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of the rod A o shall be less than the length of the rod s A in
exactly the same proportion as the expansibility of the metal
composing A o exceeds the expansibility of the rm-tal composing
s A. If the lengths of s A and A o were equal, their increments
of length would be proportional to their dilatations ; but the
length of the more dilatable rod A o, being less than that of the
less dilatable S A, in the same proportion as the dilatability of
the former is greater than that of the latter, the absolute incre-
ments of their length, will necessarily be equal, the greater dila-
tability of A o being compensated by its lesser length.

1370. Harrison's gridiron pendulum. This principle is
variously applied in different pendulums. That which is best



DILATATION OF SOLIDS. 29

known is Harrison's gridiron pendulum represented
in fig. 434. The bob P is attached to a rod of
iron P c, having a cross-piece c at the top. This
cross-piece rests upon two rods of brass, F G and
K L, which are themselves supported by a cross-
piece, B E, of iron. This latter piece is attached to
two rods of iron, B A and E D, which are themselves
attached to a cross-piece connected with the point
of suspension or knife-edge on which the pendulum
vibrates. By the expansion of the iron rods A B
and D E, the distance of the cross-piece BE from the
point of suspension is augmented ; and by the ex-
pansion of the iron rod c P the distance of the bob
p, and therefore of the centre of oscillation from the
same point, is augmented. By the expansion of
the brass rods F G and K L, the distance of the cross-
piece C from the cross-piece B E is augmented, and
therefore the bob P and the centre of oscillation
Fig. 434. proportionally raised. Thus, the distance of the
bob P and the centre of oscillation from the point
of suspension will depend upon the relative amounts of the two
dilatations of the iron and brass rods, the former having a ten-
dency to lower and the latter to raise it. By the table of ex-
pansions (1361), it appears that the linear expansion of brass for
any given change of temperature is greater than that of iron in
the ratio of i-48 to 1. If, then, the total length of the rods
A B and PC be greater than that of F G in the ratio of 1'48 to 1,
their actual dilatations will be equal, and the centre of oscil-
lation will remain at the same distance from the point of
suspension.

1371. Bars of different metals mutually attached are curved
by dilatation and contraction. If two straight bars of differ-
ently dilatable metals be soldered together, every change of
temperature will bend the combined bar into the form of a
curve, the more dilatable metal being on the convex side of the
curve when the temperature is raised, and on the concave side
of it when it is lowered.

Let the more dilatable metal be called A, and the less dilatable

B. Now, if the temperature be raised, A will become longer

than B, and, as they cannot separate, they must assume such a

form, being still in contact, as is consistent with the inequality of

c 3



HEAT.



their lengths. This is a condition which will be satisfied by a
curve in which the bar A is on the convex and the bar B on the
concave side.

If the temperature, on the other hand, be lowered, the more
dilatable metal being also the more contractible, the bar A will
be more diminished in length than the bar B, and being, there-
fore, the shorter, will necessarily be on the concave side of the
curve.

1372. Application of this principle to compensation pendu-
lums. This principle has been ingeniously applied as a com-
pensator in the pendulums of clocks and the balance-wheels of
watches.

Such a compound bar as we have just described is placed at
right angles to the rod of the pendulum, and has, at its ex-
tremities, two bobs. When the temperature rises, and the centre
of oscillation is, by expansion of the pendulum, removed to a
greater distance from the point of sus-
pension, this compensating bar is bent
into the form of a curve concave towards
the point of suspension, as represented
injrg. 435. ; and the bobs which it carries
at its extremities being brought closer
to the point of suspension, compensate for
the increased distance of the bob of the
pendulum from that point.

If, on the other hand, the temperature
falls, and the rod of the pendulum contracting brings the bob
and the centre of oscillation nearer to the point of suspension,
the compensating bar is bent into a curve, which is concave
downwards, as represented in jig. 436. ;
and the bobs which it carries being re-
moved to an increased distance from the
point of suspension, compensate for the
diminished distance of the bob of the
pendulum.

1373. In application to balance-wheels.
The balance-wheel of a watch is a me-
tallic wheel, which moves on a finely-
constructed centre, and is connected with
a fine spiral spring, from which it receives an oscillating motion,
the time of its oscillation depending partly upon the diameter




Fig. 435.




Fig. 436.



DILATATION OF GASES. 31

of the wheel. Now any change of temperature affecting the
magnitude of the wheel by expansion and contraction will cause
a change in its diameter, and a consequent change in the time
of its oscillation, and the rate of the time-piece which it
regulates.

This irregularity has been compensated by attaching to the
rim of the wheel a compound metallic arch such as that already
described. When the temperature rises, and the diameter of
the wheel is augmented, this arch, with its concavity towards
the centre of the wheel, becomes more concave, and a weight
which it carries is brought nearer to the centre of the wheel,
and this compensates for the increased magnitude of the wheel.
If, on the other hand, the temperature is lowered, and the
diameter of the wheel diminished by contraction, this arch
becomes less concave, and the weight which it carries is re-
moved to a greater distance from the centre, and this compen-
sates for the diminished diameter of the wheel.



CHAP. iv.

DILATATION OF GASES.

1374. Volume of gaseous bodies dependent on pressure and tem-
perature. It has been already shown (706), that the dimen-
sions of bodies in the gaseous state are dependent altogether upon
the pressure by which they are confined. They are capable of
expanding spontaneously into any dimensions, however great,
and of being reduced by greater pressure to any volume, how-
ever small. It follows, therefore, that whenever it be required
to determine the change of dimensions of gaseous bodies pro-
duced by change of temperature, it will be necessary to provide
means of keeping them during the experiment under a uniform
pressure, since otherwise the change of dimensions due to
change of pressure would be combined with that which is due
to change of temperature.

1375. Method of observing the dilatation of gases under uni-
form pressure. Experimental enquirers have contrived and
practised various expedients to accomplish this, one of the most
c 4



HEAT.

simple of which is that of M. Pouillet,
represented in fig. 437. An iron siphon
tube D c is formed with short legs, from
the bottom of which proceeds a pipe
with a stop-cock F, under which is placed
a cistern or reservoir G. In the legs of
the siphon D c are inserted two glass
tubes, D E and c B, of more than thirty
inches in height. The tube D E is open
at the top ; the tube c D is closed at the
top, but has a horizontal branch united to
it at B, which is connected with a tube
A B made of platinum, which terminates
in a hollow ball A, also of platinum.
Fig. 437. A stop-cock is provided in the tube B A, so

as to communicate at pleasure with the

external air. The stop-cock F being closed, and the stop-cock
in the tube B A being open, mercury is poured into the tube D E,
so as to fill the glass tubes D E and c B nearly to the top. Since
the two tubes D E and C B both communicate with the external
air, the columns of mercury in them will stand at the same level.
To determine the expansion which air suffers when raised from
the freezing to the boiling point under a uniform pressure, let
the reservoir A be immersed in a bath of melting ice, so as to
reduce the air included in it to the freezing point. Let the
stop-cock in the tube B A be then closed, and let the bulb A be
removed to a bath of boiling water. The air in the bulb ex-
panding will press down the column of mercury in B c, and will
cause the column in D E to rise ; so that the levels of the two
columns will no longer coincide. But they may be equalized
by opening the stop-cock F, and allowing mercury to flow into
the reservoir G from the siphon, until the levels in the two legs
come to the same point. When that is accomplished, the
pressure upon the expanded air included in the bulb A, and the
tube communicating with it, will be equal to that of the atmo-
sphere, and equal to that which the same air has when at the
freezing point.

The capacity of the tube c B being known, the volume which
corresponds to any length of it will be also known.

Now the increment of volume which the air has suffered by
expansion will be indicated by the height through which the



DILATATION OF GASES. 33

mercury has fallen in the tube c D. This increment, therefore,
will be the dilatation of the air included in the bulb A and the
communicating tube between the freezing and boiling points.

In the same manner, by this apparatus, the dilatation corre-
sponding to any change whatever of temperature under a given
pressure can be ascertained.

1376. Dilatation of gaseous bodies uniform and equal. It
has been proved by experiments made with this as well as a
variety of other apparatus adapted to the same purpose, that the
dilatation of all bodies in the gaseous form is perfectly uniform
throughout the whole extent of the thermometric scale, the same
increments of temperature producing, under the same pressure,
equal increments of volume. But, what is still more remark-
able, it has been found that all gases whatever, as well as all
vapours raised from liquids by heat, are subject to exactly
the same quantity of expansion by the same change of tempe-
rature.

1377. Amount of this dilatation ascertained. By the ex-
periments of M. Gay Lussac, it was demonstrated in 180-i that
1000 cubic inches of atmospheric air raised from the freezing
to the boiling point were dilated so as to make 1375 inches.
These experiments have more recently been repeated by
MM. Rudberg, Magnus, Regnault, and Pouillet. It has been
found that the dilatation is more exactly expressed by 1367
cubic inches. Thus, the increment of volume of atmospheric air
between 32 and 212 is the Y^^th, or very nearly one-third of
its volume at 32. It follows, therefore, that ten cubic inches of
atmospheric air at 32 will, if raised to the temperature of 212,
become, by dilatation, nearly 13^ cubic inches; and, for
every additional 180 of temperature which it receives, it will
undergo a like increase of volume.

1378. Increment of volume corresponding to 1. To find the
increment of volume corresponding to one degree of temperature,
we have only to divide the fraction ^Vo by 180, which gives

TST5"oW = TW

The increment of volume, therefore, which any gas or vapour
undergoes when, under the same pressure, the temperature is
raised one degree, is the 490th part of the volume which it would
have if reduced to the temperature of 32.

It follows from this, that if any volume of air at 32 be raised
to the temperature of 32 + 490 = 522, it will expand into twice
c 5



34 HEAT.

its volume ; and if it be raised to a temperature of 32 + 2 x 490
= 1012, it will be expanded into three times its volume, and
so on.

1379. Experiments of Gay Lussac, Dulong, and Petit showed
the uniformity and equality of expansion. The well-known
experiments of Gay Lussac, the results of which were in accord-
ance with those subsequently obtained by Dulong and Petit,
establish the fact, that all gases, as well as all vapours, undergo
equal changes of volume, by equal increments of temperature,
the co-efficient of the expansion of atmospheric air being common
to all.

1380. This result qualified by the researches of Rudberg and
Regnault. Rudberg first called in question the correctness of
this principle, and not only showed that the co-efficient of the
expansion of atmospheric air previously determined was inex-
act, but that other gases, though so nearly equal in their rates
of expansion to each other and to atmospheric air, were not pre-
cisely so. These researches of Rudberg have been confirmed
by those of Magnus and Regnault ; and it appears from them
that the following are the increments of volume which the un-
dermentioned gases undergo between 32 and 212, their volume
at 32 being 1-000.

Hydrogen - - 0-366

Atmospheric air - 0-367

Carbonic oxide - 0-367

Carbonic acid - 0-371

Protoxide of azote - - 0-372

Cyanogen - - 0-388

Sulphurous acid - 0-390

M. Regnault also found that the dilatation of the same gases
are not exactly the same at all pressures. Thus, under 3^ atmo-
spheres, the dilatation of hydrogen remains unvaried, but the
dilatation of air increases from 0*367 to 0-369, and that of car-
bonic acid from 0-371 to 0-385, while the dilatation of sulphu-
rous acid, under a pressure of only one atmosphere, increases
from 0-390 to 0-398.

Thus it appears that although it be certain that the gases are
subject to a small difference in their rates of dilatation, and also
that the rate of dilatation of the same gas is not absolutely the
same at different pressures, yet the inequality and variations are
such as may be disregarded for all practical purposes ; and it may



DILATATION OF GASES. 35

be assumed that all gases and all vapours dilate uniformly, and
in the same degree as atmospheric air.

1381. Formula to compute the change of volume of a gas
corresponding to a given change of temperature. The follow-
ing formulae will serve to calculate the change of volume which
atmospheric air, or any other gas which dilates equally with it,
undergoes for any proposed change of temperature.

Let v express a volume of air at 32.

Let v express its volume when raised to a temperature which
exceeds 32 by a number of degrees expressed by T.

The increment of volume, therefore, corresponding to the
increment of temperature expressed by T, will be vv; and

since the increment of volume corresponding to 1 is T^T;? the

increment corresponding to T degrees will be x T. We
shall therefore have



and consequently,



In this case the gas has been supposed to be submitted to an
increase of temperature. If it be reduced to a lower tempe-
rature, it will suffer a decrement of volume, expressed by v v ;
and if T express the number of degrees below 32 to which it

is reduced, the decrement of volume for 1 being , the de-

4yo

crement for T degrees will be as before, -7^-: x T, and we shall

4yo

have



from which we find,



If, therefore, the volume of a gas at 32 be known, its volume
at any other temperature above or below 32 may be calculated
by the following



HEAT.



RULE.

Divide the difference between the number of degrees in the
temperature and 32 by 490. Add the quotient to 1 if the
temperature be above 32, and subtract it from 1 if it be below
32. Multiply the volume of the gas at 32 by the resulting
number, and the product will be the volume of the gas at the
proposed temperature.

Table showing the changes of volume of a gaseous body
consequent on given changes of temperature.

In the columns v of the following table are expressed in cubic
inches the volumes which a thousand cubic inches of air at 32
will have at the temperatures expressed in the columns x, being
supposed to be maintained under the same pressure.



T.


V.


T.


V.


T.


V.


T.


V.


T.


V.


-50


832-7


5


924-5


40


1016-3


85


1108-2


130


1200-0


49


831-7


4


926-5


41


1018-4


86


1 i 10-2


131


l-20i-0


48


836-7


3


928-6


42


10.0-4


87


1112-2


132


1204-1


47


838-8


_2


930-6


43


1022-4


88


1114-3


133


1206-1


46


840-8




9327


44


1024 5


89


11 '6-3


134


1208-2


45


842-8





934-7


45


1026-5


90


1118-4


135


1210-2


44


844-9


1


936-7


46


1028-6


91


1120-4


136


1212-2


43


846-9


2


938-8


47


1030-6


92


1122-4


137


1214-3


42


849-0


3


940-8


48


1032-7


93


1124-5


138


1216-3


-41


851-0


4


942-9


49


10347


94


1126-5


139


1218-4


40


8531


5


944-9


50


10367


95


1128-6


140


1220-4


39


855-1


6


947-.0


51


1038-8


96


1130-6


141


1 222-4


38


857-1


7


9J9-0


52


10408


97


113-2-7


142


12-24-5


-37


859-2


8


951-0


53


1042-9


98


1134-7


143


1226-5


-36


861-2


9


953-1


54


1044-9




1136-7


144


1228-6


35


863-3


10


955-1


55


104G-9


100


113S-8


145


1230-6


34


86V3


11


957-1


56


1049-0


101


1140-8


146


1232-7


-33


867-3


12


959-2


57


1051-0


102


11429


147


1234-7


-32


869-4


13


961-2


58


1053-1


103


1144-9


148


1236-7


31


871-4


14


963-3


59


1055-1


104


1147-0


149


1238-8


30


873-5


15


965-3


60


1057-1


105


1149-0


150


1240-8


-29


875-5


16


967-3


61


1059-2


106


1151-0


151


1-242-9


28


877-6


17


'.169-4


62


1061-2


107


1153-1


152


1214-9


27


879-6


18


971-4


63


1063-3


108


1155-1


153


1246-9


26


881-6


19


973-5


64


1065-3


109


1157-1


154


1249-0


25


8837


20


975-5


65


1067-3


110


1159-2


155


1251-0


24


885-7


21


977-6


66


1069-4


111


1161-2


156


1253-0


23


887-8


22


979-6


67


1071-4


112


1 163-3


157


1255-t


-22


889-8


23


9S1-6


68


1073-5


113


1165-3


158


12-57-1


21


891-8


24


983-7


69


1075-5


114


1 167-3


159


1259-2


20


893-9


25


985-7


70


1077-6


115


1169-4


160


1261-2


19


895-9


26


987-8


71


1079-6


116


1171-4


161


1263-3


-18


898-0


27


989-8


71


1081-6


117


im-5


162


1265-3


-17


MM


28


991-8


73


1083-7


118


1175-5


163


1267-3


16


902-0


29


993-9


74


1085-7


119


1177-6


164


1269-4


-15


904-1


30


995-9


75


1087-8


120


1179-6


165


1271-4


14


906-1


31


998-0


76


1089-8


121


1181-6


166


12735


13


908-2


32


1000-0


77


1091-8


122


1183-7


167


1275-5


12


910-2


33


1002-0


78


1093-9




1185-7


168


1277-5


11


912-2


34


1004-1


79


1095-9


124


1187-8


169


1279-6


10


914-3


35


1006-1


80


1098-0


125


1189-8


170


1281-6


9


9J6-3


36


1008-2


81


11000


126


1191-8


171


1283-7


- ,-8


918-4


37


1010-2


82


1102*0


127


1193-9


172


1285-7


1 -7


920-4


38


1012-2


H3


1104-1




1195-9


173


1287-8


^6


922-5


39


I014/3


84


1106-1


129


1198-0


174


1289-8



DILATATION OF GASES.



37



T.


V.


T.


V.


T.


V.


T.


V.


T.


V.


.75


1591-8


188


1318-4


201


1344 9


2!4


1371-4


270


1485-7


176


1293-9


189


1320-4


202


1346-9


215


1373-5


280


1506-1


177


1295-9


190


1322-4


203


13490


216


1375-5


290


1526-5


178


1298-0


191


1324-5


204


1351-0


217


1377-5


300


1546-9


179


1300-0


192


1326-5


205


1353-1


218


13796






180


1302-0


193


1328-6


206


1355-1


219


1381-6


300


1546-9


1S1


1304-1


194


1330-6


207


1357-1


220


1383-7


400


1751


1R2


130'i-l


195


13326


208


1359-2






500


1955-1


183


13C8-2


196


1334-7


209


1361-2


220


1383-7


600


2159-2


184


1310-2


197


U3G-7


210


13633


230


1404-1


700


2363-3


185


1312-2


198


13H8-8


211


1365-3


240


1 4*4-5


800


2567-3


186


1314-3


199


1340-8


212


1367-3


250


1444-9


900


2771-4


187 1 13163


200 1 1342-9


213


1369-4


260


146&-3


1COO


V975-5



1382. Increase of pressure due to increase of temperature.
If air or gas be included within any limits which prevent its
expansion by increase of temperature, its elastic force or pressure
will be increased in the same proportion as its volume would
be increased if it were not thus confined. Thus, if a certain
quantity of air confined under a given pressure receive such an
increase of temperature as would cause it to expand into double
its volume, and if, after having so expanded, it be subject to
such an increased pressure as will reduce it to its primitive
volume, it will acquire double its primitive pressure. This
follows from the principles already established, that the pressure
of air and gas is universally as the volume into which they are
compressed.

1383. Formulae expressing the general relation between the
volume, temperature, and pressure. It will be convenient,
however, to establish general formulae by which the relation
between the volume and temperature of the same gases under
different pressures may be expressed, so that the volume at any
given temperature and pressure being given, the volume at any
other temperature and pressure may be obtained.

It has been already shown, that at the same temperature the
volume will be inversely as the pressure (708) ; so that, if v
and v' be two volumes at the same temperature and under the
pressures P and P', we shall have



v : v' : : P' : P



and therefore



Hence it follows, that if the same quantity of air or gas be
simultaneously submitted to changes of temperature and pressure,



38 HEAT.

the relation between its volumes, pressures, and temperatures,
will be expressed by the general formula



where T and T' express the number of degrees above or below
32 at which the temperature stands, + being used when above
and when below 32, and the pressures being expressed in
the usual manner by P and p'.

By this formula, the volume of a gas at any proposed tem-
perature and pressure may be found, if its volume at any other
temperature and pressure be given.

1384. Examples of the effects of dilatation and contraction.
The expansion and contraction of air explain a multitude of
phenomena which present themselves in the natural world, in
domestic economy, and in the arts.

1385. Ventilation and warming of buildings. In the venti-
lation and warming of buildings, the entire process, whatever
expedients may be adopted, is dependent upon this principle.
When a fire is lighted in an open stove to warm a room, the
smoke and the gaseous products of combustion, ascending the
chimney, soon fill the flue with a column of air so expanded
by heat as to be lighter, bulk for bulk, than a similar column
of atmospheric air. Such a column, therefore, will have a
buoyancy proportional to its relative lightness. This upward
tendency is what constitutes the draft of the chimney ; and
this draft will accordingly be strong and effective in just the
.same proportion as the column of air in the chimney is kept
warm. When the fire is first lighted, the chimney being filled
with cold air, there is no draft ; and, consequently, the flame
and smoke often issue into the room. According as the column
of air in the chimney becomes gradually warm, the draft is
produced and increased. The draft is sometimes stimulated by
holding burning fuel for some time in the flue, so as to warm
the lower strata of air in it.

But the most effectual method of stimulating the draft when
the fire is lighted is by what is called a blower, which is a sheet
of iron that stops up the space above the grate bars, and pre-
vents any air from entering the chimney except that which
passes through the fuel, and produces the combustion. This
soon causes the column of air in the chimney to become heated,



DILATATION OF GASES. 39

and a draft of considerable force is speedily produced through
the fire.

1386. Effect of open fire-places and close stoves. An open
chimney differs from a close stove, inasmuch as the former
serves the double purpose of warming and ventilating the room,
whereas the latter only warms, and can scarcely be said to



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