+-----------+-------+--------+---------+----------+
| t deg. C. |Joules.| s. | h | Rowland. |
+-----------+-------+--------+---------+----------+
| 0 deg. | 4.208 | 1.0094 | 0 | 0 |
| 5 deg. | 4.202 | 1.0054 | 5.037 | 5.037 |
| 10 deg. | 4.191 | 1.0027 | 10.056 | 10.058 |
| 15 deg. | 4.184 | 1.0011 | 15.065 | 15.068 |
| 20 deg. | 4.180 | 1.0000 | 20.068 | 20.071 |
| 25 deg. | 4.177 | 0.9992 | 25.065 | 25.067 |
| 30 deg. | 4.175 | 0.9987 | 30.060 | 30.057 |
| 35 deg. | 4.173 | 0.9983 | 35.052 | 35.053 |
| 40 deg. | 4.173 | 0.9982 | 40.044 | |
| 50 deg. | 4.175 | 0.9987 | 50.028 | |
| 60 deg. | 4.180 | 1.0000 | 60.020 | |
| 70 deg. | 4.187 | 1.0016 | 70.028 | |
| 80 deg. | 4.194 | 1.0033 | 80.052 | |
| 90 deg. | 4.202 | 1.0053 | 90.095 | Shaw |
| 100 deg. | 4.211 | 1.0074 | 100.158 | Regnault |
| 120 deg. | 4.231 | 1.0121 | 120.35 | 120.73 |
| 140 deg. | 4.254 | 1.0176 | 140.65 | 140.88 |
| 160 deg. | 4.280 | 1.0238 | 161.07 | 161.20 |
| 180 deg. | 4.309 | 1.0308 | 181.62 | 182.14 |
| 200 deg. | 4.341 | 1.0384 | 202.33 | |
| 220 deg. | 4.376 | 1.0467 | 223.20 | |
+-----------+-------+--------+---------+----------+
The unit of comparison in the following table is taken as the specific
heat of water at 20 deg. C. for the reasons given below. This unit is
taken as being 4.180 joules per gramme-degree-centigrade on the scale
of the platinum thermometer, corrected to the absolute scale as
explained in the article THERMOMETRY, which has been shown to be
practically equivalent to the hydrogen scale. The value 4.180 joules
at 20 deg. C. is the mean between Rowland's corrected result 4.181 and
the value 4.179, deduced from the experiments of Reynolds and Moorby
on the assumption that the ratio of the mean specific heat 0 deg. to
100 deg. to that at 20 deg. is 1.0016, as given by the formulae
representing the results of Callendar and Barnes. This would indicate
that Rowland's corrected values should, if anything, be lowered. In
any case the value of the mechanical equivalent is uncertain to at
least 1 in 2000.
The mean specific heat, over any range of temperature, may be obtained
by integrating the formulae between the limits required, or by taking
the difference of the corresponding values of the total heat h, and
dividing by the range of temperature. The quantity actually observed
by Rowland was the total heat. It may be remarked that starting from
the same value at 5 deg., for the sake of comparison, Rowland's values
of the total heat agree to 1 in 5000 with those calculated from the
formulae. The values of the total heat observed by Regnault, as
reduced by Shaw, also show a very fair agreement, considering the
uncertainty of the units. It must be admitted that it is desirable to
redetermine the variation of the specific heat above 100 deg. C. This
is very difficult on account of the steam-pressure, and could not
easily be accomplished by the electrical method. Callendar has,
however, devised a continuous method of mixture, which appears to be
peculiarly adapted to the purpose, and promises to give more certain
results. In any case it may be remarked that formulae such as those of
Jamin, Henrichsen, Baumgartner, Winkelmann or Dieterici, which give
far more rapid rates of increase than that of Regnault, cannot
possibly be reconciled with his observations, or with those of
Reynolds and Moorby, or Callendar and Barnes, and are certainly
inapplicable above 100 deg. C.
S 16. _On the Choice of the Thermal Unit._--So much uncertainty still
prevails on this fundamental point that it cannot be passed over without
reference. There are three possible kinds of unit, depending on the
three fundamental methods already given: (1) the thermometric unit, or
the thermal capacity of unit mass of a standard substance under given
conditions of temperature and pressure on the scale of a standard
thermometer. (2) The latent-heat unit, or the quantity of heat required
to melt or vaporize unit mass of a standard substance under given
conditions. This unit has the advantage of being independent of
thermometry, but the applicability of these methods is limited to
special cases, and the relation of the units to other units is difficult
to determine. (3) The absolute or mechanical unit, the quantity of heat
equivalent to a given quantity of mechanical or electrical energy. This
can be very accurately realized, but is not so convenient as (1) for
ordinary purposes.
In any case it is necessary to define a thermometric unit of class
(1). The standard substance must be a liquid. Water is always
selected, although some less volatile liquid, such as aniline or
mercury, would possess many advantages. With regard to the scale of
temperature, there is very general agreement that the absolute scale
as realized by the hydrogen or helium thermometer should be adopted as
the ultimate standard of reference. But as the hydrogen thermometer is
not directly available for the majority of experiments, it is
necessary to use a secondary standard for the practical definition of
the unit. The electrical resistance thermometer of platinum presents
very great advantages for this purpose over the mercury thermometer in
point of reproducibility, accuracy and adaptability to the practical
conditions of experiment. The conditions of use of a mercury
thermometer in a calorimetric experiment are necessarily different
from those under which its corrections are determined, and this
difference must inevitably give rise to constant errors in practical
work. The primary consideration in the definition of a unit is to
select that method which permits the highest order of accuracy in
comparison and verification. For this reason the definition of the
thermal unit will in the end probably be referred to a scale of
temperature defined in terms of a standard platinum thermometer.
There is more diversity of opinion with regard to the question of the
standard temperature. Many authors, adopting Regnault's formula, have
selected 0 deg. C. as the standard temperature, but this cannot be
practically realized in the case of water, and his formula is
certainly erroneous at low temperatures. A favourite temperature to
select is 4 deg. C., the temperature of maximum density, since at this
point the specific heat at constant volume is the same as that at
constant pressure But this is really of no consequence, since the
specific heat at constant volume cannot be practically realized. The
specific heat at 4 deg. could be accurately determined at the mean
over the range 0 deg. to 8 deg. keeping the jacket at 0 deg. C. But
the change appears to be rather rapid near 0 deg., the temperature is
inconveniently low for ordinary calorimetric work, and the unit at 4
deg. would be so much larger than the specific heat at ordinary
temperatures that nearly all experiments would require reduction. The
natural point to select would be that of minimum specific heat, but if
this occurs at 40 deg. C. it would be inconveniently high for
practical realization except by the continuous electrical method. It
was proposed by a committee of the British Association to select the
temperature at which the specific heat was 4.200 joules, leaving the
exact temperature to be subsequently determined. It was supposed at
the time, from the original reduction of Rowland's experiments, that
this would be nearly at 10 deg. C., but it now appears that it may be
as low is 5 deg. C., which would be inconvenient. This is really only
an absolute unit in disguise, and evades the essential point, which is
the selection of a standard temperature for the water thermometric
unit. A similar objection applies to selecting the temperature at
which the specific heat is equal to its mean value between 0 deg. and
100 deg. The mean calorie cannot be accurately realized in practice
in any simple manner, and is therefore unsuitable as a standard of
comparison. Its relation to the calorie at any given temperature, such
as 15 deg. or 20 deg., cannot be determined with the same degree of
accuracy as the ratio of the specific heat at 15 deg. to that at 20
deg., if the scale of temperature is given. The most practical unit is
the calorie at 15 deg. or 20 deg. or some temperature in the range of
ordinary practice. The temperature most generally favoured is 15 deg.,
but 20 deg. would be more suitable for accurate work. These units
differ only by 11 parts in 10,000 according to Callendar and Barnes,
or by 13 in 10,000 according to Rowland and Griffiths, so that the
difference between them is of no great importance for ordinary
purposes. But for purposes of definition it would be necessary to take
the mean value of the specific heat _over a given range_ of
temperature, preferably at least 10 deg., rather than the specific
heat _at a point_ which necessitates reference to some formula of
reduction for the rate of variation. The specific heat at 15 deg.
would be determined with reference to the mean over the range 10 deg.
to 20 deg., and that at 20 deg. from the range 15 deg. to 25 deg.
There can be no doubt that the range 10 deg. to 20 deg. is too low for
the accurate thermal regulation of the conditions of the experiment.
The range 15 deg. to 25 deg. would be much more convenient from this
point of view, and a mean temperature of 20 deg. is probably nearest
the average of accurate calorimetric work. For instance 20 deg. is the
mean of the range of the experiments of Griffiths and of Rowland, and
is close to that of Schuster and Gannon. It is readily attainable at
any time in a modern laboratory with adequate heating arrangements,
and is probably on the whole the most suitable temperature to select.
S 17. _Specific Heat of Gases._--In the case of solids and liquids under
ordinary conditions of pressure, the external work of expansion is so
small that it may generally be neglected; but with gases or vapours, or
with liquids near the critical point, the external work becomes so large
that it is essential to specify the conditions under which the specific
heat is measured. The most important cases are, the specific heats (1)
at constant volume; (2) at constant pressure; (3) at saturation pressure
in the case of a liquid or vapour. In consequence of the small thermal
capacity of gases and vapours per unit volume at ordinary pressures, the
difficulties of direct measurement are almost insuperable except in case
(2). Thus the direct experimental evidence is somewhat meagre and
conflicting, but the question of the relation of the specific heats of
gases is one of great interest in connexion with the kinetic theory and
the constitution of the molecule. The well-known experiments of Regnault
and Wiedemann on the specific heat of gases at constant pressure agree
in showing that the _molecular specific heat_, or the thermal capacity
of the molecular weight in grammes, is approximately independent of the
temperature and pressure in case of the more stable diatomic gases, such
as H2, O2, N2, CO, &c., and has nearly the same value for each gas. They
also indicate that it is much larger, and increases considerably with
rise of temperature, in the case of more condensible vapours, such as
Cl2, Br2, or more complicated molecules, such as CO2, N2O, NH3, C2H4.
The direct determination of the specific heat at constant volume is
extremely difficult, but has been successfully attempted by Joly with
his steam calorimeter, in the case of air and CO2. Employing pressures
between 7 and 27 atmospheres, he found that the specific heat of air
between 10 deg. and 100 deg. C. increased very slightly with increase of
density, but that of CO2 increased nearly 3% between 7 and 21
atmospheres. The following formulae represent his results for the
specific heat s at constant volume in terms of the density d in gms. per
c. c.:--
Air, s = 0.1715 + 0.028d,
CO2, s = 0.165 + 0.213d + 0.34d^2.
S 18. _Ratio of Specific Heats._--According to the elementary kinetic
theory of an ideal gas, the molecules of which are so small and so far
apart that their mutual actions may be neglected, the kinetic energy
of translation of the molecules is proportional to the absolute
temperature, and is equal to 3/2 of pv, the product of the pressure
and the volume, per unit mass. The expansion per degree at constant
pressure is v/[theta] = R/p. The external work of expansion per degree
is equal to R, being the product of the pressure and the expansion,
and represents the difference of the specific heats S - s, at constant
pressure and volume, assuming as above that the internal work of
expansion is negligible. If the molecules are supposed to be like
smooth, hard, elastic spheres, incapable of receiving any other kind
of energy except that of translation, the specific heat at constant
volume would be the increase per degree of the kinetic energy namely
3pv/2[theta] - 3R/2, that at constant pressure would be 5R/2, and the
ratio of the specific heats would be 5/3 or 1.666. This appears to be
actually the case for monatomic gases such as mercury vapour (Kundt
and Warburg, 1876), argon and helium (Ramsay, 1896). For diatomic or
compound gases Clerk Maxwell supposed that the molecule would also
possess energy of rotation, and endeavoured to prove that in this case
the energy would be equally divided between the six degrees of
freedom, three of translation and three of rotation, if the molecule
were regarded as a rigid body incapable of vibration-energy. In this
case we should have s = 3R, S = 4R, S/s = 4/3 = 1.333. In 1879 Maxwell
considered it one of the greatest difficulties which the kinetic
theory had yet encountered, that in spite of the many other degrees of
freedom of vibration revealed by the spectroscope, the experimental
value of the ratio S/s was 1.40 for so many gases, instead of being
less than 4/3. Somewhat later L. Boltzmann suggested that a diatomic
molecule regarded as a rigid dumb-bell or figure of rotation, might
have only five effective degrees of freedom, since the energy of
rotation about the axis of symmetry could not be altered by collisions
between the molecules. The theoretical value of the ratio S/s in this
case would be the required 7/5. For a rigid molecule on this theory
the smallest value possible would be 4/3. Since much smaller values
are found for more complex molecules, we may suppose that, in these
cases, the energy of rotation of a polyatomic molecule may be greater
than its energy of translation, or else that heat is expended in
splitting up molecular aggregates, and increasing energy of vibration.
A hypothesis doubtfully attributed to Maxwell is that each additional
atom in the molecule is equivalent to two extra degrees of freedom.
From an m-atomic molecule we should then have S/s = 1 + 2/(2m + 1).
This gives a series of ratios 5/3, 7/5, 9/7, 11/9, &c., for 1, 2, 3,
4, &c., atoms in the molecule, values which fall within the limits of
experimental error in many cases. It is not at all clear, however,
that energy of vibration should bear a constant ratio to that of
translation, although this would probably be the case for rotation.
For the simpler gases, which are highly diathermanous and radiate
badly even at high temperature, the energy of vibration is probably
very small, except under the special conditions which produce
luminosity in flames and electric discharges. For such gases, assuming
a constant ratio of rotation to translation, the specific heat at low
pressures would be very nearly constant. For more complex molecules
the radiative and absorptive powers are known to be much greater. The
energy of vibration may be appreciable at ordinary temperatures, and
would probably increase more rapidly than that of translation with
rise of temperature, especially near a point of dissociation. This
would account for an increase of S, and a diminution of the ratio S/s,
with rise of temperature which apparently occurs in many vapours. The
experimental evidence, however, is somewhat conflicting, and further
investigations are very desirable on the variation of specific heat
with temperature. Given the specific heat as a function of the
temperature, its variation with pressure may be determined from the
characteristic equation of the gas. The direct methods of measuring
the ratio S/s, by the velocity of sound and by adiabatic expansion,
are sufficiently described in many text-books.
S 19. _Atomic and Molecular Heats._--The ideal atomic heat is the
thermal capacity of a gramme-atom in the ideal state of monatomic gas
at constant volume. This would be nearly three calories. For a
diatomic gas, the molecular heat would be nearly five calories, or the
atomic heat of a gas in the diatomic state would be 2.5. Estimated at
constant pressure the atomic heat would be 3.5. Some authors adopt 2.5
and some 3.5 for the ideal atomic heat. The atomic heat of a metal in
the solid state is in most cases larger than six calories at ordinary
temperatures. Considering the wide variations in the physical
condition and melting points, the comparatively close agreement of the
atomic heats of the metals at ordinary temperatures, known as Dulong
and Petit's Law, is very remarkable. The specific heats as a rule
increase with rise of temperature, in some cases, e.g. iron and
nickel, very rapidly. According to W.A. Tilden (_Phil. Trans._, 1900),
the atomic heats of pure nickel and cobalt, as determined from
experiments at the boiling-points of O2, and CO2, diminish so rapidly
at temperatures below 0 deg. C. as to suggest that they would reach
the value 2.42 at the absolute zero. This is the value of the minimum
of atomic heat calculated by Perry from diatomic hydrogen, but the
observations themselves might be equally well represented by taking
the imaginary limit 3, since the quantity actually observed is the
mean specific heat between 0 deg. and -182.5 deg. C. Subsequent
experiments on other metals at low temperatures did not indicate a
similar diminution of specific heat, so that it may be doubted whether
the atomic heats really approach the ideal value at the absolute zero.
No doubt there must be approximate relations between the atomic and
molecular heats of similar elements and compounds, but considering the
great variations of specific heat with temperature and physical state,
in alloys, mixtures or solutions, and in allotropic or other
modifications, it would be idle to expect that the specific heat of a
compound could be accurately deduced by any simple additive process
from that of its constituents.
AUTHORITIES.--Joule's _Scientific Papers_ (London, 1890); Ames and
Griffiths, _Reports to the International Congress_ (Paris, 1900), "On
the Mechanical Equivalent of Heat," and "On the Specific Heat of
Water"; Griffiths, _Thermal Measurement of Energy_ (Cambridge, 1901);
Callendar and Barnes, _Phil. Trans._ A, 1901, "On the Variation of the
Specific Heat of Water"; for combustion methods, see article
THERMOCHEMISTRY, and treatises by Thomsen, Pattison-Muir and
Berthelot; see also articles THERMODYNAMICS and VAPORIZATION.
(H. L. C.)
CALOVIUS, ABRAHAM (1612-1686), German Lutheran divine, was born at
Mohrungen in east Prussia, on the 16th of April 1612. After studying at
Konigsberg, in 1650 he was appointed professor of theology at
Wittenberg, where he afterwards became general superintendent and
primarius. He died on the 25th of February 1686. Calovius was the most
noteworthy of the champions of Lutheran orthodoxy in the 17th century.
He strongly opposed the Catholics, Calvinists and Socinians, attacked in
particular the reconciliation policy or "syncretism" of Georg Calixtus
(cf. the _Consensus repetitus fidei vere lutheranae_, 1665), and as a
writer of polemics he had few equals. His chief dogmatic work, _Systema
locorum theologicorum_ (12 vols. 1655-1677), represents the climax of
Lutheran scholasticism. In his _Biblia Illustrata_ (4 vols.), written
from the point of view of a very strict belief in inspiration, his
object is to refute the statements made by Hugo Grotius in his
Commentaries. His _Historia Syncretistica_ (1682) was suppressed.
CALPURNIUS, TITUS, Roman bucolic poet, surnamed SICULUS from his
birthplace or from his imitation of the style of the Sicilian
Theocritus, most probably flourished during the reign of Nero. Eleven
eclogues have been handed down to us under his name, of which the last
four, from metrical considerations and express MS. testimony, are now
generally attributed to Nemesianus (q.v.), who lived in the time of the
emperor Carus and his sons (latter half of the 3rd century A.D.). Hardly
anything is known of the life of Calpurnius; we gather from the poems
themselves (in which he is obviously represented by "Corydon") that he
was in poor circumstances and was on the point of emigrating to Spain,
when "Meliboeus" came to his aid. Through his influence Calpurnius
apparently secured a post at Rome. The time at which Calpurnius lived
has been much discussed, but all the indications seem to point to the
time of Nero. The emperor is described as a handsome youth, like Mars
and Apollo, whose accession marks the beginning of a new golden age,
prognosticated by the appearance of a comet, doubtless the same that
appeared some time before the death of Claudius; he exhibits splendid
games in the amphitheatre (probably the wooden amphitheatre erected by
Nero in 57); and in the words
maternis causam qui vicit Iulis[1] (i. 45),
there is a reference to the speech delivered in Greek by Nero on behalf
of the Ilienses (Suetonius, _Nero_, 7; Tacitus, _Annals_, xii. 58), from
whom the Julii derived their family.[2] Meliboeus, the poet's patron,
has been variously identified with Columella, Seneca the philosopher,
and C. Calpurnius Piso. Although the sphere of Meliboeus's literary
activity (as indicated in iv. 53) suits none of these, what is known of
Calpurnius Piso fits in well with what is said of Meliboeus by the poet,
who speaks of his generosity, his intimacy with the emperor, and his
interest in tragic poetry. His claim is further supported by the poem
_De Laude Pisonis_ (ed. C.F. Weber, 1859) which has come down to us
without the name of the author, but which there is considerable reason
for attributing to Calpurnius.[3] The poem exhibits a striking
similarity with the eclogues in metre, language and subject-matter. The
author of the _Laus_ is young, of respectable family and desirous of
gaining the favour of Piso as his Maecenas. Further, the similarity
between the two names can hardly be accidental; it is suggested that the
poet may have been adopted by the courtier, or that he was the son of a
freedman of Piso. The attitude of the author of the _Laus_ towards the
subject of the panegyric seems to show less intimacy than the relations
between Corydon and Meliboeus in the eclogues, and there is internal
evidence that the _Laus_ was written during the reign of Claudius
(Teuffel-Schwabe, _Hist, of Rom. Lit._ S 306, 6).
Mention may here be made of the fragments of two short hexameter poems
in an Einsiedeln MS., obviously belonging to the time of Nero, which if
not written by Calpurnius, were imitated from him.
Although there is nothing original in Calpurnius, he is "a skilful
literary craftsman." Of his models the chief is Virgil, of whom (under
the name of Tityrus) he speaks with great enthusiasm; he is also
indebted to Ovid and Theocritus. Calpurnius is "a fair scholar, and an
apt courtier, and not devoid of real poetical feeling. The bastard style
of pastoral cultivated by him, in which the description of nature is
made the writer's pretext, while ingenious flattery is his real purpose,
nevertheless excludes genuine pleasure, and consequently genuine
poetical achievement. He may be fairly compared to the minor poets of
the reign of Anne" (Garnett).
Calpurnius was first printed in 1471, together with Silius Italicus
and has been frequently republished, generally with Gratius Faliscus
and Nemesianus. The separate authorship of the eclogues of Calpurnius
and Nemesianus was established by M. Haupt's _De Carminibus bucolicis
Calpurnii et Nemesiani_ (1854). Editions by H. Schenkl (1885), with
full introduction and _index verborum_, and by C.H. Keene (1887), with
introduction, commentary and appendix. English verse translation by
E.J.L. Scott (1891); see H.E. Butler, _Post-Augustan Poetry_ (Oxford,
1909), pp. 150 foil., and F. Skutsch in Pauly-Wissowa's
_Realencyclopadie_, iii. 1 (1897). (J. H. F.)