from his observations that its dilatation was four times
DILATATION OF LIQUIDS.
greater than that of air. M. Drion has since made very
careful experiments upon several volatile liquids, and among
them liquid sulphurous acid and chloride of ethyle.
these he has constructed the following table.
True coefficient of expansion for iC.
We may perhaps conclude that the coefficient of expansion
of a liquid is very great at those temperatures at which the
substance can only exist in the liquid state under very great
58. Contraction of liquids from their boiling-points.
Views perhaps analogous to those just mentioned in-
duced Gay Lussac to compare the contraction of different
liquids reckoned from their respective boiling-points, and
he obtained the following result for alcohol and sulphuret
Table of the contraction of alcohol and sulphuret of carbon for
successive intervals of 5C, reckoned from their boiling-points,
the volumes at these points being equal to 1000.
60 DILA TA TION OF LIQ UIDS.
It thus appears that the contractions of alcohol and
sulphuret of carbon, reckoned in this way, are as nearly as
possible the same.- Pierre and Kopp have both verified this
law of Gay Lussac, and the former has shewn
1. That amylic, ethylic, and methylic alcohol follow nearly
the same law of contraction, or, in other words, equal
volumes of these liquids at their respective boiling-points
will preserve their equality at all temperatures equidistant
from these points.
2. That the same law holds true for the bromides and
iodides of ethyle and methyle ;
3. And in one or two other cases : but that in general
two liquids formed by the combination of a common prin-
ciple with two different isomorphous elements follow different
laws of contraction starting from their respective boiling-
59. In conclusion, on comparing this chapter with the
preceding we are led to the following result :
1. Solids have a much smaller coefficient of expansion than
2. The coefficient of expansion of liquids increases with the
3. The coefficient of expansion of a liquid which only pre-
serves its state under very great pressure is probably very
DILATATION OF GASES. 6 1
Dilatation of Gases.
60. BEFORE commencing the subject of this chapter the
reader's attention must be drawn to the following experi-
ment and law.
Let there be a tube shaped as
in the accompanying figures,
of a uniform bore throughout,
and let it contain at the atmo-
spheric pressure (equal, let us
say, to 30 inches of mercury)
a volume of air AB. If this
air be shut out from the at-
mosphere by mercury, then
the surfaces B and D of the
mercury in the two limbs of
the tube will be at the same Fi S- T 5- Fig. 16.
level, since the pressure upon the surface at B is supposed
equal to that of the atmosphere pressing at D. Now if
additional mercury be poured into the tube until there is
a difference of 30 inches between the levels I? and C",
then it is evident that the air A' B' exists under the pres-
sure of two atmospheres ; for we have not only the column
of mercury C' If = 30 inches tending to press this air into
less volume, but we have in addition the pressure of the
atmosphere upon the surface C", which is also equal to 30
inches of mercury. Hence we have the air in Fig. 1 6 exist-
ing at a pressure double of that in Fig. 15. Now it is
an ascertained fact that the space occupied by the air in
Fig. 1 6 will be one- half of that occupied by it in Fig. 15 ;
and generally, provided the temperature remains the same, the
volume which a gas occupies is inversely proportional to the
62 DILATATION OF GASES.
pressure under which it exists ; or, in other words, its density
is proportional to its pressure. This law is known as that
of Boyle or Marriotte.
61. But if the temperature be increased, the air will on
this account alone tend to expand, and one of two things
will happen, i. If we wish to keep the air always occupying
the same space we must employ additional pressure ; 2 . If
we wish to keep it exerting the same pressure we must
allow it to occupy additional space. Our subject thus
naturally divides itself into two parts. In one of these
we determine the relation between the pressure and tem-
perature of a gas whose volume is constant, and in the
other we determine the relation between the volume and
temperature of a gas whose pressure is constant.
If we imagine Boyle's law to hold rigidly for gases, then
the two cases are connected with each other in a very
simple manner. For if a gas whose volume is V and pres-
sure P at oC has at f and under the same pressure the
volume V, then it is clear that were it constrained to
occupy its old volume its pressure would be P x -
Thus, whether we adhere to the method of constant
pressures or to that of constant volumes we should in this
case have precisely the same proportional change for increase
of temperature, this being in the one case change of volume,
and in the other change of pressure ; but it has been deemed
right to determine the change by both methods, and we
shall see in the sequel that the two values thus found are not
precisely the same.
62. The fact of the dilatation of gas may be easily proved
by filling a bladder nearly full of air, tying its orifice, and
heating it; it will then soon appear to be quite full, the
contained air having expanded by heat under the constant
pressure of the atmosphere.
DILATATION OF GASES. 63
Dalton in this country and Gay Lussac 1 in France were
the first who investigated the law of expansion of gases
with any considerable success, and they were both led to
the conclusion that all gases expand equally for equal incre-
ments of temperature. With regard, however, to the precise
law which connects together volume and temperature, there
was a difference in the result obtained by these two philo-
sophers. According to Gay Lussac, the augmentation of
volume which a gas receives when the temperature in-
creases i is a certain fixed proportion of its initial volume
at oC; while according to Dalton, a gas at any tem-
perature increases in volume for a rise of i by a constant
fraction of its volume at that temperature.
Gay Lussac's law may be expressed as follows.
Let F , V t denote the volumes at oC and / of a certain
quantity of gas existing at the pressure P, then these two
volumes are connected with one another by means of the
where a is the coefficient of expansion, nearly the same for
all gases, which it is the object of experiment to determine.
From this equation we derive at once the relation between
the temperature and the density (density being represented
by the mass contained in unit of volume) of air whose pres-
sure remains constant. For let D denote the mass of air that
occupies unit of volume at oC ; this mass will at / occupy
a volume equal to i -j- a /, and hence the mass of unit volume
or the density at this temperature will be
The dilatation of gases has since been investigated by
Rudberg, Dulong and Petit, Magnus, and Regnault, and the
result of their labours leaves little doubt that Gay Lussac's
method of expressing the law is much nearer the truth than
1 Charles appears to have been the first to discover that the coefficient
of dilatation is nearly the same in all permanent gases.
DILATATION OF GASES.
Dalton's. It has also been ascertained that the coefficient
of dilatation is not precisely, although very nearly, the same
for all gases. The experiments of Regnault were conducted
with very great care, and we shall now shortly describe, in
the first place, his method of ascertaining the increase of
pressure of a constant volume, of air between o c C and 100,
and, in the second place, his method of ascertaining the
increment of volume between the same limits of air of
which the pressure remains constant.
63. Relation between pressure and temperature of
air whose volume remains the same. The following
description of an apparatus, nearly the same as Regnault's,
and used by the author of this work for the same purpose,
will enable the
reader to under-
stand the me-
thod pursued in
A bulb b had
its volume at
oC and iooC
by mercury at
tures, as de-
scribed in Art.
39; and the con-
tained air was
dried by being
passed through a desiccating apparatus.
DILATATION OF GASES. 65
termination of the bulb was then attached to a tube T 7 ,
connected with another tube T' which was open to the
The lower terminations of these tubes were fitted into a
reservoir R containing mercury, and this reservoir might be
enlarged or contracted at pleasure by means of a screw S
which moved a piston out or in.
The whole apparatus was made to rest firmly on a slab of*
slate. The experiment consisted of two parts: the bulb b
was first of all surrounded by melting ice, and by means of
the screw S the mercury was forced to the height h in the tube
T 7 , and the difference of level between the surface of mercury
in the two tubes was read by means of a cathetometer, which
is an instrument for measuring vertical heights. Adding
this to the height of the barometer, which was observed at
the same moment, the whole pressure under which the air in
the bulb existed at the temperature of melting ice was thus
ascertained. Let us call this P. The bulb b was next attached
to a boiling- water apparatus, as in the figure, and by means
of the screw S the mercury was forced up to the same height
h in the tube T. Since the pressure of air increases with the
temperature, it is evident that the pressure will now be
greater, and that in consequence the mercury will be pushed
high up in the tube T'. Taking the difference of level as
before, noting the barometer, and adding the two heights
together, we get the whole pressure under which the air now
exists at the temperature of boiling water. Let this pressure
be called P'. It is now very easy to construct the formula
which must be applied.
Let the temperature of the surrounding atmosphere, as
also that of the mercury in the tubes and reservoir, be / C :
and let T denote the temperature of boiling water at the
present atmospheric pressure.
Further, let V denote the internal volume at oC of the
66 DILATATION OF GASES.
bulb b and of that portion of the capillary tube which is sub-
jected to the heating and cooling agents, and let v denote the
internal volume at oC of that portion of the tube T above
the mercury which is not subject to the influence of these
agents, but which contains air having the temperature /.
Also let K denote the coefficient of expansion for iC
of the glass, and let a denote the corresponding coefficient
-^of increase of pressure of dry air whose volume remains
constant this being what we wish to determine ; and, finally,
let us denote by D the mass of air which occupies unit
volume under unit of pressure at the temperature oC.
Then D P V (according to Boyle's law) will denote the
mass of that portion of the enclosed air existing in the bulb
(volume - = V) at the temperature oC and under the pressure
P when the bulb is surrounded by melting ice ; also (Art. 62)
will denote the mass of that portion of the
i + o/
enclosed air existing at the same time in the tube (volume
= v (i fK/)) at the temperature of the atmosphere ( = /) and
pressure P. Hence the whole mass of enclosed air will be
Now let the bulb be subjected to the temperature of
boiling water ( = T). The volume of the bulb then becomes
V (i+KT), and hence the mass of air existing in the bulb
at this temperature and under the pressure P' will be
- - ? while that existing in the tube (vol = v(i + */))
I ~T~ CL JL
at the temperature of the atmosphere ( = /) and pressure P
Hence the whole mass of enclosed air will be denoted by
(V(l + KT) V(l+Kt)l
*vi-r"; ^~ + , , , r ( 2 )
DILATATION OF GASES. 67
But since the mass of air remains unchanged, being enclosed,
we have (i) = (2) ; and hence, since D is a common factor,
j P(I+K/)J (V(l+KT) P(l+K/))
H i+a/ J = 1 i+aT 7 i+a/ )'
where everything is known but a, which may thus be easily
determined. By a similar method Regnault found that if
unity denote the pressure of a given volume of dry air at
oC, its pressure at ioo c C if confined to the same volume
will be 1.3665. The author of this work has obtained a
somewhat larger increase, but we may probably assume the
above number to represent the increase of pressure of air
of constant volume with great exactness.
64. Dilatation of air between 0C and 100C under
constant pressure. A slight alteration in the apparatus
of Fig. 1 7 enabled Regnault to make this experiment. Here
the air will of course expand and occupy part of the tube T,
and it will therefore be necessary to surround the two tubes
T, T f with water of a constant temperature, since nearly
a fourth part of the enclosed air will exist in T, and its tem-
perature must therefore be accurately known. By means of
the screw S the mercury in the two tubes is brought as
nearly as possible to the same level, both when the bulb
is in melting ice and when it is in the boiling apparatus,
so that in both these cases the pressure will be as nearly
as possible equal to that of the atmosphere. Any small
difference of level between the two tubes is read by means
of a cathetometer, and the barometer is noted; so that the
whole pressure under which the air exists in the two cases is
accurately known; but these pressures will in this experi-
ment be very nearly the same in both. By calibrating the
tube the additional volume occupied by the air at the high
temperature may be determined, and thus the coefficient of
expansion becomes known.
From these and other experiments Regnault has concluded
DILATATION OF GASES.
that while the dilatation of air between oC and iooC is
equal to .3665 of its volume at o when this dilatation is
calculated by means of the law of Boyle from the change
of pressure of air of which the volume is constant ; yet when
the dilatation is deduced directly from the change of volume
while the pressure remains constant this coefficient is some-
what increased and becomes .3670.
Dividing these results by 180, we find the coefficient which
denotes increase of pressure for i Fahr. of air whose volume
is constant = .002036.
Also, the coefficient which denotes increase of volume for
i Fahr. of air whose pressure is constant = .002039.
65. Dilatation of other gases at ordinary pressures.
Regnault has investigated this subject minutely, and has
found that different gases have notably different coefficients,
and that the coefficient of the same gas differs according as
it has been determined by the method of constant pres-
sure or by that of constant volume. He gives the following
Dilatation between oC and iooC.
Protoxide of nitrogen
o-37 J 9
It will be noticed in this list that sulphurous acid and
cyanogen, which have the greatest coefficients, are gases
which may easily be liquefied ; while, on the other hand, the
three permanent gases which have never been liquefied until
very recently have small coefficients.
DILATATION OF GASES. 69
66. Dilatation of gases existing under different
pressures. The following law has been deduced by
Regnault: "Air and all gases except hydrogen have coeffi-
cients of dilatation which increase to some extent with their
Regnault has likewise enunciated the following very im-
portant law : " The coefficients of dilatation of the different
gases approach more nearly to equality as their pressures become
feeble, in such a manner that the law which is expressed by
saying that all gases have the same coefficient of dilatation ought
strictly to be considered as a law which applies only to gas
in a state of extreme tenuity, but which is departed from as
gases become compressed, or> in other words, as their molecules
approach each other"
A gas whose molecules are so far apart as not to exert
any sensible influence upon each other may be called a
67. Air Thermometer. We are now in a position to
discuss the air thermometer, to which in our first chapter
we promised to return.
If we have a series of thermometers with different liquids,
such as mercury, alcohol, water, &c., and all enclosed in
envelopes of the same description, and if each instrument
has been accurately pointed off and graduated in such a
manner that iC denotes the one hundredth part of the
capacity of the capillary tube between o and 100, never-
theless these instruments, if plunged into the same liquid,
will not all register precisely the same temperature.
But if we restrict our choice to thermometers with the
same liquid, as, for instance, to mercurial thermometers
made of the same kind of glass, we obtain instruments
strictly comparable one with another, and these if plunged
into the same liquid will all indicate the same temperature.
But though such instruments are comparable with each
70 DILATATION OF GASES.
other, we are not yet sure if their common reading accu-
rately represents the true temperature; for there is no
obvious reason why we should prefer a mercurial to an
alcohol or ether thermometer, which would both give
slightly different indications. The mercurial thermometer
stands therefore in the following position : different instru-
ments of this kind may be made to give identical indications,
but yet we cannot rely upon these for accurately measuring
temperature; nevertheless we cannot suppose that they are
very far wrong.
Let us now employ a gas or air thermometer, and confine
in envelopes of the same material different gases of suffi-
cient tenuity and not liable to be easily condensed, and let
us suppose that we have the means of ascertaining accurately
the pressure which they exert upon their envelopes ; and
let us also in the meantime disregard the expansion of
these envelopes. Now if all these gases have a common
pressure at oC they will all have a common pressure at
iooC, or at any other temperature. If we make use of
this pressure to determine the temperature, we have thus
obtained different instruments whose indications are com-
parable with each other even although the gases with which
they are filled are different. Gases, if of sufficient tenuity, are
therefore in this respect superior to liquids ; and it only now
remains for us to determine the precise law which connects
together the temperature and pressure of a gas in order to
make a perfect thermometer. We have various reasons for
imagining the law announced in art. 62 to be correct, at
least in the case of perfect gases. This law, taken in con-
nexion with that of Boyle, asserts that if the pressure of
a constant volume of gas be unity at oC and 1.3665 at
iooC, then the pressure at 5oC will be the mean between
these two numbers, or 1.18325. Strictly speaking, it is
impossible to prove this law experimentally with precision,
DILATATION OF GASES. 71
for to do so implies the previous possession of an accurate
instrument for measuring temperature. Now we have seen
that the mercurial thermometer is not trustworthy, while
for the purpose of this proof it must be presumed that
the air thermometer does not yet exist, since in order to use
it we must have a knowledge of this very law which we wish
to prove. The case stands thus ; if we employ mercurial
thermometers (which we cannot imagine to be far wrong),
this law is found to give a near approximation to the
experimental result. There is, however, a small difference,
and the cause of this may be either that mercurial thermo-
meters are not quite right, or that the law itself is only
an approximate expression of the truth. Now, in the first
place, we know very well that mercurial thermometers are
not absolutely correct, and in the next place, with regard
to the law, its extreme simplicity is in its favour, and we
shall afterwards see that there are theoretical reasons for
supposing it to be correct, at least for perfect gases.
In fine, we apprehend that a perfect gas obeys this
law, and may be made to furnish us with a perfect ther-
mometer, and if we cannot procure a gas that is quite
perfect, yet atmospheric air deprived of moisture and car-
bonic acid is a substance sufficiently good for all practical
Although the air thermometer is in principle peculiarly
fitted for the determination of very high temperatures, yet
there are considerable mechanical difficulties in the employ-
ment of the instrument in such cases. Regnault has lately
invented two modifications of this instrument in order to
render it suitable to measure the temperature of furnaces
(see Annales de Chimie for September 1861). In one of
these vapour of mercury is the gas employed, and the
instrument is constructed as follows. There is a kind of
flask, either cylindrical or spherical, which may be either of
DILATATION OF GASES.
cast or wrought iron, of platinum or of porcelain : the mouth
is closed by a plate containing a small aperture. From 15
to 20 grammes of mercury are added to this flask, which
is then placed in that part of the furnace the temperature
of which we desire to know. The mercury soon boils,
its vapour expels the air by the orifice, and the excess
of mercurial vapour goes off by the same means. When
the apparatus has acquired the temperature of the furnace
the flask is withdrawn and made to cool rapidly, and the
mercury which remains in the flask is weighed. It may
be weighed directly, or, if it contains impurity, it is dissolved
in acid and estimated as a precipitate. This weight is
that of the vapour of mercury which filled the flask at the
temperature of the furnace, and the volume of the flask
as well as the density of mercurial vapour being known this
temperature may thus be determined.
In conclusion we append the results of a comparison made
by M. Regnault between an air thermometer and a mercurial
thermometer with an envelope of crown glass.
by air thermometer.
Temperature given by
with envelope of
APPLICATIONS OF THE LAWS OF DILATATION. 73
Applications of the Laws of Dilatation.
68. Since all the bodies around us are subject to con-
tinual change of volume owing to their varying temperature,
it is necessary to take account of this in very many opera-
tions and investigations whether of a scientific or strictly
practical nature. Let us, in the first place, proceed to
describe the influence which change of temperature exerts
on our standards of length, mass, density and time.
STANDARDS OF LENGTH.
69. Supposing that a yard is taken to denote a certain
absolute distance, and that the length of a bar is precisely
one yard at the temperature 62 Fahr., it is clear that, owing
to the expansion of the material of the bar, its length will be
greater than a yard for temperatures higher than 62, but
less than a yard for temperatures below 62.
If now we employ this bar as a standard by means of
which to measure the absolute distance between any two
points in terms of the yard as a unit of length, and if it