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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

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

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