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# An elementary treatise on heat

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occupies in the stem is accurately noted. It is clear that by
this means the volume of the liquid for each temperature
becomes known, and hence the amount of its dilatation may
be easily deduced.

48. (II) Method by specific gravity bottle. Here a
vessel, the internal volume of which is accurately known for
all temperatures, is separately filled at each temperature with
the liquid under examination, and the whole is then weighed.

DILATATION OF LIQUIDS. 45

The weight of the vessel when empty is also ascertained, and
thus the weight of liquid which it contains at each tempera-
ture becomes known. But the volume of this liquid is also
known ; hence its density, which varies as the weight of unit
of volume, becomes known, and thus the dilatation may be
determined. In this method the kind of bottle generally used
is one made of glass, having a glass stopper which fits it
accurately. The stopper is ground out of a capillary tube,
such as that used for the stem of a thermometer, and hence,
when the bottle is filled with liquid and the stopper pushed
home, any excess of liquid is forced out through the capillary
orifice. The bottle ought to be filled in this manner at a
temperature lower than that of observation, so that, when it
is subjected to the higher temperature and the liquid expands
the excess may escape by the orifice of the stopper and yet
leave the bottle quite full.

49. (Ill) Method by weighing a solid in the
liquid, or the areometric method. In this method a
solid whose volume is accurately known for each temperature
of observation is weighed immersed in the liquid at these
temperatures. The difference between the weight of this
solid in vacuo and its weight in the liquid will give us the
means of determining the relative density of the latter at the
various temperatures. This will be seen from the following
example :

Let us suppose that the volume of the solid at oC is
denoted by unity, but at iooC by 1-006. Suppose also
that the apparent loss of weight of the solid when weighed
in the liquid at o is 1800 grains, while at 100 the same is
only 1750 grains : 1800 grains is therefore the weight at o
of a volume of the liquid equal to unity, while 1750 grains
is the weight at 100 of a volume of the liquid equal
to i -006. Hence 1739*56 grains will denote the weight of
unity of volume of the liquid at 100; and hence also 1800

46 DILATATION OF LIQUIDS.

grains, which at o occupied a volume equal to unity, will
at 1 00 occupy a volume

1800

or the dilatation between these two temperatures is repre-
sented by "0347.

50. Absolute dilatation of mercury. In all these
methods the capacity of the vessel or the volume of the
solid employed must be known at the various temperatures
of observation, or, in other words, we must know its cubical
dilatation.

But the remarks in the preceding chapter (Art. 38) lead
us to conclude that in order to determine accurately the
cubical dilatation of a solid it is hardly sufficient to deter-
mine the linear dilatation of another specimen of the same
material and to multiply this by three, but the cubical
dilatation ought, if possible, to be obtained by direct experi-
ment. We have already seen (Art. 39) that in order to
accomplish this it is necessary to know the absolute dilata-
tion of some one liquid, such as water or mercury.

The problem before us is thus reduced to the determina-
tion of the absolute expansion of some one liquid, after
which that of other liquids may be easily derived. This
therefore is a determination of much importance ; and since
mercury has been chosen for the purpose, we shall now
proceed to shew how the absolute expansion of this liquid
may be found.

51. The method about to be described was first employed
by MM. Dulong and Petit. It consists in filling a U-shaped
tube with mercury, one limb being kept at a low and the
other at a high temperature. The portion of the liquid
which is heated will of course be specifically lighter than the
other, and hence the hot column must be higher than the cold
one, since the two balance each other hydrostatically. Thus

DILA TA TION OF LIQ UIDS. 47

if D, U are the two densities, and H, H' the corresponding
heights, we shall have D : I/ : : H' : H, or the heights will
vary inversely as the densities. This method is perfect in
principle, but it is almost impossible to keep a column of
mercury at a constant high temperature and at the same
time be able to observe accurately the position of the top of
the column. Regnault has however improved the apparatus
so as to overcome this obstacle, and the following sketch
will give an idea of the arrangement which he employed :
a b, a' b' are the two vertical tubes to be filled with mercury,
and these are connected together near the top by a horizontal
tube a a. At the bottom they are not connected together,
but a b is connected with the horizontal tube b c, and a' b'
with b' c'. To the extremities of these horizontal tubes two
vertical glass tubes eg, c' g' are attached, and these are
both connected with a tube h i leading to a large reservoir f
supposed to be filled with gas whose temperature is con-
stant ; hence the pressure of this gas in the tubes eg,
c f ^ is also constant. Heat is applied to the tube a b, and
by means of an agitator every part of this tube, including
the mercury which it contains, may be brought to the same
temperature throughout, and the value of this temperature is
accurately ascertained. On the other hand, the tube a b' is
exposed to a current of cold water of a known constant
temperature.

The tubes a <\$, a' b' are supposed to be filled with mercury
until above the level aa y but we shall shew in the sequel that it
is not necessary to know the height of the fluid above this level.

Now let / denote the whole pressure due to the left-
hand column of mercury, and p' the whole pressure due to
the right-hand column. The pressure at c is evidently /,
while that at / is/'. Hence the pressure at d =p pressure
of column c d, and in like manner pressure at df =p' pres-
sure of column c' d'. But the pressure at d is equal to that

48 DILATATION OF LIQUIDS.

at d', both being equal to the pressure of the gas in the
reservoir/": hence we have

/ pressure of c d =p' pressure of / d',
and therefore

p'p = pressure of column (c' d' c d) . . . ; (i)

that is to say, the difference between the pressures of the

Fig. 12.

two great vertical columns is equal to the pressure of the
column of mercury contained between the levels d* and d.
Now since the tubes a b, a V communicate together by a a',
it is evident from hydrostatical principles that the portions

DILATATION OF LIQUIDS. 49

of the two vertical columns above a a' are in equilibrium
with each other, and therefore that the pressures of these
two portions are equal. But/, or the whole pressure of the
left-hand column, = pressure of column a b + pressure of
portion above a ; and in like manner p' = pressure of column
a' b' + pressure of portion above a? . Now since the pressures
above a a are equal, it follows that

p' p = pressure of a' b' pressure of a 6, . . . (2)
and equating (i) with (2),

pressure of (</ d' cd] = pressure of a' <' pressure of a b.

We have thus obtained an expression for the difference
in pressure between two columns of mercury (a <, a' b') of
equal length but of different temperatures, and since there
is no occasion to view the top of the column, we can perfect
our arrangements for keeping the whole at the same tem-
perature throughout, while by the insertion of an air ther-
mometer alongside of a b this temperature may be measured
with great exactness. By this means therefore the relative
density of mercury at various temperatures may be deter-
mined, and its dilatation thence easily deduced l .

52. Using this method, and also a modification of it,
Regnault obtained results which enabled him to construct
a table giving the dilatation of mercury for every ten degrees
Centigrade from oC to 350. But before exhibiting this
table let us explain the distinction between the mean and the
true coefficient of dilatation, as it is quite necessary to know
this in the case of liquids which change their rate of ex-
pansion from one temperature to another.

In general language, if we take a quantity of liquid whose

1 It has occurred to the writer of this treatise that it might be possible
accurately to ascertain the length of the heated column in this experi-
ment by means of an insulated metallic pointer that would allow an
electric current to pass when it came into contact with the mercurial
column.

50 DILATATION OF LIQUIDS.

volume at oC is equal to unity, the mean coefficient of
dilatation for iC of the liquid between o and any point is
the mean rate of increase in volume of the liquid between
these two points, that is to say, it is the whole expansion
divided by the number of degrees included between the two
points.

Thus we see in the following table, second column, that
the whole dilatation of mercury between o and iooC is
.018153 > tnat i g to sav > a volume of this fluid equal to unity
at o will at 100 be equal to 1*018153. Now -018153 is the
increase for 100, and hence the mean increase for i will be
the hundredth part of this or '00018153, which accordingly
will be found in the third column opposite 100, as de-
noting the mean coefficient of dilatation of mercury between
o and that point.

On the other hand, the true coefficient of dilatation of the
liquid at any point is the rate of increase in volume of the
liquid at that point, as the temperature goes on regularly
increasing.

It has been the general custom to refer this rate of
increase to the original volume at zero, and the coefficient as
so determined is given in the fourth column of the following
table. For instance, the coefficient at 100 is found by this
column to be -00018405; that is to say, if the temperature
rises through a very small distance such as i and becomes
ioiC there will be an increase to the unit volume or
volume which it had at zero represented by -00018405. It
has, however, been remarked by Pouillet and by Tait that
this is not the proper way of representing the true coefficient,
but that the above increase of -00018405 for i should have
reference not to the volume at o but to the volume at 100.
Hence according to this definition the true coefficient at 100

00018405

will be -= -00018076.

1-018153

DILATATION OF LIQUIDS. 51

The fifth column of the following table exhibits these true
coefficients l .

Table of the absolute dilatation of Mercury.

True tempe-
rature as de-
termined by
an air thermo-
meter (/).

Whole dilata-
tion from o to
tC of a volume
of mercury
equal to
unity at o.

Mean coeffici-
ent of dilatation
between o
and /C.

Coefficient of
dilatation at
tC referred to
vol. at o.

True coeffici-
ent.

.

00017905

00017905

10

001792

00017925

00017950

00017922

20

003590

00017951

00018001

00017938

30

005393

00017976

00018051

00017955

40

007201

00018002

00018102

00017972

50

009013

00018027

00018152

00017989

60

010831

00018052

00018203

00018006

70

012655

00018078

00018253

00018024

80

014482

00018102

00018304

00018041

90

016315

00018128

00018354

00018059

100

018153

00018153

00018405

00018076

no

019996

00018178

00018455

00018092

1 20

021844

00018203

00018505

00018109

130

023697

00018228

00018556

00018125

140

025555

00018254

00018606

00018142

150

027419

00018279

.00018657

00018159

1 60

029287

00018304

00018707

00018175

170

031160

00018329

00018758

00018190

1 80

033039

00018355

00018808

00018206

190

034922

00018380

00018859

00018221

200

036811

00018405

00018909

00018237

2IO

038704

00018430

00018959

00018252

220

040603

00018456

00019010

00018267

230

042506

00018481

00019061

00018282

2 4

044415

00018506

00019111

00018297

250

046329

00018531

00019161

00018313

260

048247

00018557

00019212

00018327

270

050171

00018582

00019262

00018341

280

052100

00018607

.00019313

00018355

290

054034

.00018632

.00019363

00018370

300

055973

.00018658

.00019413

00018384

310

057917

00018683

.00019464

00018398

320

059866

00018708

.00019515

00018412

330

061820

00018733

00019565

00018426

340

063778

00018758

00019616

00018440

350

065743

00018784

00019666

00018453

1 The accuracy of Regnault's determination of the absolute expansion
of mercury has been confirmed by experiments very recently made by
Dr. A. Matthiessen by a quite different method.

52 DILATATION OF LIQUIDS.

It will be seen from this table that the true coefficient of
dilatation of mercury increases with the temperature.

53. Dilatation of water. The determination of the dila-
tation of water is also a point of much importance ; but before
proceeding to this subject it will be necessary to notice a very
striking peculiarity which this fluid exhibits with reference
to its change of volume through heat.

If ice-cold water, or water at oC, be heated, it does not
at first expand as might be supposed, but contracts for about
4C, and after that begins to expand. It thus exhibits a
point of maximum density. This beha-
viour was illustrated by Hope by means
of the following ingenious apparatus.

Fig. 13 represents a glass cylinder
rilled with water at an ordinary tempe-
rature, and having holes made for the
insertion of two thermometers, one near
the top and the other near the bottom.
The middle of the cylinder is surrounded
by an envelope filled with a freezing
F - mixture. At first, as the temperature

falls, the lower thermometer is very much
affected, while the upper one falls but slowly. This con-
tinues until a temperature about 4C is reached, when
the lower thermometer ceases to fall, remaining stationary
for some time. On the other hand, the upper one begins
to fall more rapidly, and continues doing so until it reaches
the freezing-point. This behaviour is explained by sup-
posing that water attains a point of maximum density at
about 4C, below which it expands instead of contracting.
At first therefore the particles of water contiguous to the
freezing mixture becoming denser descend, and are replaced
by warmer particles from beneath : and this process goes on
until the water below A is reduced to 4C ; while, on the

DILATATION OF LIQUIDS. 53

other hand, that above the freezing mixture is not so cold.
But as the action proceeds, the water contiguous to the
freezing mixture having already attained its point of maxi-
mum density, becomes specifically lighter instead of heavier,
and rising upwards rapidly cools the upper thermometer:
all this while the lower thermometer remains stationary at the
point which corresponds to the maximum density of water.
Many observers have made experiments with the view of
determining the temperature corresponding to the maximum
density of water, and by the mean of all their determinations
this appears to be as nearly as possible equal to 4C, or
3 9. 2 Fahr. Various watery solutions also possess their own
points of maximum density: but a very extensive series of
researches made by M. Pierre tends to shew that for other
liquids such points do not exist.

54. Having made these remarks on this peculiarity of
water and watery solutions, let us now exhibit a table
framed by M. Despretz, in which the volume and density
of water is given at the various temperatures from 9 to
iooC. It may appear anomalous that this table should
descend below the freezing point of water, but we shall
afterwards see (Art. 98) that this liquid, if kept perfectly
still, may be brought to a lower temperature than its usual
freezing-point without assuming a solid state.

These experiments were made according to the method
first described in this chapter, or the method by thermo-
meters (Art. 47).

54

DILATATION OF LIQUIDS.

Table of the density and volume of Water, from gC to loo ^
according to M. Despretz (the density and volume at 4 taken
as unity].

Tempe-
rature

Volume.

Density.

Tempe-
rature.

Volume.

Density.

-9

i-ooi 631 i

0-998371

15

i-ooo 875 i

0-999 125

-8

i-ooi 373 4

0-998628

16

i-ooi 02 1 5

0.998 979

-7

i -ooi 135 4

0-998 865

17

i-ooi 206 7

0-998 794

-6

1-000918 4

0-999 82

18

I.QOI 39

0-998 612

-5

i. ooo 698 7

0-999 302

19

i-ooi 58

0-998 422

-4

i. ooo 561 9

-999 437

20

i-ooi 79

0-998 213

-3

1-000422 2

0-999577

21

1-00200

0-998 004

2

1-0003077

0-999 692

22

I .QO2 22

0-997 784

I

i-ooo 213 8

0-999 786

23

1-00244

0-997 5 66

i-ooo 126 9

0-999873

24

1-002 71

0-997 297

I

1-000073 o

0999927

25

1-002 93

0-997078

2

1-000033 i

0-999966

26

1-003 21

0-996 800

3

i-ooo 008 3

0999999

27

1-003 45

0-996 562

4

i. ooo ooo o

I-OOOOOO

28

1-003 74

0-996 274

5

I -OOO OO8 2

0-999 999

29

1-00403

0-995 986

6

1-0000309

0-999 969

30

1-00433

0-995 688

7

1-0000708

0-999929

4

1-00773

0-992 329

8

I-OOO 121 6

0-999 878

50

I-OI2 05

0-988093

9

i-ooo 187 9

0-999 812

60

1-01698

-9 8 3 303

10

i-ooo 268 4

0-999731

70

1-022 55

0-977 947

ii

i-ooo 359 8

0-999 640

80

1-02885

0-971959

12

i-ooo 472 4

0-999527

9

1-03566

0-965 567

13

1-000586 2

0.999 4 J 4

100

1-043 15

0.958 634

H

I'ooo 714 6

0-999285

Matlhiesseris experiments. Matthiessen has recently made
a series of very valuable experiments on the expansion by
heat of water ; the method employed by him being the areo-
metric method, or that of weighing a solid in the liquid.

It was remarked in Art. 50 that in such a method the
cubical dilatation of the solid employed must be known at
the various temperatures of observation. It was also re-
marked that in order to determine accurately the cubical
dilatation of a solid it is hardly sufficient to determine the
linear dilatation of another specimen of the same material
and to multiply this by three, but the cubical dilatation

DILATATION OF LIQUIDS. 55

ought, if possible, to be obtained by direct experiment. This,
however, cannot be done unless we know the absolute dilata-
tion of some one fluid such as water or mercury.

The uncertainty introduced into a determination of the
expansion of water by the areometric method when we
deduce the cubical expansion of an auxiliary solid from its
linear expansion becomes, however, very small when the solid
is one which has a very small coefficient of expansion com-
pared with water. For this reason Matthiessen employed
pieces of glass cut from rods of which in the first place he
determined the linear expansion with great accuracy.

The method which he adopted for weighing accurately
these pieces of glass in water of different temperatures is
exhibited in Fig. 14. The piece of glass to be weighed
was -attached to the end of a fine platinum wire hung from a
balance-pan above. The hot water box below was made
of zinc double sided and encased in wood. The covers were
cut in two to allow them to be put on or taken off with-
out disturbing the fine wire. The stirrer was a square piece
of zinc soldered to the wires RR. A draft pipe leading into
a chimney where gas was burning served to draw off the
steam formed at high temperatures, thereby preventing its
condensation on the platinum wire. It was found by ex-
periment that this draft had not the slightest influence on
the weighings.

A silver cylinder filled with distilled water stood in the
middle of the box, and in it the glass was weighed. The
water in the box was heated by steam supplied by means
of a small boiler heated by gas at some distance from the
apparatus, and by carefully regulating the gas it was found
that any temperature between 50 and iooC might be kept
constant for some time.

The following table exhibits Matthiessen's results obtained
by this process as compared with those of Kopp, Despretz,

56"

DILATATION OF LIQUIDS.

and Pierre, all of whom determined the expansion of water
in glass vessels, and also with those of Hagen, who used

Fig. 14.

the same method as Matthiessen but without the same pre-
cautions.

DILATATION OF LIQUIDS.

57

Expansion of water according to different observers.

Kopp.

Despretz.

Pierre.

Hagen.

Matthiessen

4 C

I-OOOOOO

I -OOOOOO

I -000000

I- OOOOOO

I -OOOOOO

10

1-000247

i 000268

1-000271

1-000269

1-000271

J 5

1-000818

1-000875

1-000850

1-000849

1-000892

20

1-001690

1-001790

1-001717

1-001721

1.001814

30

1-004187

1-004330

1-004195

1-004250

1-004345

40

1-007654

1-007730

1-007636

1-007711

1-007730

50

1-011890

1-012050

1-011939

i 011994

1-011969

60

1-016715

1-016980

1.017243

1-017001

1-016964

70

1.022371

1-022550

1-023064

1-022675

1-022648

80

1-028707

1-028850

1-029486

1-028932

1-028953

90

I-035524

1-035660

1-036421

1-035715

1-035813

100

1-043114

1-043150

1-043777

1-042969

1-043159

It will be seen from this table that the determinations of
Matthiessen and Despretz agree very well together. Mat-
thiessen also redetermined the coefficient of expansion of
mercury by weighing it (enclosed in a bucket) in water of
different temperatures. By this means he determined its
mean coefficient of dilatation between o and iooC to be
.0001812, a number closely agreeing with Regnault's, which
was .0001815.

55. Dilatation of other liquids. The dilatations of
a great many liquids have been carefully determined by
M. I. Pierre, and he has embodied his results in expressions
of the following kind

3 /= + 0/4-3/2 + ^/3.

where 8 t represents the dilatation of unit volumes from o to
/C, and where a, t>, c are constants depending on the nature
of the substance. This expression, he finds, generally repre-
sents the expansion of a liquid with considerable accuracy.
We derive from his results the following table, in which the
approximate temperature of the boiling-point of each liquid
is compared with its coefficient of dilatation at oC.

DILATATION OF LIQUIDS.

Name of liquid.

Approximate
temp, of boiling
point in Centi-
grade degrees.

Coefficient of
dilatation for
IC at the tem-
perature oC.

Chloride of ethyle
Oxide of ethyle (sulphuric ether)
Bromide of ethyle
Iodide of methyle
Sulphuret of carbon .

II-0

35-5
40.7

43-8

A*7Q

.001575
001513
.001338

001200
OOII4O

Formiate of oxide of ethyle . .
Terchloride of silicon ....
Acetate of oxide of methyle . .
Bromine

52-9

59-o

59-5
63-0

001325
OOI29I
001296
OOIO38

Methylic alcohol .

66.3

OOIl86

Iodide of ethyle

*7O-O

.OOII42

Acetate of oxide of ethyle . . .
Alcohol

74.1

78.3

001258
OOIO4Q

Terchloride of phosphorus . .
Butyrate of oxide of methyle . .
Bichloride of tin .

78-3

102- 1

I I C('d.

OOII29
.OOI24O
OOII32

Amylic alcohol ....

ili'S

000890

Terchloride of arsenic ....
Bichloride of titanium ....
Terbromide of silicon ....
Terbromide of phosphorus . .
Mercury (JRegnault} .

133-8
136-0

153-4
175-3

3^OO

000979
.000943
000953
000847

ooo i 7Q

56. It will be gathered from this table that in general
liquids with high boiling-points expand less at the tempera-
ture oC than volatile liquids which boil at a low temperature,
and it may be inferred that there is some connexion be-
tween the coefficient of expansion of a liquid and its
volatility. This leads us to consider the dilatation of very
volatile liquids.

57. Dilatation of volatile liquids, When we come
to liquids, such as carbonic acid, which can only exist in
this state at ordinary temperatures under very great pressure,
we have reason to think that their coefficient of dilatation is
extremely great. Thilorier in 1835 had made the curious
remark that liquid carbonic acid presents the anomaly of
a liquid more dilateable than any gas, and he concluded

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