Henry S. (Henry Smith) Carhart.

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Font size ature TI during the first operation.

(2) A quantity of heat H 2 communicated by the
working substance to B at the temperature T during
the third operation.

(3) The performance by the substance of work equal
to the area ABCD.

94. Reversibility of Garnet's Engine (M., 149; S.,

351) Let us now suppose all the preceding operations

to be reversed, or that the engine is worked backwards, or
is reversed in all its physical and mechanical actions.

Beginning at the higher temperature and at volume v 15
let the cylinder be placed on and let the substance ex-
pand along the adiabatic line J.D, while the temperature
falls from TI to T 2 . Next place the engine on B and allow
the substance to expand isothermally along DC. During
this latter expansion heat H> will be taken from the colder
body B ; and by the two expansions the body has done
work denoted by the area AD Cv. 2 'vi .

Now place the engine on C and compress adiabatically
till the temperature rises from T 2 to TI. Then removing
it to A, compress the substance isothermally along BA till
it again returns to its initial volume and pressure. During
the last compression, heat HI has been given out to A at
the higher temperature T^and work has been done in

140 HEAT.

compressing the substance adiabatically and isothermally
in the two compressions equal to the area CBAv^v.f.

In this reverse action of the engine more heat has been
given out to A at the higher temperature than has been
drawn from B at the lower temperature, and more work
has been done on the engine than by it equal to the area
ABCD. It is possible then to convey heat from a colder
body to a hotter one, but only at the expense of mechanical
work.

95. Garnet's Principle. - Heat may be transferred
from a hot body to a cold one either directly by conduc-
tion, or indirectly by means of an artificial engine, in such
a way that part of the heat is converted into mechanical
work ; but heat never flows from a cold body to a hot one,
and it can be thus transferred only by artificial means and
at the expense of mechanical work.

What is known as Carnot's principle, derived from a
consideration of his reversible engine, is as follows : " If a
given reversible engine, working between the upper tem-
perature Tj and the lower temperature T a , and receiving
a quantity HI of heat at the upper temperature, produces
a quantity w of mechanical work, then no other engine,
whatever be its construction, can produce a greater quan-
tity of work when supplied by the same amount of heat
and working between the same temperatures."

Suppose an engine M to have a higher efficiency than
this reversible one. Let it be coupled to a reversible
engine N working backwards. Then since M converts a
larger portion of the heat HI into mechanical work than N
requires to restore the heat HI from the refrigerator to the
source, the two engines constitute an automatic arrange-
ment by which M, by the use of heat HI , supplies to N

THERMODYNAMICS. 141

sufficient energy to enable it to restore to the source more
heat than J?i; or, in other words, the coupled engines
would run perpetually, transferring heat continuously
from colder bodies to hotter ones. Such an operation is
denied by experience, and is inadmissible. Therefore no
engine can be more efficient than the ideal reversible one
of Carnot.

96. The Second Law of Thermodynamics. - - The
second law of thermodynamics expresses a conception
derived from Carnot's reversible engine, and is stated by
Clausius as follows :

" It is impossible for a self-acting machine, unaided by
any external agency, to convey heat from one body to
another at a higher temperature."

Lord Kelvin gives it in a slightly different form :

"It is impossible, by means of inanimate material
agency, to derive mechanical effect from any portion of
matter by cooling it below the temperature of the coldest
of the surrounding objects."

These statements apply only to the performance of
engines working in a complete cycle. Without this
limitation it is evident that the heat of a body, that of
a compressed gas for example, may be converted into
work by cooling it below surrounding objects.

Since the quantities of heat taken in and given out by
a reversible engine depend only on the temperatures of
the source and the cooler, the ratio of the two tempera-
tures may be made equal to that of the quantities of heat to
form a scale of temperature. Then ff l /ff^= TJT Z . Such
a scale agrees with that of a perfect gas thermometer.

142

CHAPTER X.

THE KINETIC THEORY OF GASES.

97. Molecular Hypotheses. The comparative sim-
plicity of the laws relating to gases has stimulated inquiry
into a kinetic theory to account for them on simple
dynamical principles. The results are encouraging to the
extent that they exhibit satisfactory agreement between
the deductions from theory and the laws established by
experiment.

Certain preliminary hypotheses relating to molecular
motion in gases are assumed, though not without justifica-
tion. Since it cannot be assumed that all like molecules
even have the same velocity, the statistical or average
method is adopted, which applies the reasoning to certain
groups of molecules whose velocities do not differ by more
than a very small quantity from a mean value. It is then
possible to discover definite relations between the physical
properties of such a group without knowing anything
about the performance of individual molecules.

Some of the hypotheses are the following :

(1) Molecules of the same gas are alike, and are
separated by intervals which are very great compared
with the size of the molecules. This inference is drawn
from the fact that when a gas is heated so as to become lu-
minous the colors emitted are independent of the pressure ;
that is, the colors depend on the nature of the molecules

THE KINETIC THEORY OF GASES. 143

and not on the distance between them ; for if the molecu-
lar distances were relatively small, mutual action would
ensue, and this action would depend on the pressure
which changes the intervals between the molecules.

(2) The molecules of a gas move in straight lines
between mutual encounters. Their motion for any excur-
sion is uniform and rectilinear. The phenomena of diffu-
sion exhibit rectilinear motion.

(3) All molecules of the same gas have equal masses,
and the average kinetic energy is the same for all mole-
cules at the same temperature.

(4) When two * sets of molecules of different kinds
are placed in the same enclosure, kinetic and thermal
equilibrium ensues. The average kinetic energy of trans-
lation for one set is then the same as for the other ; this
statement may be extended to any number of sets. If
m l and m>> are the two molecular masses, then

2 2

\$mv~ is called the average kinetic energy of agitation of a
single molecule. The velocity v is " the square root of
the mean square " of all the molecules whose velocities
differ by only a small amount. The squares of the rates
of diffusion of different gases through small pores are in-
versely as their molecular masses. Thus, hydrogen dif-
fuses four times as fast as oxygen. This should be the
case if the two gases have the same molecular kinetic
energy at the same temperature as assumed.

98. Theory of the Pressure of a Gas (M., 319 ; P.,
69). Let a molecule of mass m approach the side of a
cubical box of unit volume with a normal velocity u. If it
rebounds with the same velocity, the change in momentum

144

HEAT.

will be Zmu. If the molecule moves backwards and for-
wards between two opposite sides of the box with veloc-
ity u, it will strike each side %u times a second, since the
space traversed between two successive impacts on the
same wall is two linear units. Hence the total change
of momentum of the molecule per second with respect to
the wall is

2mu x-ii mu 2 . ^

If the unit cube contains n such molecules, then the press-
ure, which is the rate of change of momentum, will be

p = "Zmu 2 = mLu~.

But if u 2 is the mean of the squares of all
the velocities normal to the face of the
cube,, then nu 2 = 2w 2 , and
p mnu~ t

In general a molecule may be moving in
any direction with a velocity v. If we
suppose that u is the velocity normal to
the plane between A and B (Fig. 43), and u^ u 2 , the
two other rectangular components, then

v- = u 2 + ui + u\.

If now V- denotes the mean of the squares of all the
molecular velocities of the different groups, with corre-
sponding meanings for U\ U\, TJ\, then

v* = u-+ u\+ m.

But since the molecules do not accumulate in any part
of the enclosure, as many passing on the average across
the plane between A B in one direction as the other, the
pressure in all directions will be the same, or

Fig. 43.

U* = U\=Ul = -
3

THE KINETIC THEORY OF GASES. 145

Therefore when the molecules are moving in all directions
within the cube, the pressure on each face of unit area
will be

P = m nT-^

While we may not know the absolute mass of each mole-
cule nor the number in unit volume, yet the product mn
of the two is the mass in unit volume, or the density.
Hence

3

The pressure is therefore one-third the product of the
density of the gas and the mean square of the molecular
velocity.

99. Mean Square of the Velocity of Hydrogen.
From the preceding expression Joule calculated the square
root of the mean square of the velocity of hydrogen as
follows :

The data are, d = 0.0000896 ; p = 1033.3 x 980 dynes.
Hence

/ 3 x 1033.3 x 980
: V - 0.0000896 ~~ 184 ' 133 CmS ' per SeC nd '

or in round numbers 184,000. This is the value for the
hydrogen molecule between impacts at C. and 76 cms.
pressure.

100. Deduction of Boyle's Law. If v denotes now
the volume containing unit mass of the gas, then

1 V- 1

= - V 1 .

6 V 6

Since heat is energy of motion, the mean square V 1 is a

146 HEAT.

function of the temperature of the gas only. Consequently
pv at any one temperature is a constant; this is Boyle's
law " raised from the rank of an experimental fact to that
of a deduction from the kinetic theory of gases."

101. Law of Gay-Lussac. Consider two gases in
thermal equilibrium. Then for the two we have

PI = m&iVi and p 2 -m#w\.

o o

If the pressures are equal

m\niV\ = m 2 n 2 v% .
But since they have the same temperature

for the mean kinetic energy of translation of the molecules
is the same for each gas at the same temperature.
Dividing the two equations member by member and

HI n 2 ,

or equal volumes of all gases at the same temperature and
pressure contain the same number of molecules. This is
known as the law of Gay-Lussac or of Avogadro. While
this demonstration cannot be considered as stringent, it
shows that this hypothesis is entirely in harmony with the
kinetic theory of gases.

If we put d } = m^ and d> = m 2 n 2 , then since % n we
have

or the densities of two gases at the same temperature and
pressure are directly proportional to their molecular
masses or weights.

THE KINETIC THEORY OF GASES. 147

102. Total Molecular Energy. The mean kinetic
energy of agitation of a molecule is Jra V 1 . But its energy
may be due partly to the vibration of its parts and to
rotation. Clausius and others have assumed that the
energy of internal agitation tends toward a value having a
constant ratio to the energy of agitation of the molecule
as a whole. The whole energy will then be proportional
to the energy of agitation, and may be written

Then the total kinetic energy of the gas contained in
unit volume of n molecules is

But since p = mn V 1 .

The energy per unit mass may be found by multiplying
the energy per unit volume by the number of units of
volume containing unit mass, or

103. Specific Heat at Constant Volume. Since the
product pv is proportional to the absolute temperature, the
last equation shows that the energy per unit mass is also
proportional to the temperature on the absolute scale.
The specific heat at constant volume is the increase in the
energy of unit mass for one degree increase of tempera-
ture. Hence in dynamical units

148 HEAT.

that is, the entire energy divided by the number of degrees
of temperature gives the energy corresponding to one
degree.

Now since pv / T is a constant for gases obeying the
laws of Boyle and Charles, it follows that the specific heat
at constant volume must be constant for any gas, what-
ever its pressure and temperature. This conclusion
is in harmony with the experimental results of Reg-
nault (34).

For different gases the specific heat is directly propor-
tional to the volume v containing unit mass, or inversely
proportional to the density and directly proportional to ft.
Since /3 is nearly the same for several gases, the specific
heat of these gases is inversely as their densities, or
inversely as their molecular weights ; and therefore the
product of specific heat and molecular weight is the
same for all such gases. This is the law of Dulong and
Petit.

1O4. Ratio of the Two Specific Heats The thermal

capacity of any mass M of a gas at constant volume con-
sists of the energy of the molecular motion of translation
plus the energy of the internal motions of the molecules
for one degree of temperature. If E denotes this internal
energy for one degree, then in dynamical units

1 MV 2

-^=2-^ + *

Also, since the work done in expanding unit mass of a
gas under constant pressure is pv / T for one degree rise
of temperature, we may write for the thermal capacity
under constant pressure,

IMF* pvM

~ ~~'

T1IE KINETIC TBSORY OF GASES. 149

From (100) pv = V\ Therefore,

Therefore, to find the ratio of the two specific heats,

I Mr 2 5 MV*

2 T 3 T 6 T

But E is necessarily positive ; hence 7 must always be
less than f or 1.667, which would be its value if E were
zero. As E increases, 7 approaches unity. These con-
clusions are justified by experiment as shown by the fol-
lowing table:

7 7

Mercury .... 1.666 Chloroform . . . 1.200

Oxygen .... 1.404 Methyl ether . . 1.113

Nitrogen .... 1.410 Ethyl ether . . . 1.029
Ammonia . . . 1.300

The value of 7 approaches its upper limit only in the
case of mercury, which is the only monatomic gas exam-
ined. The simple constitution of such a gas would lead
to the anticipation that its internal molecular energy
is negligible as compared with the energy of molecular
translation. In all other gases the internal energy is very
appreciable, and it increases as the number of atoms in the
molecule increases. As the molecule becomes more com-
plex, its internal energy represents a larger fraction of the
heat applied to warm the gas.

150 ELECTRICITY AND MAGNETISM.

ELECTRICITY AND MAGNETISM.

CHAPTER XI.

ELECTRIC CHARGES.

105. Electricity and Electrification. - - The simple
elementary phenomenon that a piece of amber, rubbed
with a flannel cloth, acquires the property of attracting bits
of paper, pith, or other light bodies has been known since
about 600 B.C. But it appears not to have been known
for the following 2,200 years that any bodies except amber
and jet were capable of this kind of excitation. About
1600 Gilbert, an English physician, discovered that a large
number of substances possess the same property. These
bodies he styled electrics, but the word electricity to des-
ignate the invisible agent concerned in the phenomena
appears to have been introduced by Boyle in 1675.

Electrical phenomena are now well understood, but the
nature of electricity remains obscure. It was long supposed
to be a kind of subtle fluid ; in later times philosophers
were disposed to consider it a form of energy transformable
into heat and light. But it is now quite certain that, while
it may be a form of attenuated matter, like the ether, it is
not energy. " It is quite true that electricity under press-
ure or in motion represents energy, but the same thing is
true of water or air, and we do not therefore deny them to
be forms of matter." When a body is electrically excited

ELECTRIC CHARGES. 151

it is said to be electrified, and electrification' is always a
result of work done in charging with electricity. Electri-
fication, or electricity under pressure, is therefore a form
of potential energy, just as air under pressure and water
elevated above the earth represent potential energy. But
air and water- on the one hand and electricity on the other
are not energy, but only its vehicles or receptacles.

Electricity, like matter and energy, appears to be inde-
structible. Its distribution is subject to control; it may
be put under electric pressure, or be endowed with kinetic
activity; it may represent energy of stress or energy of
motion ; but when its energy has been spent in producing
physical effects, its quantity has suffered no diminution.
It has simply been strained and moved like matter. The
only way to charge a body is to pass to it electricity
from outside ; none can be created or generated and none
destroyed.

106. Division of the Subject. The study of electric
currents began near the close of the eighteenth century,
and the earlier observed phenomena relating to them were
widely differentiated from the older manifestations of
electrostatic charges. It has therefore long been customary
to divide the entire subject into three grand divisions, viz. :
static electricity, magnetism, and current electricity. But
since all the phenomena of electrostatics can be produced
by means of electricity set in motion and put under stress
by batteries or dynamo-electric machines, it is apparent
that electricity, however excited, is one and the same
agent. At the same time magnetism is inseparable from
electric currents and must be studied in connection with
them. While, therefore, the general phenomena and laws of
electrostatics, or electricity in equilibrium under pressure,

152

ELECTRICITY AND MAGNETISM.

are conveniently studied together, it should be clearly
perceived that this is merely a matter of convenience, and
that such a classification is not imposed by fundamental
differences. Electric charges and electric sparks may now
be produced as well from one source of electricity as
another; and magnetism may be evoked by electrostatic
discharges, by electric convection, and by electric currents.
Nevertheless it will be convenient to study first the facts
and principles applying especially to electrostatics, and
then those relating to electric currents and their magnetic
effects. A fourth division for purposes of classification
comprises the study of periodic or undulatory disturbances
propagated through the ether as waves of electromagnetic
origin. This subject is the most difficult one of all, but
possesses for us surpassing interest. It includes the
electromagnetic theory of light, elaborated by Maxwell
and confirmed experimentally by Hertz.

1O7. Attraction and Repulsion.

Fig. 44,

Support a small
pith-ball by a
silk fibre (Fig.
44) and present
to it a warm
glass tube ex-
cited by rubbing
with a piece of
silk. The pith-
ball will first be
attracted, but if
it be allowed to
come in contact
with the electri-
fied glass, it will

ELECTRIC CHARGES. 153

then be strongly repelled. If a stick of sealing-wax,
electrified by rubbing with flannel, be used instead of the
glass tube, the results will be exactly similar.

Two facts are clearly exposed by this experiment : (1)
A body may be charged by contact with an electrified
body. (2) When one body is charged by contact with
another the two repel each other.

Boyle discovered that the attraction between the electri-
fied and the unelectrified body is mutual. Excite a glass
tube and lay it in a light wire stirrup supported by a silk
thread (Fig. 45). If the hand be presented to it, it may
be made to swing round by the attraction. Force, whatever
its origin, is of the nature of a stress in the medium, and
action and reaction are equal (I., 42).

1O8. Two Kinds of Electrification. Not all electri-
fied bodies repel each other. If a second excited glass
tube be presented to the one hung in
the stirrup (Fig. 45), there will be
mutual repulsion between them. On
the contrary, -an excited stick of
sealing-wax will attract the pith-ball
charged by contact with an electrified
glass tube ; and if the pith-ball be
charged by contact with the rubbed
sealing-wax, it will be repelled by
the sealing-wax, but attracted by the F 45

glass tube rubbed with silk.

So if two or three pith-balls, hung by silk fibres (Fig.
46), be touched either with an excited glass tube or a
stick of electrified sealing-wax, they will fly apart by
mutual repulsion. It is, therefore, inferred that there are
t\\o kinds of electrification, or that electricity manifests

154 ELECTRICITY AND MAGNETISM.

itself under two opposite aspects, analogous to the oppo-
site properties possessed by the two poles of a magnet.
The electricity excited by rubbing glass with silk Du Fay
called vitreous electricity ; and the electricity excited on
such substances as sealing-wax, resin, amber, shellac, and
hard rubber when rubbed with flannel, he called resinous

electricity. The former Frank-
lin called positive and the latter
negative electricity ; and this clas-
sification is better than Du Fay's,
since glass does not always show
positive nor resin negative elec-
trification. The result of friction
depends on the rubber as well as
on the material rubbed.

From such experiments as the
foregoing is derived the first law
of electrostatics, viz., bodies similarly electrified repel and
those oppositely electrified attract one another.

The student should guard against the inference, from
the expression " two kinds of electrification," that there
are two kinds of electricity, called positive and negative,
respectively. Positive and negative forces constituting a
stress are not essentially different forces, nor are positive
and negative rotations different except in respect to alge-
braic sign. Yet in both cases the forces and motions may
annul each other, as equal quantities of positive and nega-
tive electricity neutralize each other. The terms positive
and negative are applied to electricity merely for the pur-
pose of enabling us to describe concisely, and, to that
extent, to explain certain electrical phenomena.

109. Conductors and Insulators.' Gilbert concluded

ELECTRIC CHARGES. 155

that some bodies were capable of electrical excitation and
others were not. To substances like metals which gave
no sign of electrification when held in the hand and rubbed
he gave the name " non-electrics." In 1729, however,
Stephen Gray discovered that Gilbert's "non-electrics"
convey away the ** electric virtue " as fast as it is excited,
and therefore show no signs of electrification. If a metal
rod be held by a glass handle, it can be excited by rubbing
it with silk. Gray succeeded in conveying electric charges
a distance of seven hundred feet by means of a hempen
thread suspended by silk loops, and Du Fay carried them
to nearly double this distance by means of moistened
thread. Ever since Gray's discovery bodies have been
divided with respect to their power of conveying elec-
tricity into conductor* and non-conductors, or insulators.
The latter Faraday preferred to call dielectrics. It should
be noted, however, that all bodies can be arranged in a
graded series having the best conductors at one end and
the poorest at the other. None conduct perfectly and none
insulate perfectly. Pure copper, silver, and other metals
are the best conductors ; and the best insulators are silk,
shellac, glass, and quartz. More definite data on the
specific resistance of various conductors will be given
in treating of electric currents.

110. Electric Field and Lines of Force. The old
mathematical notion of action at a distance has now been
abandoned ; and when there is attraction or repulsion be-
tween separated bodies, the action is conceived to take place
through the agency of the intervening medium. This con-
ception, developed by Faraday and elaborated by Maxwell
in its application to electricity, has been very fruitful in
discovery, and bears every mark of conforming to the truth.

158 ELECTRICITY AND MAGNETISM.

In harmony with this view, a region within which the
medium is under stress is said to be a field of force ; and
an electric field is one in which the forces acting are electric
in their origin. For concreteness the stress in the medium
is said to act along lines of force. An electric field may
be completely specified by giving at every point in it the

Online LibraryHenry S. (Henry Smith) CarhartPhysics for university students (Volume 2) → online text (page 10 of 28)