The specific heat of liquids, of powders, and of sub-
stances soluble in water may be determined by sealing
them in thin glass or metal tubes and proceeding as before.
The slowness with which they may then acquire the tem-
perature of the water increases the correction for radiation
and reduces the accuracy.
In the case of solids of poor conductivity and soluble in
water, another liquid of known specific heat in which they
are insoluble may be used in the calorimeter.
There are three other methods of measuring specific heat.
The first is founded on the mass of ice which a known
mass of the substance will melt. The second depends on
the relative rate of cooling of equal masses of water and of
the substance. The third is based on determining the
amount of steam condensed in raising the temperature of
the body through any observed range of temperature. The
last method has lately been developed into one of great
OF THE QUANTITY OF HEAT. 47
scientific value and accuracy. The details will be found
in Preston's Theory of Heat, p. 236.
^3O. Variation of Specific Heat with Temperature
(S., 3O7 ; P., 258). The specific heat of a substance in
general increases with the temperature. This increase
becomes quite large in solids near the temperature of
fusion. The law governing the variation of specific heat
with temperature has not yet been discovered; but the
specific heat of any substance may be expressed by the
s = a + It + <-t*
in which a, b, c, etc., are constants determined by experi-
ment. Such a formula is used only to express the results
of a series of experiments, and cannot be regarded as con-
taining any law which holds beyond the range of the
The following table embodies the results of Dulong and
Petit's experiments :
SUBSTANCE. MEANT SPECIFIC HEAT.
Between and 100 C. Between 0" and 300 C.
Iron 0.1098 0.1218
Glass 0.177" 0.1990
Coppor 0.0949 0.1013
Zinc 0.0927 0.1015
Silver 0.0557 0.0611
Antimony 0.0507 0.0549
Platinum 0.0355 0.0355
Bismuth n.n308 ....
For higher temperatures platinum has since been found
to exhibit a variation, but it is less marked than with other
metals. For this reason, a piece of platinum may be used
to determine the temperature of a furnace. When it has
acquired the temperature of the furnace, it is quickly re-
moved and plunged into a known mass of ice-cold water.
By noting the rise of temperature of the water, it is easy to
calculate the approximate temperature of the platinum and
hence of the furnace. Such an instrument for measuring
high temperatures is called a pyrometer.
According to Hirn the thermal capacity of alcohol attains
the value 1.11389 at 160 C., a value superior even to that
of water at 100 C.
31. Specific Heat of Carbon (P., 26O). A few sub-
stances, notably carbon, exhibit large variations of specific
heat with temperature. Weber conducted a series of careful
experiments on the specific heat of diamond, and found the
following formula for the mean specific heat between
and 200 C. :
s = 0.0947 + 0.000497* - 0.00000012* 2 ... (a)
The total quantity of heat required to raise one gramme
of diamond from to * C. is then
q = 0.0947*4 0.000497* 2 - 0.00000012* n ...()
The mean specific heat between and * C. is obtained by
dividing q by *. If the specific heat at any definite tem-
perature is required, it is necessary to find the limiting
value of the mean specific heat as the range of temperature
is indefinitely diminished ; or
8 - d i,
w r here dq is the indefinitely small quantity of heat required
to raise the temperature of unit mass through the indefinitely
small range of temperature dt. Therefore !
1 The formula is obtained by finding from (b) the differential coefficient *?
MEASUREMENT OF THE QUANTITY OF HEAT. 49
8 = 0.0947 + 0.000994* - 0.00000036* 2 . . . <V)
At 200 C. the thermal capacity of diamond is therefore
nearly three times as great as at C.
\Veber showed further that the specific heat of carbon at
600 is about seven times as great as at 50 C. As the
temperature rises it approaches a maximum value of about
The following are the specific heats of carbon in its dif-
ferent states of aggregation :
Animal charcoal 0.2608
Wood charcoal 0.2415
Coke ! 0.2008
32. Specific Heat of Water (P. 262). Water has
the highest thermal capacity of any known substance except
hydrogen, unless it be a mixture of water and twenty per
cent of alcohol, which Dupre' and Page found to have a
thermal capacity five per cent higher than water.
The thermal capacity of water is nearly twice as great
as that of ice (0.504), and more than twice as great as that
of steam under constant pressure (0.477). Generally
speaking, the specific heat, of a substance when liquid is
higher than when solid.
The heat which will warm a gm. of water one degree
will warm 9 gms. of iron, or 18 gms. of silver, or 28 gms.
of platinum or gold, or 31 gms. of lead one degree.
The distribution of large quantities of heat in buildings
by means of hot water is made possible because of the
large thermal capacity of this agent. " The vast influence
which the ocean must exert as a moderator of climate here
suggests itself. The heat of summer is stored up in the
ocean, and slowly given out during winter. This is one
cause of the absence of extremes in an island climate."
Water exhibits a marked peculiarity in the variation of
its specific heat with temperature. The formula of Reg-
nault, which is still often quoted, indicates a gradual in-
crease of specific heat as the temperature rises from the
freezing to the boiling point. But the experiments of
Rowland, in his exhaustive investigation of the dynamical
equivalent of heat, were the first ones of sufficient accuracy
to show that the specific heat of water first decreases from
to about 30 C., and then a gradual increase begins.
Rowland's conclusion has been confirmed by Griffiths and
by Bartoli and Stracciati, who found a minimum value
for the specific heat of water at 20 C. The precise posi-
tion of this minimum is difficult of determination, since
the change in the specific heat near this point is very
, 33. Atomic Heat of Simple Bodies (P., 256 ; S., 313).
- In 1819 Dulong and Petit made experiments on simple
substances to determine whether their specific heats could
be connected by any simple law. From an examination of
the specific heats of such substances as iron, lead, gold,
silver, etc., these physicists concluded that the atoms of all
simple substances have the same thermal capacity. The
number of atoms of simple substances in the same mass
is inversely as the atomic weight. If therefore the thermal
capacity of the atom is the same, the specific heat must be
inversely proportional to the atomic weights, " or the prod-
uct of the specific heat by the atomic weight is the same
for all the elementary substances."
This law has been found to hold approximately true for
most of the elements which occur in the solid state at
OF THE QUANTITY OF HEAT. 51
ordinary temperatures, if the specific heats be taken at
temperatures sufficiently below the point of fusion. For
thirty-two of these substances the mean product is 6.38 and
the extremes are 6.76 and 5.7. The atomic weight of
hydrogen is the unit.
Since the specific heats of solids are not constant, but
vary with the temperature and the physical state, it is to
be expected that the product of the atomic weights and
the specific heats will exhibit a similar variation from
34. Specific Heat of Gases. - - The specific heat of
a gas may be measured in two different ways. It may
be measured under the condition of a constant pressure or
of a constant volume. The former is called the specific
heat under constant pressure and the latter the specific heat
at n*t<mt volume. The two are by no means the same.
In the latter all the heat applied goes to increase the mo-
lecular kinetic energy, while in the former the gas does
work in expanding by heat under a constant pressure ; and
heat must be supplied not only to increase the kinetic
energy of the molecules to the same extent as when the
volume is kept constant, but in addition enough to do the
external work. The specific heat under constant pressure
is therefore greater than the specific heat at constant
volume. The ratio of the one to the other for air is about
1 .41 . The importance of this ratio, to which reference has
already been made in Sound (I., 120), will be discussed in
a later chapter.
It has been found very difficult to measure the specific
heat of gases at a constant volume, and till quite recently
the difficulties have not been surmounted.
The specific heat of a gas under a constant pressure has
been determined by conducting the dry gas at a uniform
flow and constant pressure through two spirals. In the
first it is heated to a known temperature, and in the latter
it is cooled to the temperature of the bath. The heat given
up in the second spiral, or series of chambers, is determined
by measuring the rise of temperature of a known mass of
water, or by passing the gas through till the temperature
becomes stationary, when the heat gained from the gas
equals the heat lost by radiation. The mass of gas flowing
through is determined by measuring the change of pressure
taking place in the known constant volume of the gas-
holder. The experimental difficulties are largely due to
the small density of gases, so that a large volume must be
passed through the calorimeter to produce a measurable
change of temperature. This requires time, and the errors
due to conduction and radiation are greatly augmented.
The following are Regnault's conclusions respecting the
specific heat of gases under constant pressure :
1. The specific heat of all approximately perfect gases,
like air, does not vary with the temperature.
2. The thermal capacity of a given mass of such a gas
does not vary with its pressure ; and therefore the thermal
capacity of a given volume of such a gas is proportional to
3. The thermal capacity of equal volumes of the simple
gases which are not easily condensible are equal. This
equality does not hold for easily condensible gases.
4. The specific heat of easily condensible gases increases
with the temperature, like that of solids and liquids.
The specific heat of air is sensibly constant for all tem-
peratures between 30 and 225 C., and under pressures
from 1 to 10 atmospheres. The specific heat of carbon
dioxide is about doubled at 2.000 C.
MEASUREMENT OF THE QUANTITY OF HEAT. 53
The table is from Regnault's results.
SPECIFIC HEAT OF SIMPLE GASES.
Hydrogen 3.4090 Oxygen 0.2175
Nitrogen 0.2438 Chlorine 0.1210
Air 0.2374 Bromine 0.0555
SPECIFIC HEAT OF COMPOUND GASES.
Ammonia 0.5084 Carbon dioxide . . . 0.2169
Carbon monoxide . . 0.2450 Hydrochloric acid . . 0.1852
Hydrogen sulphide . . 0.2432 Sulphur dioxide . . . 0.1544
1. If 3 kilos, of iron (specific heat, 0.11) at 95 C. are put into
3 litres of water at 10 C., what will be the rise of temperature of
2. The specific heat of mercury is ^th. If 10 kilos, of mercury
be cooled from 100 to 25 C. in 1 kilo, of water, at what tempera-
ture was the water before the addition of the mercury ?
3. A.mass of 500 gms. of copper at 98 C. put into 500 gms. of
water at C., contained in a copper vessel weighing 150 gms.,
raises the temperature of the water to 8. 3 C. Find the specific heat
4. If 20 gms. of iron at 98 C. (specific heat, 0.11) are immersed
in 80 gms. of water at 10 C., contained in a copper vessel whose
mass is 15 gms., find the resulting temperature, the specific heat of
copper being 0.095.
5. 250 gms. of turpentine, enclosed in a copper vessel whose
mass is 25 gms., are heated to 100 C. and immersed in 589 gnis. of
water at 13 C. in a copper calorimeter weighing 110 gms. The tem-
perature rises to 27.5 C. Assuming the specific heat of copper to be
0.1, find that of turpentine (Glazebrook's Heat).
35. The Fusing Point. When heat is applied to a
crystalline solid, its temperature rises till it reaches the
point where it begins to pass into the liquid form. The
temperature then remains sensibly constant till the entire
mass has fused or melted, when with continued application
of heat it rises again. Conversely when the temperature
falls, a stationary point is again reached where the crystal-
lization or solidification sets in, and the body continues to
give up heat while the temperature remains fixed. Under
the same conditions of pressure the two stationary tem-
peratures coincide, and this point is called the normal
fusing point of the substance under the given conditions.
Above this temperature the substance will be in the liquid
state, and below it in the solid state.
This temperature is called the normal fusing point be-
cause under different conditions the fusing point may be
different. Thus ice melts normally at C., but under
pressure it melts at a lower temperature, and water may be
cooled .several degrees below zero before it freezes. Other
substances present similar abnormal features, and the liquid
state may persist at a temperature considerably below the
normal point of solidification.
The melting point of ice is sharply marked, and there
is no appreciable difference of temperature between the
melting ice and the water into which it passes. This is gen-
erally true of crystalline substances, but the case is very
different with amorphous solids, like wax, glass, and iron,
which cannot be said to have a definite melting point.
Such substances soften and become plastic before reaching
a more or less viscous liquid state. It is because of this
property that glass can be bent, moulded, drawn out into
rods and tubes, or blown into various forms. Similarly the
softening of wrought iron at a temperature far below the
liquefying point permits the metal to be rolled, forged,
and welded. In the fusion of wax the outer portions are
softer than the interior and presumably at a higher tem-
perature. The experiments of Person go to show that ice
begins to soften and to increase in specific heat between
2 and C., and that there is a certain very small range
of temperature within which ice softens and melts. The
difference between it and wax is then one of degree. Ice
represents one extreme of this transition state, and wrought
iron perhaps the other.
In general, however, crystalline bodies have a definite
fusing point, or a temperature at which they may exist
either as a solid or a liquid ; while amorphous bodies pass
gradually from the solid to the liquid state.
36. Condition of Instability (P., 27O). A liquid
which has a definite point of solidification, or whose pas-
sage from the solid to the liquid state is abrupt, may be
slowly and carefully cooled several degrees below the
normal freezing point without solidifying. This condition
is an unstable one, and if the under-cooled liquid be jarred,
or if a solid fragment of the same substance be dropped
into it, solidification will at once set in, with the disengage-
ment of heat. The temperature then rises to the normal
Fahrenheit observed that water sealed in a glass bulb
remained liquid at a temperature below freezing, but on
breaking off the stem rapid solidification followed. Gay-
Lussac found that water placed in a small vessel and
covered with oil remained liquid down to 12 C. Depretz
cooled water down to 20 C. in fine capillary tubes, and
Dufour obtained a like result by suspending small drops of
water in a liquid of the same density, with which it would
On the other hand, the surface of very still water freezes
sooner than one which is disturbed by the wind. A
running stream freezes less readily than a placid one.
There is no evidence, however, that the temperature of
running water is ever below C. The surface layers of
still water cool down to the freezing point by rapid radi-
ation, while the poor conductivity of water (64) prevents
the replenishing of the heat from below.
This property of under-cooling is not peculiar to water.
It has been observed also in the case of phosphorus. If an
over-saturated solution of sodium sulphate, prepared by
dissolving the salt in hot water, be placed in a clean flask,
it will remain liquid on cooling if undisturbed. But a
slight jar, or the introduction of a small crystal of the salt,
will start the solidification. When the unstable equilibrium
is disturbed crystallization proceeds rapidly with a rise of
temperature. The potential energy of the unstable liquid
mass is converted into heat.
37. Change of Volume during Fusion. In passing
from the liquid to the solid state bodies undergo a change of
volume. In most cases the volume diminishes. Ice, bismuth,
type metal, and cast iron are among the exceptions. Cast
iron and type metal expand on solidifying, and this expansion
causes them to fill every little line and crevice of the mould.
The powerful expansion of ice is attested by the bursting
of water-pipes and the rending of rocks by frost. If a
short piece of gas-pipe, with a screw cap fitted to each end?
be completely filled with water and placed in a freezing
mixture, it will burst with a loud report when the water
Major Williams at Quebec filled a 12-inch shell with
water and closed it with a wooden plug driven in with a
mallet. When the shell was exposed in the air at 28 C.
the stopper was
projected to a
distance of 300
feet, and a cylin-
der of ice about
8 inches in
I e n o- tli pro- A
truded from the
some of the
liquid till actu-
ally relieved of pressure by the giving way of the wooden
plug. The time required for the water to follow the
plug a distance of 8 inches was the interval from the
liquid to the solid state.
The change of volume of ice has been followed by Erman
from the solid to the liquid state by enclosing it in a large
bulb like a thermometer and taking readings on the long
stem. The continuous change in volume is represented in
Fig. 15, where AB represents the expansion of ice. At
II C. there is a rapid diminution in rolume, which con-
tinues after the whole mass is liquefied, but at a reduced
rate, up to 4 0., the temperature of the maximum density
of water. Beyond this point the liquid at first dilates
rapidly along DE, and then the uniform expansion of the
liquid sets in along the line EF. The slope of the line AB
is greater than that of EF, or ice expands by heat more
rapidly than water. It is probable from the later experi-
ments of Kopp that the change in volume at zero is much
more abrupt than that found by Erman.
In the case of phosphorus the dilatation in the solid state
is^ less rapid than in the liquid ; while for a fusible alloy,
consisting of one part of tin, one of lead, and two of bis-
muth, the coefficient of expansion of the solid is the same
as that of the liquid.
Numerous examples of substances investigated go to
show that there is generally an anomalous dilatation at
the fusing point, but that the curve connecting volume
and temperature is probably in all cases continuous.
38. Influence of Pressure on the Melting- Point (P.,
275; T., 119). That the melting point is affected by
pressure was deduced from theory by James Thomson in
1849. His conclusions were verified by his brother, Lord
Kelvin, in the same year. The theoretical conclusion was
as follows : Bodies which contract on melting have their
melting points lowered by pressure, while those which
expand have their melting points raised by the same means.
Such a result might have been anticipated from the simple
consideration that if a substance like water expands on
freezing, any pressure which prevents this expansion at the
same time prevents congelation. But if the substance con-
tracts on solidifying, then increase of pressure is favorable
to this change of state.
Thomson calculated that the freezing point of water
should be lowered by 0.0075 of a degree Centigrade for an
increase of pressure of one atmosphere ; and the experi-
ments of Dewar, later than those of Lord Kelvin, show a
mean reduction of 0.007*2 degree Centigrade per atmos. up
to 700 atmos. Mousson by enormous pressure lowered the
freezing point of water to 20 C.
A rough numerical statement of the result is that under
a pressure of one ton per square inch, or of 144,600 gms.
per square cm., ice melts one degree Centigrade below its
The converse conclusion has also been verified by Bun-
sen, who found that paraffin wax, which melted at 46.3 C.
under atmospheric pressure, melted at 49.9 C. when the
pressure was raised to 100 atmos.
Lord Kelvin has shown that the rigidity of the earth is
greater than if it were composed of glass. This conclusion
is derived from the phenomena of the tides, which show
that the earth does not yield appreciably to the forces
which raise them. The great rigidity, and therefore the
solidity, of the earth can be accounted for if it be assumed
that the materials composing the earth have their fusing
point raised by pressure. It has been ascertained that this
is true of ordinary l
39. Regelation. The phenomenon of the re-freezing
of water from ice melted by pressure, when the pressure is
relieved, is called revelation. It was first noticed by Fara-
day. Familiar illustrations of it are the hardening of a
snowball under the pressure of the hands, and the passage
of snow into compact ice in a roadway, where it is com-
pressed by vehicles and the hoofs of horses. Frozen foot-
forms may often be seen to persist in compact ice after the
loose snow has melted, and the bottom of a snowbank is
not infrequently compressed into clear ice.
60 HE A r.
Unless the pressure is very great, this solidification occurs
only when the snow is soft or near the melting point. The
pressure applied then reduces the freezing point and melts
those portions of the snow that are subjected to stress,
while the water again freezes when the pressure is removed.
If two pieces of ice at C. be firmly pressed together, they
will adhere by freezing after the pressure is relieved.
This may be done even under the surface of warm water.
If there is a small range of temperature within which lique-
faction takes place, as Person supposed, then the interior of
a lump of ice is at a slightly lower temperature than the
surface ; and when two such surfaces are pressed together,
even lightly, they are brought sufficiently near together to
give them the temperature of the interior of the block, and
as soon as the stress is removed the intervening film of
Bottomley's experiment on regelation is instructive.
Support a stout bar of ice horizontally by wooden supports
at the two ends, and hang on it a weight by means of a
copper wire passed over the ice at the middle. The press-
ure will melt the ice under the wire, and the water passing
around it and relieved of the stress will freeze. In this
way the wire will cut its way through the ice and the
weight will fall, but the bar of ice will remain intact,
though the track of the wire through it remains visible.
It is well to put a non-conducting link between the weight
and the wire to prevent the flow of heat upward from the
When a large body of ice melts in the spring, it will
sometimes be found to have a columnar structure consisting
of long slender prisms standing vertically. These can be
readily detached a foot in length without making more
than a small hole through the weak ice. It seems not
improbable that this peculiar structure has been caused by
lateral pressure and incipient melting.
Regelation has been invoked to explain the motion of a
glacier down its uneven, tortuous channel. A glacier
makes its way down its course by very irregular move-
ments. Ice is undoubtedly to some extent plastic, but it
is quite probable that regelation plays an important r61e in
glacial motion. The ice melts where it is subjected to the
pressure of enormous masses above it. This relief by press-
ure at many points permits the ice to accommodate itself
to changes in the channel, and a slow ice-flow is permitted.
As soon as the pressure is relieved at any surface the water
again freezes. The motion thus takes place by alternate