Physics for university students (Volume 2) online

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-