Online Library → Henry S. (Henry Smith) Carhart → Physics for university students (Volume 2) → online text (page 9 of 28)

Font size

the radiation are then equal to each other.

In illustration of the law that good reflectors are bad

radiators, if a pot of red-hot lead be examined in the dark

RADIATION AND ABSORPTION. 123

the dross will appear more luminous than the metal which

is cleared of it.

Also, if a piece of platinum foil, having on it a figure in

ink, be heated in a dark room by a Bunsen flame held

under it, the part blackened by the ink will appear

brighter than the rest if it be viewed from the tarnished

side ; but if it be viewed from the reverse side the figure

in ink will be seen as a darker portion than the adjacent

parts. Since the tarnished surface radiates more than the

bright surface, it is cooler and appears dark by contrast on

the reverse side. A striking experiment of Balfour Stew-

art to illustrate the same fact consists in heating to red-

ness a piece of stoneware of a black and white pattern.

When viewed in the dark the black part will shine much

more brightly than the white, presenting a curious re-

versal of the pattern.

Again, whatever substances may be put into a bright

coal fire, they will not alter the nature of the light given

out after they have attained the temperature of the fire.

A piece of red glass, for example, transmits red from the

hot coals and radiates the greenish light which it absorbs

when cold. Hence the light which it radiates exactly

makes up for what it absorbs.

A transparent piece of tourmaline cut parallel to the

axis absorbs nearly all the light polarized in a plane

parallel to the axis of the crystal. If the extension of

Prevost's theory is true, such a plate when heated red hot

should emit light polarized in the same plane as the light

which it absorbs. This conclusion has been shown to be

true in the following manner :

A hollow iron bomb, with a small hole extending

through opposite sides, is heated red hot in a fire, a plate

of tourmaline having previously been placed on a pedestal

124 HEAT.

within so as to be supported at the centre of the bomb.

After removal from the fire the apparatus is placed in the

dark. The light received by the eye, viewing the tourma-

line through the hole, then comes only from the tourmaline

itself, since no light enters the opposite hole and none is

transmitted from the iron. When examined by means of

a polariscope, this light is found to be polarized in a plane

at right angles to the light which the crystal transmits ;

or, in other words, the light emitted is polarized in the

same plane as the light absorbed.

84. Law of Cooling (S., 23O ; M., 246^ Newton's

law of cooling is that the rate of cooling of a heated body

is proportional to its excess of temperature over that of

the surrounding medium. This law holds only approx-

imately for small differences of temperature and fails

entirely when the excess is large.

The most elaborate investigations on this subject are

those of Dulong and Petit. They were conducted by the

use of a large thermometer within a spherical shell of

copper, blackened on the inside and exhausted of air.

The first conclusion reached was that, for a given excess

of temperature of the thermometer above that of the en-

closure, the rate of cooling in a vacuum increases in a

geometrical series when the temperature of the enclosure

increases in an arithmetical series, and the ratio of the

geometrical series is the same whatever be the excess of

temperature. Thus, if the excess of temperature be 200 C.,

the rate of cooling for the enclosure at was 7.40 ; at

20, 8.58 ; at 40, 10.01 ; at 60, 11.64 ; at 80 U , 13.45. The

average ratio of these successive numbers, and of others

found by the same experimenters, was 1.165, while the

temperature of the enclosure increased by equal steps of

20 C.

RADIATION AND ABSORPTION. 125

The formula of radiation obtained by Dulong and Petit,

which does not express the facts with great exactness, is

R = ma' + &,

where R is the quantity of heat radiated in unit time from

unit area of the surface at the temperature , m is a con-

stant depending on the substance and the nature of the

surface, a is a constant equal to 1.0077 for the Centigrade

scale, and k is a constant not yet determined.

From an examination of the data of Dulong and Petit,

Stefan concluded that the radiation emitted is proportional

to the fourth power of the absolute temperature, or

R = n (273 + 4 ,

where n is a constant and t is the temperature of the radi-

ating body. A similar expression holds for the rate of

cooling if the specific heat of mercury be assumed to be

constant. If t is the temperature of the enclosure and t'

the excess of temperature of the thermometer, then the

rate of cooling will be the difference between the radia-

tion of the thermometer and the counter radiation of the

walls of the enclosure, and we may write :

Rate of cooling = n (273 + t + ty n (273 + t)*.

This formula has been deduced theoretically by Boltzmann,

and is in better agreement with more recent experiments

than that of Dulong and Petit.

The rate of convective cooling in a gas was expressed

by Dulong and Petit as follows :

r=apV,

where a and b are constants for any given gas, p is the

pressure, and t the excess of temperature of the cooling

body over the gas. This rate is independent of the nature

and surface of the body, but varies with its form and

dimensions.

126 HEAT.

CHAPTER IX.

THERMODYNAMICS.

85. First Law of Thermodynamics. A short account

of the experiments of Rumford and Davy has already been

given in Chapter I. They go to show that heat implies

motion of the invisible particles of matter, and that heat

is the energy of this motion. The science of thermody-

namics is based on two fundamental laws relating to the

conversion of heat into work. The first law is the prin-

ciple of Conservation of Energy applied to heat. It

postulates the equivalence between heat and energy, and

may be expressed as follows :

When work is transformed into heat or heat into work,

the 'quantity of work is dynamically equivalent to the

quantity of heat.

It has also been expressed in this way :

"When equal quantities of mechanical effect are pro-

duced by any means whatever from purely thermal sources,

or are lost in purely thermal effects, equal quantities of

heat are put out of existence, or are generated " (Kelvin).

This law has been confirmed in a variety of ways :

1. The experiments of Joule, Rowland, and others in

generating heat by the expenditure of work.

2. The experiments of Him and others, showing that

when work is done by a heat-engine heat disappears. Hirn

made a fair calculation of the ratio between the two.

THEE MOD YNAMICS.

127

3. Investigations on the specific heat of air and other

gases under the two conditions of constant pressure and

constant volume permit of the calculation of the ratio

between the units of heat and of work. This calculation

was first made by Dr. Julius Mayer in 1842.

The limits of this book will restrict the discussion to the

first of these investigations.

86. Joule's Experiments (P., 575). - The investiga-

tions of Joule to determine the dynamical equivalent of

heat, or the ratio be'tween the

units of heat and of work, are

examples of the highest class

of experimental research. Rum-

ford made a rough\ calculation

of the mechanical work ex-

pended in heating a pound of

water one degree ; Joule in-

vestigated this relation by a

long series of varied and elab-

orate experiments which left

little for subsequent investiga-

tors, except the refinement of

details and an increase in the

scale on which the experiments

were conducted. The results

of all his experiments were fairly concordant, and a brief

description of the latest one of 1878 must suffice here.

The plan was to heat water by churning it with paddles,

and to find the ratio between the work expended in turning

the paddles and the number of heat units generated.

Hence both the work done and the heat generated had to

be measured.

Fig. 38.

128 HEAT.

The former was accomplished by an arrangement devised

by Him. The calorimeter h (Fig. 38), containing the

water, was supported on a hollow cylindrical vessel w,

which floated in water in v. It was thus free to turn

around a vertical axis, and the pressure was taken off the

bearings. The paddles within the calorimeter were carried

on a vertical axis b, about which the calorimeter could also

turn. A piece of box-wood was inserted in the axis at o

to prevent the conduction of heat downward from the

bearing c. There was a horizontal fly-wheel at f, and

the paddles were turned by the hand-wheels d and e.

To prevent the turning of the calorimeter by the friction

of the water, two thin silk strings were wound in a groove

around it, and, passing over two light pulleys, carried

weights &, k. These weights were adjusted till they

remained stationary, while the shaft and paddles revolved

at a suitable uniform speed, which was recorded by the

counter g. The weights then gave the torque necessary

to keep the calorimeter at rest, or the moment of the force

exerted by the paddles on the water. To measure the

work transmitted, it was then only necessary to multiply

this moment by the angular velocity of the shaft.

Let w be the mass of each weight, r the radius of the

groove in the calorimeter, and n the number of rotations

per second. Then since the work done is the same as if

the axle and paddles were at rest, and the calorimeter was

made to turn n times per second by the fall of the weights,

the energy expended can be readily calculated. In one

turn the weights would descend a distance 2-Trr. Hence

in n turns the work is

2-Trr x n x 2wg = farnrwg.

2-Tm is the angular velocity of the axle, and 2rwg is the

moment of the couple made by the two weights.

THERMODYNAMICS. 129

To measure the heat generated, let M be the mass of

water and m the water equivalent of the calorimeter and

paddles, and let t be the rise in temperature. Then the

heat generated is (M + m) t. The ratio of the work done

to the heat generated is

m)t

Corrections for ' radiation and other losses are required.

Joule's experiments proved that this ratio, which is the

work done to produce a unit of heat, is constant. It is

called Joule's equivalent, and is represented by the letter J.

The fundamental equation expressing this law is

W=Jff,

where W is the number of units of work and J5T the num-

ber of units of heat.

Joule's final value for J in gravitational units was

1390.59 ft.-lbs. or 423.85 kilogramme-metres. That is,

the heat which will raise a kilogramme of water 1 C.

will, if applied mechanically, lift 423.85 kilogrammes 1

metre high at sea-level. Of course the gramme can be

substituted in this expression without other change.

87. Rowland's Experiments (P., 583). In 1879

Rowland extended the work of Joule by a series of

exhaustive experiments which leave nothing to be desired.

His object was to reduce the temperatures to those of

the air thermometer, and to increase the rate at which

the work was done and the heat was generated.

Rowland's plan was the same in principle as Joule's, the

chief differences being that the paddles were turned from

below by power derived from a steam engine, and the

revolutions were recorded on a chronograph. On the

130 HEAT.

same chronograph were recorded the transits of the mer-

cury over the divisions of the thermometer. The rate at

which heat was generated in Rowland's apparatus was 50

times as great as in Joule's. Joule's rate of increase of

temperature was only 0.62 C. per hour, while Rowland's

was 35. The correction for radiation was thus reduced

in the inverse ratio of the rates.

For the sake of comparison, Rowland reduced Joule's

results to the air thermometer and the latitude of Bal-

timore, where his own experiments were conducted.

Combining the results, he deduced 426.75 from Joule's

experiments, and 427.52 gramme-metres from his own,

both at 14.6 C. His series of experiments at different

temperatures shows that the specific heat of water is a

minimum at about 30 C.

To reduce Rowland's result to C.G.S. units, the above

quantity must be changed to gramme-centimetres and then

multiplied by the value of g at Baltimore, which is 980.05.

Hence

J= 427.52 x 100 x 980.05 = 4.19 x 10 7 ergs,

or one calorie is equivalent to 4.19 x 10 7 ergs.

88. The Relation between J and R. The constant R

in the equation for a perfect gas, pv = RT, is numerically

equal to the dynamical equivalent of the difference be-

tween the two specific heats of a gas (34). The demon-

stration is as follows : If v be the volume of unit mass of

the gas at absolute temperature T 7 , then v/T is the increase

in volume, or the expansion, for one degree, &ndpv/Tis the

work done by the gas during the expansion under pressure

p (I., 44). The specific heat at constant volume S v is the

heat required to raise the temperature of unit mass one

degree when the volume is kept constant ; while the specific

THE R MOD YNAMICS. 131

heat under constant pressure S p is the heat required to

raise the temperature of the same mass one degree when

the pressure is kept constant. Since there is no internal

work, the latter will exceed the former by the thermal

equivalent of the work done in expanding under constant

pressure. Hence we may write

R may be evaluated if the density is known. Let d be

the density of the gas ; then since v is the volume of unit

mass, dv = 1, and R = p/Td.

For air d = 0.001293 when p = 76 cms. of mercury =

1033.3 gms. per square cm. 1033.3 x g dynes. Therefore

R _ 1033.3 xg _ 2 927

"0.001293x273"

For any other gas the value of R may be found by

dividing the value of R for air by the relative density of

the gas.

S p for air is 0.2374 (Art. 34) ; if the ratio between the

two specific heats be assumed to be 1.41, in accordance with

the best experimental results, then the above equation ex-

pressing the relation between J and R will give for J the

value of 42,420 gramme-centimetres, or 4.16 x 10 7 ergs.

89. Coefficient of Elasticity of a Gas (M., 1O6). -

Before proceeding to the second law of thermodynamics

it is desirable to introduce some topics subsidiary to it.

Since the working medium for the conversion of heat into

work is usually a gas or a vapor, a few propositions relat-

ing to them are necessary.

The coefficient of elasticity of a fluid is the ratio be-

tween any small increase of pressure and the resulting

132

HEAT.

voluminal compression. Let V be the initial volume and

v the diminution in volume due to an increment of press-

ure p. Then v/V is the compression per unit of volume.

The quotient of the increment of pressure by this com-

pression is the coefficient of elasticity of volume ; or, in

symbols,

v V

Since voluminal compression is only a ratio, the coefficient

of elasticity is a quantity of the same kind as a pressure.

Let volumes be rep-

resented by abscis-

sas and corresponding

pressures by ordinates

(Fig. 39). Then to

volume FP will cor-

respond pressure LP.

If now the pressure be

increased to MQ, the

volume will decrease

to G-Q. The coordi-

nates of the point P

represent the initial

and those of Q the

final condition of the

body with respect to volume and pressure, the temperature

remaining constant.

Join P and Q and produce the line to its intersection

E with the axis of pressures. Then will FE represent

the coefficient of elasticity. For

FE FP

THERMODYNAMICS. 133

VP V

But RQ is the increment of pressure, and _ _ _.

Hence

If therefore the relation between the volume and pressure

of a gas under the condition of a constant temperature be

represented by a curve traced by the point P, then the

coefficient of elasticity for any point P may be found by

drawing PE tangent to the curve at P and a horizontal

line P F ; the portion FE of the axis of pressures included

between PE and PF will represent the coefficient of elas-

ticity on the same scale as the pressures.

If the temperature is not constant, but is increased by

the compression, the effect will be to increase the increment

of pressure for any given decrement of volume. Hence

the corresponding coefficient of elasticity will be increased.

It is therefore evident that a gaseous substance has two

coefficients, one corresponding to constant temperature and

the other to the case where no heat is allowed to escape or

to enter during compression or expansion. The first is ap-

plicable to long continued stresses ; the second to rapidly

changing or alternating forces, as in the vibrations consti-

tuting sound, in which there is insufficient time for the

equalization of temperature by conduction and radiation.

The ratio of these two elasticities is the same as that of the

two specific heats.

90. Isothermal Lines (M., 108; S., 438). If the

ordinates of the curve traced by P represent pressures and

the abscissas volumes of a gas at constant temperature,

then the curve expresses the relation between p and v and

134

HEAT.

is called an isothermal line (Fig. 40). If the temperature

be increased to jp+l and be kept at this value, another

isothermal line will be obtained lying wholly above the

one for T. In this way any number of isothermal lines

may be drawn corresponding to regular intervals of tem-

perature. From such a diagram it is evident that, when

two out of the

three quantities

*jt>, v, T, are given,

the third may

be found graphi-

cally.

If the sub-

stance follows

Boyle's law,

then for a con-

stant tempera-

ture pv is a con-

stant, and this

product is rep-

resented in the

figure by the

area OFPL. If

this area is con-

stant the curve is known as a rectangular hyperbola.

The isothermal line corresponding to any temperature is

therefore a rectangular hyperbola.

It is a property of this hyperbola that if a tangent to the

curve be drawn through any point P till it meets Op in E,

then OF equals FE. But FE equals the coefficient of

elasticity of the gas and OF is the pressure. Hence the

coefficient of a perfect gas obeying Boyle's law is numeri-

cally equal to the pressure. This result was reached in

another way in the theory of sound (I., 118).

Fig. 40.

THERMODYNAMICS. 135

91. Adiabatic Lines. It remains to consider the

properties of a gas under the condition that no heat enters

or leaves it during the expansion or compression. If the

point traces a line expressing the relation between volume

and pressure in this case, it is called an adiabatic line.

When adiabatic lines cross isothermal lines, they are always

inclined to the horizontal at a greater angle than the

isothermal lines, because as the gas expands the pressure

diminishes more rapidly than for an isothermal line, since

the temperature is reduced by the work done in expanding

under pressure.

The equation to an adiabatic line is

pvy = a constant. 1

1 Let dQ be the quantity of heat required to raise unit mass of a perfect gas

through the temperature difference dT under constant pressure/?. This heat is

all expended in changing the temperature and doing external work. The quantity

required for the former purpose is S v dT. If the volume increases by a quantity

dv under pressure p, the work done is pdv, and the heat required is (pdv)/J.

Hence the whole heat necessary to effect the transformation is

When a gas expands adiabatically no heat enters or leaves it. and dQ =0.

Therefore

.

J

Differentiating the equation pc = RT, we have

Substituting in the last equation the value of dT obtained from this one, and

replacing R by its value J (S p ) from Art. 88, we have

Sjpdv + Sjdp^Q.

If y denotes the ratio S /$, > then

y dv + dp =Q

v p

I ntegrating, y log v + log p = constan t,

or, pv y = constant.

136

HEAT.

92. Carnot's Cycle (M., 138). If a volume of gas v l

at pressure pi and temperature T is allowed to expand

isothermally to the condition w/ and p/ represented by the

point B (Fig. 41), then work has been done against

external forces equal to the area ABv l / v 1 (I., 44). If now

the gas expands adiabatically from condition B at tempera-

ture TI to condition C at temperature T 2 , then the gas

does work represented by the area BCv./v/.

Suppose now the

gas to be compressed

isothermally along

the line CD. Then

the work is done on

the gas with loss of

heat, or is negative,

and it is represented

by the area CDv.,v./.

Lastly let the gas be

compressed adiabati-

cally from condition

- D to condition A.

Then the work done

on the gas raises its

Fig 4I

temperature from T 2 to T and equals the area

The algebraic sum of the several parts of the work is

then the area ABCD, enclosed between the two isother-

mals and the two adiabatics.

The working substance has returned to its initial vol-

ume, pressure, and temperature, and has gone through

an operation called a cycle. It is known as Carnot's Cycle.

The advantage gained by supposing the working substance

carried through a complete cycle of operations is that there

is then no balance of work done by or against internal

THERMOD YNAMICS.

137

forces, as there might be if the substance were not left in

its initial state.

If If i is the quantity of heat supplied at the higher

temperature TI , and H 2 the heat lost to surrounding bodies

at the lower temperature 21, then

Heat utilized H, H T } T* w

Heat supplied

= -7=. = efficiency.

Fig. 42.

93. Carnot's Engine. Carnot's engine is an ideal

one designed to embody the series of operations described

in the last ar-

ticle. Suppose

D, the working

substance (Fig.

42), to be con-

tained in a

cylinder imper-

vious to heat

except through

its bottom,

which is a s-

sumed to be a perfect conductor. Let A and B be two

stands, the temperatures of which are maintained at the

values 2\ and T 2 respectively. C is another stand the

top of which is supposed to be perfectly non-conducting.

Suppose the working substance D at the temperature of

the hot stand T^ and that its volume and pressure are

represented by Vi and _p,, the coordinates of the point A

on the isothermal line AB in the diagram of the last

article. Then we shall have the following operations :

First Operation. Place the cylinder containing the

working substance D on A and allow the piston to rise.

Heat flows in through the bottom of the cylinder to keep

138 HEAT.

the temperature of the working substance at the point jPj,

and the substance expands along the isothermal line AB

to the point B. During this operation the substance is

doing work by its pressure against the piston. It is

positive and is denoted by the area ABvJvi . During this

operation a quantity of heat HI has passed from A into

the substance.

Second Operation. The cylinder is now transferred to

the non-conducting stand O and the substance is allowed

to 'expand adiabatically, thus losing heat till its tempera-

ture falls from TI to T. 2 . Its expansion is represented by

the adiabatic line BO. The work done by the substance

during this process is equal to the area BOv 2 'vi.

Third Operation. The cylinder is next placed on the cold

body B, and the piston is pressed down till the volume

and pressure are represented by the coordinates of D.

Heat passes out through the bottom of the cylinder, the

substance remaining at the temperature T z . Its compres-

sion is represented by the isothermal line <7D, and the

work done on it equals the area ODv 2 v./; this work is nega-

tive. During this operation a quantity of heat H has

flowed from the working substance into the cold body B.

Fourth Operation. Finally place the cylinder on and

force the piston down. The temperature rises and the

relation of the volume and the pressure will be represented

by the adiabatic line DA. Continue the operation till the

temperature has risen to that of the hot body T v . Then

work equal to the area DAv^v- 2 is done on the substance,

and is negative.

The substance has thus passed through a series of opera-

tions by which it has finally been brought back in all

respects to its initial state. When the piston is rising the

substance is doing work ; this is the case in the first and

THERMODYNAMICS. 139

second operations. When the piston is sinking it is per-

forming work on the substance ; this is the case in the third

and fourth operations. The useful work done by the sub-

stance is the difference between the positive and negative

work, and is represented by the area ABCD.

The physical results at the end of the cycle are the

following :

(1) A quantity of heat jffi taken from A at the temper-

In illustration of the law that good reflectors are bad

radiators, if a pot of red-hot lead be examined in the dark

RADIATION AND ABSORPTION. 123

the dross will appear more luminous than the metal which

is cleared of it.

Also, if a piece of platinum foil, having on it a figure in

ink, be heated in a dark room by a Bunsen flame held

under it, the part blackened by the ink will appear

brighter than the rest if it be viewed from the tarnished

side ; but if it be viewed from the reverse side the figure

in ink will be seen as a darker portion than the adjacent

parts. Since the tarnished surface radiates more than the

bright surface, it is cooler and appears dark by contrast on

the reverse side. A striking experiment of Balfour Stew-

art to illustrate the same fact consists in heating to red-

ness a piece of stoneware of a black and white pattern.

When viewed in the dark the black part will shine much

more brightly than the white, presenting a curious re-

versal of the pattern.

Again, whatever substances may be put into a bright

coal fire, they will not alter the nature of the light given

out after they have attained the temperature of the fire.

A piece of red glass, for example, transmits red from the

hot coals and radiates the greenish light which it absorbs

when cold. Hence the light which it radiates exactly

makes up for what it absorbs.

A transparent piece of tourmaline cut parallel to the

axis absorbs nearly all the light polarized in a plane

parallel to the axis of the crystal. If the extension of

Prevost's theory is true, such a plate when heated red hot

should emit light polarized in the same plane as the light

which it absorbs. This conclusion has been shown to be

true in the following manner :

A hollow iron bomb, with a small hole extending

through opposite sides, is heated red hot in a fire, a plate

of tourmaline having previously been placed on a pedestal

124 HEAT.

within so as to be supported at the centre of the bomb.

After removal from the fire the apparatus is placed in the

dark. The light received by the eye, viewing the tourma-

line through the hole, then comes only from the tourmaline

itself, since no light enters the opposite hole and none is

transmitted from the iron. When examined by means of

a polariscope, this light is found to be polarized in a plane

at right angles to the light which the crystal transmits ;

or, in other words, the light emitted is polarized in the

same plane as the light absorbed.

84. Law of Cooling (S., 23O ; M., 246^ Newton's

law of cooling is that the rate of cooling of a heated body

is proportional to its excess of temperature over that of

the surrounding medium. This law holds only approx-

imately for small differences of temperature and fails

entirely when the excess is large.

The most elaborate investigations on this subject are

those of Dulong and Petit. They were conducted by the

use of a large thermometer within a spherical shell of

copper, blackened on the inside and exhausted of air.

The first conclusion reached was that, for a given excess

of temperature of the thermometer above that of the en-

closure, the rate of cooling in a vacuum increases in a

geometrical series when the temperature of the enclosure

increases in an arithmetical series, and the ratio of the

geometrical series is the same whatever be the excess of

temperature. Thus, if the excess of temperature be 200 C.,

the rate of cooling for the enclosure at was 7.40 ; at

20, 8.58 ; at 40, 10.01 ; at 60, 11.64 ; at 80 U , 13.45. The

average ratio of these successive numbers, and of others

found by the same experimenters, was 1.165, while the

temperature of the enclosure increased by equal steps of

20 C.

RADIATION AND ABSORPTION. 125

The formula of radiation obtained by Dulong and Petit,

which does not express the facts with great exactness, is

R = ma' + &,

where R is the quantity of heat radiated in unit time from

unit area of the surface at the temperature , m is a con-

stant depending on the substance and the nature of the

surface, a is a constant equal to 1.0077 for the Centigrade

scale, and k is a constant not yet determined.

From an examination of the data of Dulong and Petit,

Stefan concluded that the radiation emitted is proportional

to the fourth power of the absolute temperature, or

R = n (273 + 4 ,

where n is a constant and t is the temperature of the radi-

ating body. A similar expression holds for the rate of

cooling if the specific heat of mercury be assumed to be

constant. If t is the temperature of the enclosure and t'

the excess of temperature of the thermometer, then the

rate of cooling will be the difference between the radia-

tion of the thermometer and the counter radiation of the

walls of the enclosure, and we may write :

Rate of cooling = n (273 + t + ty n (273 + t)*.

This formula has been deduced theoretically by Boltzmann,

and is in better agreement with more recent experiments

than that of Dulong and Petit.

The rate of convective cooling in a gas was expressed

by Dulong and Petit as follows :

r=apV,

where a and b are constants for any given gas, p is the

pressure, and t the excess of temperature of the cooling

body over the gas. This rate is independent of the nature

and surface of the body, but varies with its form and

dimensions.

126 HEAT.

CHAPTER IX.

THERMODYNAMICS.

85. First Law of Thermodynamics. A short account

of the experiments of Rumford and Davy has already been

given in Chapter I. They go to show that heat implies

motion of the invisible particles of matter, and that heat

is the energy of this motion. The science of thermody-

namics is based on two fundamental laws relating to the

conversion of heat into work. The first law is the prin-

ciple of Conservation of Energy applied to heat. It

postulates the equivalence between heat and energy, and

may be expressed as follows :

When work is transformed into heat or heat into work,

the 'quantity of work is dynamically equivalent to the

quantity of heat.

It has also been expressed in this way :

"When equal quantities of mechanical effect are pro-

duced by any means whatever from purely thermal sources,

or are lost in purely thermal effects, equal quantities of

heat are put out of existence, or are generated " (Kelvin).

This law has been confirmed in a variety of ways :

1. The experiments of Joule, Rowland, and others in

generating heat by the expenditure of work.

2. The experiments of Him and others, showing that

when work is done by a heat-engine heat disappears. Hirn

made a fair calculation of the ratio between the two.

THEE MOD YNAMICS.

127

3. Investigations on the specific heat of air and other

gases under the two conditions of constant pressure and

constant volume permit of the calculation of the ratio

between the units of heat and of work. This calculation

was first made by Dr. Julius Mayer in 1842.

The limits of this book will restrict the discussion to the

first of these investigations.

86. Joule's Experiments (P., 575). - The investiga-

tions of Joule to determine the dynamical equivalent of

heat, or the ratio be'tween the

units of heat and of work, are

examples of the highest class

of experimental research. Rum-

ford made a rough\ calculation

of the mechanical work ex-

pended in heating a pound of

water one degree ; Joule in-

vestigated this relation by a

long series of varied and elab-

orate experiments which left

little for subsequent investiga-

tors, except the refinement of

details and an increase in the

scale on which the experiments

were conducted. The results

of all his experiments were fairly concordant, and a brief

description of the latest one of 1878 must suffice here.

The plan was to heat water by churning it with paddles,

and to find the ratio between the work expended in turning

the paddles and the number of heat units generated.

Hence both the work done and the heat generated had to

be measured.

Fig. 38.

128 HEAT.

The former was accomplished by an arrangement devised

by Him. The calorimeter h (Fig. 38), containing the

water, was supported on a hollow cylindrical vessel w,

which floated in water in v. It was thus free to turn

around a vertical axis, and the pressure was taken off the

bearings. The paddles within the calorimeter were carried

on a vertical axis b, about which the calorimeter could also

turn. A piece of box-wood was inserted in the axis at o

to prevent the conduction of heat downward from the

bearing c. There was a horizontal fly-wheel at f, and

the paddles were turned by the hand-wheels d and e.

To prevent the turning of the calorimeter by the friction

of the water, two thin silk strings were wound in a groove

around it, and, passing over two light pulleys, carried

weights &, k. These weights were adjusted till they

remained stationary, while the shaft and paddles revolved

at a suitable uniform speed, which was recorded by the

counter g. The weights then gave the torque necessary

to keep the calorimeter at rest, or the moment of the force

exerted by the paddles on the water. To measure the

work transmitted, it was then only necessary to multiply

this moment by the angular velocity of the shaft.

Let w be the mass of each weight, r the radius of the

groove in the calorimeter, and n the number of rotations

per second. Then since the work done is the same as if

the axle and paddles were at rest, and the calorimeter was

made to turn n times per second by the fall of the weights,

the energy expended can be readily calculated. In one

turn the weights would descend a distance 2-Trr. Hence

in n turns the work is

2-Trr x n x 2wg = farnrwg.

2-Tm is the angular velocity of the axle, and 2rwg is the

moment of the couple made by the two weights.

THERMODYNAMICS. 129

To measure the heat generated, let M be the mass of

water and m the water equivalent of the calorimeter and

paddles, and let t be the rise in temperature. Then the

heat generated is (M + m) t. The ratio of the work done

to the heat generated is

m)t

Corrections for ' radiation and other losses are required.

Joule's experiments proved that this ratio, which is the

work done to produce a unit of heat, is constant. It is

called Joule's equivalent, and is represented by the letter J.

The fundamental equation expressing this law is

W=Jff,

where W is the number of units of work and J5T the num-

ber of units of heat.

Joule's final value for J in gravitational units was

1390.59 ft.-lbs. or 423.85 kilogramme-metres. That is,

the heat which will raise a kilogramme of water 1 C.

will, if applied mechanically, lift 423.85 kilogrammes 1

metre high at sea-level. Of course the gramme can be

substituted in this expression without other change.

87. Rowland's Experiments (P., 583). In 1879

Rowland extended the work of Joule by a series of

exhaustive experiments which leave nothing to be desired.

His object was to reduce the temperatures to those of

the air thermometer, and to increase the rate at which

the work was done and the heat was generated.

Rowland's plan was the same in principle as Joule's, the

chief differences being that the paddles were turned from

below by power derived from a steam engine, and the

revolutions were recorded on a chronograph. On the

130 HEAT.

same chronograph were recorded the transits of the mer-

cury over the divisions of the thermometer. The rate at

which heat was generated in Rowland's apparatus was 50

times as great as in Joule's. Joule's rate of increase of

temperature was only 0.62 C. per hour, while Rowland's

was 35. The correction for radiation was thus reduced

in the inverse ratio of the rates.

For the sake of comparison, Rowland reduced Joule's

results to the air thermometer and the latitude of Bal-

timore, where his own experiments were conducted.

Combining the results, he deduced 426.75 from Joule's

experiments, and 427.52 gramme-metres from his own,

both at 14.6 C. His series of experiments at different

temperatures shows that the specific heat of water is a

minimum at about 30 C.

To reduce Rowland's result to C.G.S. units, the above

quantity must be changed to gramme-centimetres and then

multiplied by the value of g at Baltimore, which is 980.05.

Hence

J= 427.52 x 100 x 980.05 = 4.19 x 10 7 ergs,

or one calorie is equivalent to 4.19 x 10 7 ergs.

88. The Relation between J and R. The constant R

in the equation for a perfect gas, pv = RT, is numerically

equal to the dynamical equivalent of the difference be-

tween the two specific heats of a gas (34). The demon-

stration is as follows : If v be the volume of unit mass of

the gas at absolute temperature T 7 , then v/T is the increase

in volume, or the expansion, for one degree, &ndpv/Tis the

work done by the gas during the expansion under pressure

p (I., 44). The specific heat at constant volume S v is the

heat required to raise the temperature of unit mass one

degree when the volume is kept constant ; while the specific

THE R MOD YNAMICS. 131

heat under constant pressure S p is the heat required to

raise the temperature of the same mass one degree when

the pressure is kept constant. Since there is no internal

work, the latter will exceed the former by the thermal

equivalent of the work done in expanding under constant

pressure. Hence we may write

R may be evaluated if the density is known. Let d be

the density of the gas ; then since v is the volume of unit

mass, dv = 1, and R = p/Td.

For air d = 0.001293 when p = 76 cms. of mercury =

1033.3 gms. per square cm. 1033.3 x g dynes. Therefore

R _ 1033.3 xg _ 2 927

"0.001293x273"

For any other gas the value of R may be found by

dividing the value of R for air by the relative density of

the gas.

S p for air is 0.2374 (Art. 34) ; if the ratio between the

two specific heats be assumed to be 1.41, in accordance with

the best experimental results, then the above equation ex-

pressing the relation between J and R will give for J the

value of 42,420 gramme-centimetres, or 4.16 x 10 7 ergs.

89. Coefficient of Elasticity of a Gas (M., 1O6). -

Before proceeding to the second law of thermodynamics

it is desirable to introduce some topics subsidiary to it.

Since the working medium for the conversion of heat into

work is usually a gas or a vapor, a few propositions relat-

ing to them are necessary.

The coefficient of elasticity of a fluid is the ratio be-

tween any small increase of pressure and the resulting

132

HEAT.

voluminal compression. Let V be the initial volume and

v the diminution in volume due to an increment of press-

ure p. Then v/V is the compression per unit of volume.

The quotient of the increment of pressure by this com-

pression is the coefficient of elasticity of volume ; or, in

symbols,

v V

Since voluminal compression is only a ratio, the coefficient

of elasticity is a quantity of the same kind as a pressure.

Let volumes be rep-

resented by abscis-

sas and corresponding

pressures by ordinates

(Fig. 39). Then to

volume FP will cor-

respond pressure LP.

If now the pressure be

increased to MQ, the

volume will decrease

to G-Q. The coordi-

nates of the point P

represent the initial

and those of Q the

final condition of the

body with respect to volume and pressure, the temperature

remaining constant.

Join P and Q and produce the line to its intersection

E with the axis of pressures. Then will FE represent

the coefficient of elasticity. For

FE FP

THERMODYNAMICS. 133

VP V

But RQ is the increment of pressure, and _ _ _.

Hence

If therefore the relation between the volume and pressure

of a gas under the condition of a constant temperature be

represented by a curve traced by the point P, then the

coefficient of elasticity for any point P may be found by

drawing PE tangent to the curve at P and a horizontal

line P F ; the portion FE of the axis of pressures included

between PE and PF will represent the coefficient of elas-

ticity on the same scale as the pressures.

If the temperature is not constant, but is increased by

the compression, the effect will be to increase the increment

of pressure for any given decrement of volume. Hence

the corresponding coefficient of elasticity will be increased.

It is therefore evident that a gaseous substance has two

coefficients, one corresponding to constant temperature and

the other to the case where no heat is allowed to escape or

to enter during compression or expansion. The first is ap-

plicable to long continued stresses ; the second to rapidly

changing or alternating forces, as in the vibrations consti-

tuting sound, in which there is insufficient time for the

equalization of temperature by conduction and radiation.

The ratio of these two elasticities is the same as that of the

two specific heats.

90. Isothermal Lines (M., 108; S., 438). If the

ordinates of the curve traced by P represent pressures and

the abscissas volumes of a gas at constant temperature,

then the curve expresses the relation between p and v and

134

HEAT.

is called an isothermal line (Fig. 40). If the temperature

be increased to jp+l and be kept at this value, another

isothermal line will be obtained lying wholly above the

one for T. In this way any number of isothermal lines

may be drawn corresponding to regular intervals of tem-

perature. From such a diagram it is evident that, when

two out of the

three quantities

*jt>, v, T, are given,

the third may

be found graphi-

cally.

If the sub-

stance follows

Boyle's law,

then for a con-

stant tempera-

ture pv is a con-

stant, and this

product is rep-

resented in the

figure by the

area OFPL. If

this area is con-

stant the curve is known as a rectangular hyperbola.

The isothermal line corresponding to any temperature is

therefore a rectangular hyperbola.

It is a property of this hyperbola that if a tangent to the

curve be drawn through any point P till it meets Op in E,

then OF equals FE. But FE equals the coefficient of

elasticity of the gas and OF is the pressure. Hence the

coefficient of a perfect gas obeying Boyle's law is numeri-

cally equal to the pressure. This result was reached in

another way in the theory of sound (I., 118).

Fig. 40.

THERMODYNAMICS. 135

91. Adiabatic Lines. It remains to consider the

properties of a gas under the condition that no heat enters

or leaves it during the expansion or compression. If the

point traces a line expressing the relation between volume

and pressure in this case, it is called an adiabatic line.

When adiabatic lines cross isothermal lines, they are always

inclined to the horizontal at a greater angle than the

isothermal lines, because as the gas expands the pressure

diminishes more rapidly than for an isothermal line, since

the temperature is reduced by the work done in expanding

under pressure.

The equation to an adiabatic line is

pvy = a constant. 1

1 Let dQ be the quantity of heat required to raise unit mass of a perfect gas

through the temperature difference dT under constant pressure/?. This heat is

all expended in changing the temperature and doing external work. The quantity

required for the former purpose is S v dT. If the volume increases by a quantity

dv under pressure p, the work done is pdv, and the heat required is (pdv)/J.

Hence the whole heat necessary to effect the transformation is

When a gas expands adiabatically no heat enters or leaves it. and dQ =0.

Therefore

.

J

Differentiating the equation pc = RT, we have

Substituting in the last equation the value of dT obtained from this one, and

replacing R by its value J (S p ) from Art. 88, we have

Sjpdv + Sjdp^Q.

If y denotes the ratio S /$, > then

y dv + dp =Q

v p

I ntegrating, y log v + log p = constan t,

or, pv y = constant.

136

HEAT.

92. Carnot's Cycle (M., 138). If a volume of gas v l

at pressure pi and temperature T is allowed to expand

isothermally to the condition w/ and p/ represented by the

point B (Fig. 41), then work has been done against

external forces equal to the area ABv l / v 1 (I., 44). If now

the gas expands adiabatically from condition B at tempera-

ture TI to condition C at temperature T 2 , then the gas

does work represented by the area BCv./v/.

Suppose now the

gas to be compressed

isothermally along

the line CD. Then

the work is done on

the gas with loss of

heat, or is negative,

and it is represented

by the area CDv.,v./.

Lastly let the gas be

compressed adiabati-

cally from condition

- D to condition A.

Then the work done

on the gas raises its

Fig 4I

temperature from T 2 to T and equals the area

The algebraic sum of the several parts of the work is

then the area ABCD, enclosed between the two isother-

mals and the two adiabatics.

The working substance has returned to its initial vol-

ume, pressure, and temperature, and has gone through

an operation called a cycle. It is known as Carnot's Cycle.

The advantage gained by supposing the working substance

carried through a complete cycle of operations is that there

is then no balance of work done by or against internal

THERMOD YNAMICS.

137

forces, as there might be if the substance were not left in

its initial state.

If If i is the quantity of heat supplied at the higher

temperature TI , and H 2 the heat lost to surrounding bodies

at the lower temperature 21, then

Heat utilized H, H T } T* w

Heat supplied

= -7=. = efficiency.

Fig. 42.

93. Carnot's Engine. Carnot's engine is an ideal

one designed to embody the series of operations described

in the last ar-

ticle. Suppose

D, the working

substance (Fig.

42), to be con-

tained in a

cylinder imper-

vious to heat

except through

its bottom,

which is a s-

sumed to be a perfect conductor. Let A and B be two

stands, the temperatures of which are maintained at the

values 2\ and T 2 respectively. C is another stand the

top of which is supposed to be perfectly non-conducting.

Suppose the working substance D at the temperature of

the hot stand T^ and that its volume and pressure are

represented by Vi and _p,, the coordinates of the point A

on the isothermal line AB in the diagram of the last

article. Then we shall have the following operations :

First Operation. Place the cylinder containing the

working substance D on A and allow the piston to rise.

Heat flows in through the bottom of the cylinder to keep

138 HEAT.

the temperature of the working substance at the point jPj,

and the substance expands along the isothermal line AB

to the point B. During this operation the substance is

doing work by its pressure against the piston. It is

positive and is denoted by the area ABvJvi . During this

operation a quantity of heat HI has passed from A into

the substance.

Second Operation. The cylinder is now transferred to

the non-conducting stand O and the substance is allowed

to 'expand adiabatically, thus losing heat till its tempera-

ture falls from TI to T. 2 . Its expansion is represented by

the adiabatic line BO. The work done by the substance

during this process is equal to the area BOv 2 'vi.

Third Operation. The cylinder is next placed on the cold

body B, and the piston is pressed down till the volume

and pressure are represented by the coordinates of D.

Heat passes out through the bottom of the cylinder, the

substance remaining at the temperature T z . Its compres-

sion is represented by the isothermal line <7D, and the

work done on it equals the area ODv 2 v./; this work is nega-

tive. During this operation a quantity of heat H has

flowed from the working substance into the cold body B.

Fourth Operation. Finally place the cylinder on and

force the piston down. The temperature rises and the

relation of the volume and the pressure will be represented

by the adiabatic line DA. Continue the operation till the

temperature has risen to that of the hot body T v . Then

work equal to the area DAv^v- 2 is done on the substance,

and is negative.

The substance has thus passed through a series of opera-

tions by which it has finally been brought back in all

respects to its initial state. When the piston is rising the

substance is doing work ; this is the case in the first and

THERMODYNAMICS. 139

second operations. When the piston is sinking it is per-

forming work on the substance ; this is the case in the third

and fourth operations. The useful work done by the sub-

stance is the difference between the positive and negative

work, and is represented by the area ABCD.

The physical results at the end of the cycle are the

following :

(1) A quantity of heat jffi taken from A at the temper-

Online Library → Henry S. (Henry Smith) Carhart → Physics for university students (Volume 2) → online text (page 9 of 28)