Physics for university students (Volume 2) online

Mercury is a very suitable thermometric
sul >stance, for it fulfils most of the necessary requirements.
It can be readily procured in a state of purity. Its co-
efficient of expansion is large and nearly uniform between
the limits within which it remains liquid, and those limits
represent a wide range of temperature. It readily transmits
heat through itself, so that all the mercury in the ther-
mometer rapidly comes to the same temperature. It requires



Fig. I.



14 HEAT.

less heat to raise the temperature of any mass of mercury
through any range than is required for equal masses of
most other liquids that is, its thermal capacity is small.
When therefore it is brought into contact with a warmer
body, at whose expense its temperature rises, this body in
general loses but little heat and its temperature is not
changed by the application to it of the instrument intended
to measure its temperature. Moreover, it does not stick to
the tube so much as other liquids, and it is opaque and can
easily be seen as a fine thread in the bore. On the other
hand, mercury is very heavy, and its weight brings great
stress on the bulb. Also its meniscus is not the same
when the column rises as when it falls, and on a falling
temperature the column is known to descend with an
irregular jerky movement.

10. The Two Fixed Points on a Thermometer. For
the purpose of making different thermometers comparable,
it is necessary to have fixed points of temperature which
are invariable and easy of reproduction. The two points
universally employed are the temperature of melting ice
and the temperature of steam from water boiling under the
pressure of a standard atmosphere (I., 101). The former
is called for brevity the freezing point, and the latter the
boiling point. The employment of these two points as
standards of reference was first suggested by Hooke ; they
were adopted by Newton in 1701.

The first point is obtained by placing the thermometer
in a vessel filled with pounded ice at the melting tempera-
ture. It is desirable that the interstices between the lumps
of ice should be filled with water while all excess drains
off. The thermometer must be completely immersed in
the ice and water, and must remain there till the mercury



TEMPERATURE AND ITS MEASUREMENT. 15

becomes stationary in the tube. The top of the column is
then marked by a fine scratch on the glass.

To determine the boiling point, the thermometer is
passed through a hole in the top of a tall vessel, the bottom
of which contains boiling water. The thermometer must
be completely enveloped in steam, no part of it touching
the water. When it has acquired the temperature of the
steam it is drawn up till the top of the mercury thread is
visible and the point is marked by a scratch. The upper
portion of the tall vessel is made double so that the steam
may circulate round the inner tube containing the thermom-
eter as a steam jacket to keep the steam up to the boiling
point at every part of the thermometer. The bulb of the
thermometer is not allowed to touch the water, because
the temperature at which water boils varies somewhat
with the material of the containing vessel, while the steam
escaping from boiling water is always at the same tem-
perature for the same pressure. If the atmospheric pressure
is not 760 mms. a correction must be applied, the boiling
point rising 1 C. for every 26.8 mms. increase of pressure.

11. Thermometer Scales (T., lO3). - The distance
between the two fixed points on a thermometer must be
subdivided into some convenient number of divisions, each
of which represents one degree of temperature. The
volume of the capillary bore of the tube between the
freezing point and the boiling point represents the total
expansion of the mercury from the one temperature to the
other. A degree of temperature is then that rise of
temperature which causes the mercury to expand some
definite fraction of its entire expansion between the
freezing and boiling points.

Three scales of uniform graduation are in common use :



16 HEAT.

Fahrenheit's Scale. Fahrenheit about 1714 constructed
the first thermometers with a uniform graduation of the
scale, and this scale is still the one most commonly used in
English-speaking countries. The distance between the
two fixed points is divided into 180 parts of equal volume.
The freezing point is marked 32, and the boiling point is
therefore 212. The graduation is usually continued
below 32. One degree F. is that rise of temperature
which causes i-Jo^ri of the expansion in volume between
the freezing and boiling points.

The Centigrade Scale. Celsius of Upsala divided the
scale between the fixed points into 100 equal parts. The
freezing point he marked and the boiling point 100.
This scale is obviously simpler than that of Fahrenheit,
and is in general use among scientific men in connection
with the metric system of measurement. One degree C.
is longer than 1 F. in the ratio of 9 to 5.

Reaumur's Scale. In this scale the freezing point is
marked 0, and the boiling point 80. It is in use for
domestic purposes on the continent of Europe, but has
little to- commend it, except that it avoids Fahrenheit's
fault of a misplaced zero.

In all three scales the graduation is often extended below
zero and above the boiling point.



Fahrenheit 3 g

Centigrade

Reaumur

Fig. 2.



212



100



12. Comparison of Thermometer Scales (G., 12 ;
T., 1O5). To compare corresponding readings on the
three scales, let us suppose the three attached to the same



TEMPERATURE AND ITS MEASUREMENT. 17

thermometer (Fig. 2). Let A be the freezing point,
B the boiling point, and P the head of the mercury column ;
also let F, <?, and R be the readings on the three scales
respectively corresponding to the point P.

Then, since AP is the same fraction of AB measured by
either scale,

F-Z'2 = C R
180 ~100~80*

The readings on either scale below zero must be treated
as negative. It must be noted also that the zero of Fahren-
heit's scale is displaced 32 in comparison with the zero of
the other two. For Fahrenheit readings therefore 32 must
be subtracted algebraically to find the number of degrees
between the freezing point and the reading. Thus, 50 F.
is 50 32 =18 above freezing; and -10 F. is -10
-32= -42, or 42 below freezing.

13. Change of Zero (P., 115). A thermometer should
not be graduated for several months after filling with mer-
cury. It has been found that the volume of the bulb
slowly decreases for a long period after being strongly
heated. Glass is in some degree plastic, and a gradual
molecular readjustment goes on after it has been strained
or heated. This decrease of the capacity of the bulb raises
the zero point on the stem. The correction at the zero
point even on standard thermometers may often amount to
as much as 0.T C., though it rarely equals 1 C.

Besides this progressive and permanent change, there is
another temporary one which may be observed after a
thermometer has been heated in boiling water. It is there-
fore customary to determine first the freezing point and
then the boiling point. If the freezing point is determined
immediatelv after immersion in boiling water, it will be



18 HEAT.

found that it may have been depressed as much as 0.3 C.,
and it will not recover its former value until ten days or
more have elapsed.

If the fixed points have been found with the thermometer
in a horizontal position, it should be used horizontally ; or
if they have been found in a vertical position, the ther-
mometer should be used vertically. The reason is that the
hydrostatic pressure of the mercury column compresses
the mercury and enlarges the bulb in the vertical position,
and so lowers all the readings. For a similar reason the
readings of unprotected deep-sea thermometers are too
high, because the bulb is compressed by the pressure of
the water.

14. The Alcohol Thermometer. Since mercury freezes
at 38.8 C. and boils at about 350 C., the mercury ther-
mometer cannot be employed for temperatures beyond
these limits. For temperatures lower than 38 C. absolute
alcohol has often been used because it freezes only at about

- 130 C. and its dilatation is even greater than that of mer-
cury. But since the dilatation of alcohol is not uniform
at different temperatures, the alcohol thermometer must be
graduated by comparison with a standard mercurial ther-
mometer. It can be used only in a vertical position, bulb
downward, because the alcohol wets the tube, and time
must be allowed after a fall of temperature to permit the
liquid to run down.

15. The Air Thermometer (M., 46; S., 7O). For
high temperatures and for accurate scientific purposes some
form of air thermometer is often used. If a volume of
gas V (} be heated from to 1 under a constant pressure
and its increase of volume be v, then its dilatation will be



TEMPERATURE ANU ITS MEASUREMEXT. 19

the same volume v for an equal rise of temperature at any
other part of the scale. This law, called the law of
Charles, is not rigorously exact, but gases approach it more
and more closely at low pressures and high temperatures,
or, in other words, in a highly rarefied state. Within cer-
tain limits, however, all gases, sufficiently removed from
their condensing points, may be regarded as
expanding equally. The ratio v / Fi fbr one
degree Centigrade was found by Kegnault to
be 0.00866.") for air.

This property of uniform expansion may be
employed in the construction of a thermometer;
The first air thermometer was made by Galileo
before 1597. The air was contained in a bulb
from which a tube descended to a bottle filled
with a colored liquid (Fig. 3), or was bent
twice at right angles and terminated in an open
bulb. This thermometer is filled by heating the
bulb before the stem is inserted in the liquid.
On cooling, the air contracts and the liquid
rises in the stern. Then if the temperature
changes, the liquid column moves. But un-
fortunately the instrument is also affected by R 3
any change in atmospheric pressure, and can
therefore be used only as a thermoscope unless it be greatly
modified and made more complicated.

The first use to which the air thermometer was applied
was by physicians to obtain the temperature of the human
body. The patient took the air bulb in his mouth, and the
extent to which the liquid column descended indicated to
the observer whether the patient had a fever.

The simplest form of air thermometer is the one employed
by Boyle in 1665. It was composed of a glass bulb from



20 HEAT.

which rose a long stem containing a drop of mercury or
sulphuric acid to separate the air within from the external
atmosphere (Fig. 4). As the temperature rises, the air
within expands and drives the liquid index before it.

The dilatation of air is about twenty times as great as
that of mercury for the same range of temperature.
Hence a thermometer filled with air is much more sensitive
than one filled with mercury. For any given
range of temperature it has been found that air
and mercury thermometers agree closely, though
not exactly.

It is worth while to point out that the only
reason we have for asserting that the thermal value
of the successive degrees of a well-calibrated mer-
cury thermometer are the same is that they cor-
respond closely with those of the air thermometer.
But, strictly speaking, it is impossible to prove
the law of Charles with precision, for its experi-
mental demonstration implies the possession of an
accurate instrument for measuring temperature.
There are, however, theoretical reasons for believ-
ing this law to be exact when the gas is in a state
of extreme tenuity and the molecules are so far
apart as to exert no influence upon one another. It is
then called a perfect gas.

The practical methods of using air as a thermometric
substance are described in memoirs and large treatises. A
description of one form will be found in a later chapter
(26).

16. The Absolute Zero (M , 48, 213).-^ The air
thermometer in the form of a straight tube of uniform
bore may be employed to illustrate the meaning of the




TEMPERATURE AND ITS MEASUREMENT. 21

"absolute zero of temperature," or, better, the "zero of
absolute temperature."

Let a long narrow tube be closed at one end, and let air
be confined and separated from the external air by a short
cylinder of oil, mercury, or sulphuric acid. We shall
assume that the pressure on this enclosed air is maintained
constant.

Let the point marked F (Fig. 5) be the position
of the surface of the enclosed air or index cylinder
when the tube is in melting ice, and let B mark
the position of the index for the temperature of
boiling water. The question then arises, What
temperature will be indicated at the bottom of the
tube, if the uniform graduation is carried down
there, and what is its meaning?

The first question is easily answered. Let x
equal the length AF on the same scale as FB is
KM) divisions. Then we know that the volume of
the portion of the tube between A and F is to the
volume of AB as 1 is to 1.3665, since 0.3665 is the
dilatation of air for 100 C. Then

xix+100 ::1 : 1.3665.

Whence x = 272.85, or in whole numbers 273. The
bottom of the tube will then be marked 273.
This point is called " absolute zero." The meaning p .
of it is that if the law of Charles should continue
to hold down to the temperature 273 C., the volume of
the gas would become zero, or the air would be entirely
devoid of heat. Now, while it is not supposed that the
contraction of a gas would continue at the same rate down
to any such temperature, still this is a convenient point
from which to reckon temperatures, because the volume
of a perfect gas is simply proportional to its temperature



22 jrEAT.

measured on this scale. Temperatures on the Centigrade
scale are converted into corresponding readings on the
absolute scale by adding 273.

It is important to know that the scale of the air ther-
mometer agrees almost exactly with that derived from
thermodynamical considerations. The agreement has been
experimentally verified between the limits C. and 100 C.

PROBLEMS.

1 . Convert the following readings on the Fahrenheit scale into
the corresponding degrees Centigrade : 60, 28, 20.

2. Convert the following readings on a Centigrade thermometer
into degrees of the Fahrenheit scale: 15, 10, 20.

3. At what temperature will the Fahrenheit and Centigrade
scales read the same?

4. At what temperature will the reading of the Fahrenheit scale
be double that of the Centigrade ?

5. At what temperature will the reading of the Centigrade scale
be double that of the Fahrenheit?

6. If a thermometer scale were marked 10 at the freezing point
and 60 at the boiling point, what would 40 on this scale mean in
Centigrade degrees ?

7. A thermometer tube with uniform bore has 5 C. divisions to a
cm. ; how many F. divisions to the cm. would there be?

8. The testing of a Centigrade thermometer shows that the
freezing point reads -}- 0-6 and the boiling point 101. What is the
meaning of 50 on this scale if the tube is uniform ?

9. The latent heat of fusion of ice on the Centigrade scale is 80;
find it on the Fahrenheit scale.



EXPANSION. 23



CHAPTER III.

EXPANSION.

17. The Cubical Dilatation of Solids (S., 27 ; P., 157).

- The expansion of solids and liquids has already been

alluded to in the last chapter. The property of a thermo-

metric substance which is utilized to indicate temperature

is its increase in volume with heat.

Let F" be the volume of a body at zero and F^its volume
at t. Then if the increase of volume v for an increase of
one degree in temperature is constant at different parts of
the scale, we have



or F= F + vt = F (1 + It) = I \ (1 4- kf).

*0

Tin- constant k is called the coefficient of cubical ex-
pansion. It is equal to v / F , or the expansion per unit of
volume when the temperature rises from to 1 C. This
is sometimes called the zero coefficient. If, for example,
1 c.c. of iron at becomes 1.003546 c.c. at 100 C., then
0.00003546 denotes the mean coefficient of cubical dilata-
tion of iron between these two temperatures.

While the 'equation VV Q (1 + fa) is a very near aj>-
proximation, it is not rigorously exact. Each substance
has its own constant k.



24 HEAT.

Since the volume of any mass of a substance is inversely
as its density, we may write



Whence



td

This formula is the basis of a method of measurement
which depends on the determination of the density of a
solid at different temperatures.

The general law of the dilatation of solids assumes that
they expand when heated and recover their initial volume
when restored to their initial temperature ; that is, that
under a constant pressure the volume is a function of the
temperature.

Neither of these assumptions is rigorously correct. It
has been found that Rose's fusible metal expands to a
maximum, after which, if the temperature be increased, it
contracts. So also Fizeau found that iodide of silver con-
tracts regularly when heated between 10 and 70 C. It
has since been determined that it reaches a point of maxi-
mum density at 116 C., at which point on cooling it passes
from the amorphous into the crystalline state.

Neither is it true that the restoration of an antecedent
temperature always restores a body to the corresponding
volume. If some bodies, like glass, are cooled suddenly,
the molecules have insufficient time to arrange themselves
in accordance with their mutual attractions. Hence certain
stresses are set up which may produce a slow change in
volume as they adjust themselves to zero.

The purpose of annealing glass and metals by slow cool-
ing is to give time for the forces of cohesion to adjust
themselves without constraint. The annealed body is then
much tougher. It is not much in error to say that when



EXPANSION. 25

bodies are heated and then very slowly cooled, they return
to the same volumes at the same temperatures.

18. Linear Expansion. If the distance between two
transverse parallel lines on a metallic bar is 1 Q at a tempera-
ture of and I at , the increase in length is I 1 . This
linear expansion is found to be nearly proportional to the
length and to the rise of temperature; and the constant
which defines this proportionality, and which depends upon
the nature of the body, is called the coefficient of linear
expansion. If this coefficient is denoted by a, then



I IQ = al t, or a =



l-lo



Whence I = k (1 + a).

It is obvious from the equation for a that the coefficient
of linear expansion is the increase which occurs in unit
length of a solid when the temperature rises from to 1
C. This is very nearly the same as the mean coefficient
between and 100 C. It is the ratio of the increase in
length for one degree to the total length at 0. In the
metric system it is the increase in the length of one cm.
due to a rise of temperature of one degree C.




Fig 6.



The expression 1 + at is called the expansion-factor. It
is the ratio of the final to the initial length.

A simple method of showing the expansion of a wire in



26 HEAT.

length is illustrated in Fig. 6. The wire, which should be
about one metre long, is rigidly attached at one end A to
the stand, and at the other is fastened to a small screw-eye
in the long, light wooden pointer BO. The pointer is free
to turn around a smooth pin at B, a point very near the
screw eye. Heat the wire by passing through it a current of
electricity from some appropriate source. The expansion
will be indicated by a wide sweep of the pointer. The
wire will cool quickly when the current is off, and the
pointer will return to its initial position.

The expansion of a bar may be conveniently illustrated
by supporting one end A rigidly, as by a weight (Fig. 7),
while the other end rests on a thin, straight sewing-needle,

which in turn lies on
a sheet of plate glass.
A slender pointer of
straw or foil may be
attached to the eye
of the needle by a bit

Fig. 7. of sealing wax, and it

should be counterbalanced.

When the bar is heated by a lamp or a Bunsen burner
it lengthens, and the free end advancing rolls the needle.
The movement of the pointer indicates the expansion. It
should return to its original position when the bar cools.

19. Relation between the Coefficients of Length and
of Volume (P., 199). The coefficient of volume-expan-
sion is three times that of linear expansion ; for the volume
of a cube, whose side is 1 at zero, is II (1 + at) 3 at t.
This volume is also V (1 + kfy. But V Q = 11. Hence

l + kt=(l + at) 3 =I + 3at + 3aV + a*t 3 .




)




EXPANSION. '2 1

Hut since a is a very small quantity, its higher powers
may be neglected in comparison with the first, or

1 + fo = 1 + 3a#,
and k = 3a nearly.

This relation assumes that the body is isotropic, or has
the same physical properties and expands equally in all
directions. In the case of crystals this is true only for
those of the regular cubic system, which do not cause
double refraction of light (I., 226). These dilate uniformly
in all directions in the same manner as amorphous bodies.

In general crystals have three rectangular axes of dilata-
tion, and the linear coefficients in these three directions
are not identical : the voluminal coefficient is then equal to
the sum of the three linear coefficients. It follows that a
crystalline sphere at one temperature ceases to be spherical
at any oth^r temperature, and a cubical portion of a crys-
talline body at one temperature will not remain cubical
when the temperature changes, unless the crystal belongs
'to the cubic system.

( 'rystals belonging to the rhombic system have an axis
of crystalline symmetry, and the two coefficients of expan-
sion perpendicular to this axis are equal, or the crystal has
the same properties in all directions perpendicular to the
axis of symmetry. In this case

k = a, -f- 2a., .

Here a l is the coefficient of expansion parallel to the axis,

and a, is the coefficient perpendicular to it.

Optically biaxial crystals dilate unequally in the direc-
tion of the three principal axes. Iceland spar and beryl
expand in the direction of their principal axis, but contract
transversely with rise of temperature.

Mitscherlich concluded that the effect of heat on crystals



28



HEAT.



is a tendency to separate the molecules in the direction in
which their distance is the least, so as to equalize their
distances, and to give to the crystal identical properties in
all directions.

If such crystals as quartz are strongly heated, their
unequal expansion in different directions causes them to
burst into small pieces.

2O. Measurement of Linear Expansion. All meth-

r ods of determining

coefficients of linear
expansion involve the
- exact measurement of
the change in length
of a body, or some
definite portion of it,
produced by a known
change of tempera-
ture. The variations
among them consist in
the methods adopted
to measure this change
of length.

By the "interfer-
ential method " of Professors Morley and Rogers the small
difference in the length of the two bars compared is
measured by counting the corresponding number of wave-
lengths of monochromatic light of known refrangibility.
The instrument by which such measurements are made
was invented by Professor Michelson, and is called the
" interferential comparator." The elements of it are shown
in plan in Fig. 8, where 6 is a plate of plane parallel glass
so silvered in front that half of an incident ray of light



V>



Fie. 8.



EXPANSION. 29

from S is transmitted and half is reflected. To the near
ends of the bars are attached plane mirrors 5 and 8, silvered
in front, and to the remote ends, 4 and 9, with the silvered
portion extending out at one side of the bar. At 7 is a
plate of plane parallel glass of the same thickness as 6,
but unsilvered.

In using the apparatus a ray of monochromatic light from
S is incident at 6. Half of it is reflected and goes to the
mirror 5, from which it is reflected back to 6, where half of
this reflected portion is transmitted and passes to the eye
of the observer at I. The transmitted half of the incident
ray at 6 is reflected from 8 back to 6, where half of it is
reflected and enters the eye along with the other com-
ponent from 5. Since the mirror 6 is silvered on the side
facing S, the portion of the light which returns from 8
traverses the glass 6 three times, while the first portion
reflected from 5 traverses it but once. Hence the plane
plate 7 is introduced to equalize the thickness of glass