Physics for university students (Volume 2) online

traversed by the two components which enter the eye of
the observer.

If now the two rays have travelled exactly equal dis-
tances from the first incidence at 6 to the eye, they will
interfere, because a difference of phase of half a wave-
length has been impressed on them by the fact that one has
suffered internal and the other external reflection at the
mirror 6 (I., 214). Any difference of path of the two
portions of the incident ray reflected from 5 and 8 will
produce a difference of phase at /; and when this differ-
ence of path amounts to an even number of half wave-
lengths for the particular color employed interference will
result.

The form of interferential comparator shown in Fig. 9
was devised by Professor Morley for the determination of



30 HEAT.

the absolute coefficient of expansion of metals between the
freezing and the boiling points of water. The bars to be
compared are mounted as shown. Plate 14 is moved by a
weight which keeps it in contact with a cross-plate actuated
by a precision screw. By means of the interference phe-
nomena described, 5 and 8 and then 4 and 9 are made
equidistant from 6. The motion of the bar 8 and 9 in
passing from the first position to the second can be




Fig. 9.

measured by counting the interference bands during the
motion. A microscope and graduated scale shown at 10
are used to measure the length corresponding to any
observed number of wave-lengths of the monochromatic
light.

If now one of these bars be kept at a constant tempera-
ture and the other one be compared with it in the way de-
scribed, first at the freezing point and then at the boiling-
point, the expansion for 100 degrees C. will be measured in
terms of a particular wave-length of light as a unit of
length.

21. Dilatation of Liquids. Liquids in general are more
expansible than solids. In the case of liquids and gases



EXPANSION.



31



the only expansion to deal with is volume expansion. The
approximate formula V= F~ (1 + kt~) holds as in the case
of solids.

An instrument like a thermometer is well suited to
measure the apparent expansion of a liquid, or the excess
of its expansion over the volume expansion of the glass
envelope. If the absolute coefficient of expansion of the
glass is known, the absolute expansion of the liquid can be
deduced from the apparent expansion.

The absolute coefficient of ex-
pansion of mercury has been de-
termined by Regnault with great
accuracy by means of the principle,
that the heights of two liquids in
communicating tubes above their
common surface of separation are
inversely as their densities (I.,
80). The actual investigation
involved some modifications and
many minute details.

Two vertical iron tubes, ab and
a'b', about 150 cms. long, were con-
nected near their upper ends by a horizontal cross-tube aa 1
(Fig. 10). The cross-tube joining the lower ends b and b*
was interrupted at its middle, and two vertical glass
tubes were inserted and connected with each other and
with a reservoir filled with air, the pressure of which
could be varied at pleasure. When the two columns of
mercury filling the apparatus are at different temperatures,
the mercury will stand at different heights d and d' in the
glass tubes ; while their upper surfaces near a and a' will
be at levels to produce equilibrium at the upper horizontal
cross-tube by hydrostatic pressure. The tubes were all




Fig. 10.



32 HEAT.

enclosed in water jackets, and the two glass tubes were at
the same temperature.

The pressures at d and d' are the same, because the two
surfaces of mercury are in contact with air under pressure.
We may therefore place the pressures on the two sides at
d and d' equal to each other. Since the pressure of the
short column above a is equal to the one above a', bemuse
they are in equilibrium through the tube aa', we need to
consider only the long columns from a and a' respectively
down to the horizontal plane through bb'. Let If be this
common height, and let h and h' be the heights of d and d 1
respectively above the same level through bb'. Also let t
be the temperature of the mercury in ab, and t' the temper-
ature of the mercury in all the other tubes. Then

H h JT h'

I + kt I + kt' ~ 1 + ~M 1 + M '

The division of H by the expansion-factor reduces it to the
height at zero, and this multiplied by the density at zero
gives pressure. The same is true of the other terms. But
the density is a common factor and disappears.
From this equation

H _ff-(h'-h)
1 + kt T+to'

h-h'



Hence k=



HV-t^H+h-W)



It is not necessary to see the tops of the long columns,
since the parts above aa' are in equilibrium. H is deter-
mined from the apparatus itself, though a correction is
needed for a change in temperature. In addition the two
temperatures and the difference of level between d and d'
must be observed.



EXPANSION.

By means of this apparatus Regnault made measure-
ments which enabled him to draw up a table of the dilata-
tion of mercury for every 10 from to 350 C. (Appendix,
Table I.).

22. Dilatation of Water ^B., 293 ; S., 52; P., 176).
Water shows the anomalous property of contracting
when heated at the freezing point. This contraction con-
tinues up to 4 C. ; at this point expansion sets in, so that
the greatest density of water is at a temperature of 4, and
its density at 8 is nearly the same as at C.

This peculiar behavior of water is illus-
trated by Hope's apparatus (Fig. 11). It
consists of a glass jar with a tubulure near
the top and the bottom to admit thermom-
eters. About its middle is placed an annular
reservoir. If the vessel is filled with water
at about 10 C'., the upper thermometer will
show at first a slightly higher temperature
than the lower one. If now the trough at
the middle be filled with a freezing mixture,
the first effect will be the gradual fall of the ^-
lower thermometer to 4 C'. without much
change of the upper one. After the lower thermometer
becomes stationary, the upper one falls rapidly till its
temperature is reduced to zero and ice forms at the surface.
The water at 4 ('. sinks to the bottom, while that below
4 is lighter and rises to the top, where the freezing first
takes place. For this reason ice forms at the surface of a
body of cold water which freezes from the surface down-
ward, instead of from the bottom upward.

The relation between the volume and the temperature of
water near the freezing point may be determined by means




34



HEAT.



120



of a large thermometer filled with distilled water. If the
apparent volumes of the water in glass are plotted as ordi-
nates and the corresponding temperatures as abscissas, the
curve is approximately a parabola (abc, Fig. 12). The
vertex is somewhat above 4 C. This is then the tempera-
ture of the least apparent volume. But the observations

for this curve include the dilatation

of both the glass and the water.
The real volume-temperature curve
of water may be found by adding to
the ordinates of this one the expan-
sion of the glass. For this purpose,
if the glass is assumed to expand
uniformly for the small range of tem-
perature included within the obser-
vations, it is only necessary to draw
a line OD, making with the axis of
temperatures an angle whose tan-
gent, expressed in terms of the two
scales, is the dilatation of the glass
V for one degree. If the vertical ordi-
nates between OX and OD are
added to the corresponding ones of
abc, the result is the curve adf.
The point of least volume, or great-
est density, will correspond to the shortest ordinate between
OD and the curve abc. This may be found by drawing a
tangent to the curve parallel to OD. This tangent touches
the curve at 6, and this is the point of least volume. It
corresponds very closely to 4 C.

When the pressure is increased above one atmosphere,
the temperature of maximum density of water recedes
toward zero. Amagat found the mean rate of recession to




EXPANSION. 35

be about 0.025 degree C. per atmosphere. At 144.8 atmos-
pheres the temperature of greatest density was 0.6 C.

Table II. in the Appendix contains the volumes and
densities of water from to 100 C. deduced from Rosetti's

experiments.

23. Dilatation of Gases Law of Charles (P., 186;
S., 61; G., 1OO). The law first enunciated by Charles in
17 s " and confirmed later by Rudberg and Regnault is the
following : The volume of a given mass of any gas, under
constant pressure, increases from the freezing to the boiling
point by a constant fraction of its volume at zero. This is
therefore known as the law of Charles. For the Centi-
grade scale the constant fraction is 0.3665 for dry air.
This is equivalent to 0.003665 for one degree C. A near
approximation is ^^. Hence 30 c.c. at become about
41 c.c. at 100 C. "

It follows from this law that the formula of dilatation,
which has already served for solids and liquids, may be-
applied to a gas under constant pressure, or

v v {) (1 + let).

The investigations of Regnault and others have shown that
this law, like that of Boyle, is not absolutely exact, but is
a close approximation to the truth.

For a perfect gas obeying Boyle's law (I., 103), the
product pv of the pressure and volume, for a constant
temperature, is a constant. This product is then some
function of the temperature, or

JH> =/().

It is obvious from this expression that the changes pro-
duced by the application of heat to a gas may be investi-
gated by observing the changes of volume under constant



36



HEAT.



pressure, or the changes of pressure at constant volume.

These two methods have been found to give nearly, though

not absolutely, identical results.

The method of a constant volume is more readily applied

than the other to determine the laws relating to gases.

Regnault's apparatus consisted essentially of a large glass

bulb of some 600 to 800 c.c. capacity, connected with an

open mercury manometer (Fig. 13). At the point h was

a mark, and the mer-
cury was kept at this
height by enlarging
or contracting the size
of the reservoir at the
bottom by means of
the screw S, which
moved a piston out
or in, or by some
equivalent method.
The bulb b was first
placed in melting ice,
the mercury in T was
brought to the point




L \



T'



Fig. 13.



and the difference
between the levels of
the mercury in T and T was measured. By adding the
height of the barometer, the pressure on the gas in the bull)
was determined.

The bulb was then enveloped in steam and the operations
were repeated to determine the total pressure at 100 C.
Then, knowing the several temperatures and the volume of
the bulb at the different temperatures employed, as well
as that of the stem, it was possible to calculate the coeffi-
cient of increase of pressure. Regnault found that for dry



EXPANSION. 37

air an initial pressure of one atmosphere at C. became
1.3665 atmospheres at 100 C.

With slight modifications in the operations, Regnault
found the dilatation in volume under constant pressure.
Between and 100 C. the increase in volume was 0.3670.

The table exhibits the results with several gases.

COEFFICIENTS OF DILATATION AND PRESSURE BETWEEN
AND 100 C.

Gas. Constant pressure. Constant volume.

Hydrogen 0.003661 0.003667

Air 0.003670 0.003665

Nitrogen . 0.003668

Carbon monoxide 0.003669 0.003667

Carbon dioxide 0.003710 0.003668

Nitrous oxide 0.003719 0.003676

Sulphur dioxide 0.003903 0.003845

Cyanogen 0.003877 0.003829

The easily liquefiable gases at the bottom of the list
have a somewhat larger coefficient of dilatation than those
which are liquefied with great difficulty. Regnault con-
cluded from his elaborate investigations

(1) That all gases have not the same coefficient of
expansion, and that for the same gas there is a slight
difference between the coefficient under constant pressure
and that at constant volume.

(2) That the coefficient of all gases, except hydrogen,
increases with the initial pressure of the gas.

(3) That the coefficients of the gases investigated
approach equality as the pressure decreases.

These conclusions correspond with the fact that all gases
depart more or less from Boyle's law ; but as they are more
highly rarefied by reduced pressure, they approximate more
nearly to the ideal limit of exact obedience to this law.



tr ,
i



HEAT.

24. Volume of a Mass of Gas proportional to Abso-
lute Temperature. Under the condition of a constant
pressure, the law of expansion of a perfect gas is such that
increments of volume are proportional to increments of
temperature, or

tt n = A (v v,,),

where A is a constant. If now the temperature of the
least volume of the gas be taken as the zero of the scale
(16), and the temperature on this scale be denoted by T,
then is zero, and

T=A (w-t> ).

For an ideal gas following Boyle's law rigorously, the
volume would become zero at the zero of this absolute
scale, or

T=Av.

Hence, under a constant pressure, the volume of a given
mass of such a gas is proportional to the temperature on
the absolute scale.

The zero of this scale can be calculated from the
formula of Art. 23,

v = v n (1 + M) = v {} (1 + 0.0036650-

To find the value of t on the Centigrade scale at which the
volume v becomes zero, we have

= 1 + 0.003665*,
or t = - 273.

25. The Laws of Boyle and Charles combined.
The application of the law of Charles enables us to com-
bine both it and the law of Boyle into one expression, viz.,
that the product of the volume and pressure of any mass
of a gas is proportional to its absolute temperature. This
result may be reached in the following manner :

Let v , j0o, T^ be the volume, pressure, and absolute



EXPA XSION. 39

temperature of the gas under standard conditions, as, for
example, C. and 7(30 mms. pressure.

Also let v, ])i and T be the corresponding quantities at
temperature T.

Then, applying Boyle's law to increase the pressure to
the value p, the temperature remaining constant, we have

V H : v' : : p : p .

By changing the pressure from p to p the volume has
changed to v'.

Next apply the law of Charles, keeping the pressure con-
stant at the value /?, and starting with volume v'. Then

v' : v : : T lt : T.

It must l)e observed that these changes have taken place
by two independent, successive steps.

From the first proportion ?.; and from the second

v' p

1 = 2 . Multiplying the two equations together member
by member, we have

! = SL, a constant;
"



T,, " T

or the product pv is proportional to T, the temperature on
the absolute scale. We may therefore write



where R is a constant. We see from this expression that
in a perfect gas, following these two laws, both the press-
ure at constant volume and the volume under constant
pressure vary directly as the absolute temperature.




26. The Constant- Volume Air Thermometer. Pro-
fessor Jolly has devised a constant-volume air thermometer,
which is similar in principle to Regnault's apparatus for



40



HEAT.



the determination of the coefficient of dilatation of gases.
It is shown in Fig. 14. The capillary tube is bent twice
at right angles, and at B is joined to another tube of larger
diameter, on which a mark is made near the junction with
the capillary. OE is a glass tube of the same diameter as
BD, and the two are connected by a
piece of strong, flexible rubber tubing,
which permits CE to be raised or
lowered so as to keep the level of the
mercury at B. CE may be clamped
in any position by the screw clamp S.
The difference in level of the mercury
at B and E, added to the height of the
barometer, both corrected for temper-
ature, gives the pressure of the air
in the thermometer. The air in the
bulb must be very dry and free from
carbonic acid.

For ordinary measurements the dif-
ference of level of B and E may be
obtained with sufficient accuracy by
means of a scale engraved on a strip
of glass before it is silvered. This

o

scale is mounted on the frame sup-
porting the thermometer and tubes.
In reading, the observer avoids parallax by reading the
point on the scale touched by the line, joining the top of
llit; mercury column and its image in the mirror.
From the relation

pv = RT,

it is obvious that the pressures of a fixed volume of gas
are proportional to the corresponding absolute temperatures,
since R is a constant.




Fig. 14.



EXPANSION. 41

If, therefore, p be the pressure at C. and p the press-
ure at some higher temperature t 3 C., then since the abso-
lute zero is 273 degrees below zero C., we may write

273 : 273 + t . : p, : p.



Whence t = 273 ( P. _ 1 \ .

\^o /



If we employ Regnault's coefficient 0.003665, the abso-
lute zero is 272 C .85 C. instead of -273\

The pressure at zero must be determined by surrounding
the bulb of the thermometer with ice and taking readings.
Any other temperature is then measured by observing the
pressure necessary to keep the mercury at the fiducial
point near B.

PROBLEMS.

1. A glass flask holds 200 e.c. of water at C. How much will
it hold at 100 C. ? The coefficient of linear expansion for glass is
0.0000083.

2. The density of a piece of silver at is 10.5. Find its den-
sity at 100 C. if its coefficient of cubical expansion is 0.0000583.

3. The volume of a mass of copper at 50 C. is 500 c.c. ; find its
volume at 300 C. Coefficient of linear expansion, 0.0000565.

4. A brass pendulum keeps correct time at 15 C., but at 35 C.
it loses 16 seconds a day. Find the linear coefficient of expansion
of brass.

5. A solid displaces 500 c.c. when immersed in water at C. ;
but in water at 30 C. it displaces 503 c.c. ; find its mean coefficient
of cubical expansion.



42 HEAT.



CHAPTER IV.

MEASUREMENT OF THE QUANTITY OF HEAT.

27. Unit Quantity of Heat. - Heat as a physical
quantity is subject to measurement. For this purpose no
knowledge of the ultimate nature of heat is required, but
the methods of measurement are based on some established
property or effect attributed to heat. Twice as much heat
is required to raise the temperature of two grammes of
water one degree as of one gramme one degree. The
thermal element of such a comparison is limited to an
observation of temperatures. The measurement of heat is
called Calorimetry.

Heat, like other physical quantities, must be expressed
in terms of some unit. The unit quantity of heat is the
heat required to raise the temperature of unit mass of
water one degree. If the unit of mass is the gramme and
the unit of temperature the degree Centigrade, the unit of
heat is called the calorie.

The number of units required to raise the temperature
of m gms. of water 1 C. is then m calories ; and since the
heat necessary to effect the same increase of temperature
of 1 gm. of water at any part of the scale is nearly the
same, the heat which will warm 50 gms., for example, one
degree is almost the same as the heat required to raise
1 gm. 50 degrees. This is demonstrated by mixing equal
masses of water of different temperatures and observing



OF THE QUANTITY OF HEAT. 43

whether the temperature of the mixture is the mean of
the two contributing temperatures. It is found that the
quantity of heat given out by the warmer mass in cooling
through any range raises the cooler mass through the same
range.

Since the heat which will warm one gramme of water one
degree at different temperatures is not rigorously the same,
the definition of unit quantity is often as follows : The unit
quantity of heat is the heat required to raise the tempera-
ture of 1 gm. of water from 4 C. to 5 C. The same
quantity of heat is given out by 1 gm. of water in cooling
from 5C. to 4 C.

28. Thermal Capacity. The thermal capacity of a
body is the number of heat units required to raise its tem-
perature one degree. The thermal capacity of any body of
water is numerically equal to its mass in grammes, since
the thermal capacity of unit mass of water is the heat unit.
But the case is very different with other substances. If
equal masses of mercury at 80 C. and water at 20 C. be
mixed, the temperature of the whole will be only about
22 C. The heat which the mercury gives up in cooling 58
degrees will heat the water only about 2 degrees, or the
thermal capacity of water is about thirty times that of
mercury.

This difference in thermal capacities may be further shown
as follows : Take a number of metal balls of equal mass,
such as lead, tin, zinc, copper, and iron, and place them in
boiling water. By means of fine attached wires place them
all simultaneously on a flat cake of paraffin supported at
the edges, and observe the extent to which the paraffin is
melted by cachball. If the plate is not too thick the iron,
copper, and zinc balls may melt through, but they will not



44 HEAT.

go through in exactly the same time. The tin ball will
not sink into the wax so deeply, while the lead will melt
less than any of the others. The thermal capacity of the
lead ball is the smallest, while that of the iron one is the
greatest of the series.

The thermal capacity of a substance is the heat required
to raise the temperature of unit mass of it one degree.
When the unit of heat is denned as above, the thermal
capacity of unit mass is numerically equal to the specific
heat of a substance.

The specific heat of a substance is generally defined as
the ratio between the thermal capacities of equal masses of
the substance and of water. Since specific heat is a ratio,
it is independent of the unit of measurement employed.
The thermal capacity of a body is the product of its specific
heat and its mass.

Liquids exhibit differences of specific heat similar to those
of solids. If one kilo, of bisulphide of carbon at C. be
mixed with one kilo, of water at 60 C., the temperature
of the mixture will be about 48. 25 C. The number of calo-
ries lost by the water in cooling 11.75 degrees is 1,000 x
11.75 or 11,750 ; hence the thermal capacity of the kilo, of

carbon bisulphde is ^ or 240, and the thermal capacity
48.25

of 1 gm. of it is 0.240. This is therefore its specific
heat. If heat be applied at the same rate to equal masses
of water and carbon bisulphide, the temperature of the
latter will rise about four times as rapidly as that of the
former.

29. Specific Heat by the Method of Mixtures (P.,
221; G., 34). The last example illus4rates roughly
the method of determining specific heats by the method



MEASUREMENT OF THE QUANTITY OF HEAT. 45

of mixtures. It is desirable to describe the method some-
what more fully for the purpose of illustrating the thermal
principles involved.

Let A l and A., be two bodies of masses m l and m. 2 , tem-
peratures ti and ,, and specific heats 81 and 2 . If they
are placed in contact they will arrive at some intermediate
temperature t. The quantity of heat lost by A. 2 will be
ri-_*-2 (t. 2 f), and the quantity gained by A will be
mi8l (t ti). If we assume that the only interchange of
heat going on is between A l and A 2 , the heat lost by A 2 will
be equal to that gained by A\ , and consequently

w*a*2 (t-i = niiSi (t ti).

If .4., be a mass of water, its specific heat by definition is
unity, and therefore



This equation gives the mean specific heat between the
temperatures ^ and t. 2 obtained by means of the water
calorimeter.

It has been assumed that the thermal equilibrium be-
tween A l and A 3 is reached without loss of heat to other
bodies during the period of equalization of temperatures.
In practice there will be interchange of heat with other
bodies. There will be some loss by radiation, and the
heat given to the calorimeter and its fittings must be taken
into account. The thermal capacity of the calorimeter is
usually expressed in terms of the quantity of water which
the number of heat units expressing that capacity would
heat one degree. This is called its " water equivalent."
The gain of heat by the calorimeter and its fittings must
be added to that gained by the water.



46 HEAT.



Let the water equivalent be m. Then the heat acquired
by the calorimeter and its contents will be

w(*-*i) + w, (*-,),
and we have m 2 s 2 (t 2 ) = (m + m{) (t t^) ,



or



771. (t, _

The correction appears in the formula as an addition to
the water in the calorimeter.

To correct for radiation, Rumford arranged the experi-
ment so that the initial temperature of the water in the
calorimeter shall be as much below that of the surround-
ing air as the final temperature is above it. Then the heat
gained by absorption during the first part of the experiment
will be nearly equal to that lost by radiation during the