

337. In the next place, the whole heat abstracted from

342 RELA TION BETWEEN HE A T

the source of higher temperature / is that required to heat
the body as it increases in volume from v to v-\-dv, this
operation being performed at the temperature /, while, on
the other hand, when the body is finally restored to this
temperature / it has only the volume v.

Heat corresponding to the volume v -f- dv is therefore taken
from source /, while heat corresponding to the volume v
is restored to this source.

Hence the whole heat abstracted from source / is that
required to increase the volume of the body at the constant
temperature / from v to v + dv.
This may be called M dv.

338. Let us now suppose the cycle of operations to be
reversed, or the engine to be worked backwards starting
from c and from the lower temperature / r.

First, let the body contract in volume without giving or
receiving heat until it becomes of the temperature /.

Next, let it be further contracted at the constant tempera-
ture / through the volume dv.

Thirdly, let it expand without giving or receiving heat till
it falls to the temperature / r. And

Fourthly, let it expand at the constant temperature / r
through the volume dv.

It will be seen that in this cycle the body goes from c to b
and from b to a having work done UPON /// while it goes from
a to d and from d to c having work done BY it.

The work done UPON it is represented therefore by the
larger area abckg, and that done BY it by the smaller area
ckgady and on the whole there is a surplus of work done
UPON the body denoted by the parallelogram abed. At the
same time it will be noticed that in the second operation,
where the body is contracted from v + dv to v (going from
b to a) at the temperature /, there is a surplus supply of heat
brought to the source equal to M dv.

AND MECHANICAL EFFECT. 343

339. Thus when this engine was worked in a direct
manner we had work produced equal to the area abed, while
heat was drawn from the source equal to M dv. Now when
the engine is reversed we have work spent equal to the same
area abed] while heat is brought to the source equal as before
to M dv. A body acted upon in this manner forms there-
fore a reversible engine, and we are entitled to apply to
it the reasoning of Art. 334, and to conclude that whatever
be the substance employed between the limits of temperature
/ and / r if the heat drawn from the source or M dv remain
constant, the work done, or the area abed, will also remain
constant; in other words, for the same temperature-limits

the ratio between M dv and the area abed, or : ->

area abed

is constant, whatever be the substance used.

340. The area abed is easily found thus :

Produce cd to cut ag in e. Now area <2\$<:<^ = area abfe,
since they are on the same base and between the same
parallels, but area abfe = aex perpendicular distance between
ae and bf=aexgi. Hence area abed = aexgt.

Now if the operation denoted by the body going from
e to dj namely the contraction of volume at the fixed tem-
perature / r, be continued until the original volume v is
reached, that is, up to the line ag which limits this volume,
we should be brought to the point e ; eg would thus denote
the pressure of the body at temperature / r and at volume
v ; while ag is the pressure at the same volume, but at tem-
perature /.

Hence ae denotes the change of pressure of the body at
constant volume v while the temperature falls from / to
/__ r . If we consider the pressure to be a function of the

temperature /, ae will thus be represented by r ; that is to
say, by the differential coefficient of this function multiplied

344 RELATION BETWEEN HEAT

by the small temperature-change r. Again, gi is evidently
the change of volume or dv. Hence

Area abcd = aexgi = -^T dv.

341. Carnot's function. Hence also

Mdv M dv M

which :

area abed' dp dp

is constant for the same temperature / and range T whatever
be the substance used, but if in going from one substance to
another we always adhere to the same value of r, or keep
the fall of temperature the same, it will thus become a
constant multiplier, and we may therefore dispense with it

M

and assert that - is constant for the same temperature
#

dt
whatever be the substance used. In other words, = \$ (/) ;

dt

that is to say, it is a function of the temperature only, and
does not vary with the nature of the substance.

This very important common property of all bodies was
first discovered by Carnot.

342. Probable form of this function. Since there-
fore this function is the same for all substances, in. order
to determine its form let us take some one body whose
laws are best known. Let us, for instance, take a perfect
gas, and consider in the first place the relations which
subsist between the temperature and pressure of such a gas
whose volume is v. This is very easily found, for if P
denote the pressure of this gas at oC, its pressure at f will
be/>(i+o/). (Art. 63.)

Now we have reason to conclude from the experiment of

AND MECHANICAL EFFECT. 345

Joule (Art. 331) that the molecular heat of the particles of a
perfect gas whose temperature is constant is independent of
the volume of this gas, being the same for a great volume
as for a small one. The heat absorbed when such a gas
expands at a constant temperature is therefore the exact
equivalent of the work done in expansion.

Hence the mechanical equivalent of M dv, that is to say
of the heat absorbed while the gas increases in volume at
the constant temperature / from v to v + dv, will be denoted
by the work done, that is, by pdv or P (i + af) dv.

Hence if we use J to denote the multiplier by which our
heat-unit must be multiplied in order to produce the me-
chanical equivalent (that is to say, /= 424 (Art. 328)), we
shall have

-

= P

M _ P (i+q/) __ i+o/
~~

But again, since / = P (i + a/), we have ~- = Pa ; and hence

dp JPa ~7o~

dt

343. This therefore is most probably the true value of

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