+-----------+---------------+------+-------------+----------------+
|Flow of Hg.| Rise of Temp. |Watts.| Heat-loss. | Specific Heat. |
+-----------+---------------+------+-------------+----------------+
| gm./sec. | d[theta] | EC | hd[theta] | Per gm. deg. |
| 8.753 | 11.764 |14.862| 0.655 | \ .13780 joules|
| 4.594 | 12.301 | 7.912| 0.865 | / .03297 cals. |
+-----------+---------------+------+-------------+----------------+
It is assumed as a first approximation that the heat-loss is
proportional to the rise of temperature _d[theta]_, provided that
_d[theta]_ is nearly the same in both cases, and that the distribution
of temperature in the apparatus is the same for the same rise of
temperature whatever the flow of liquid. The result calculated on
these assumptions is given in the last column in joules, and also in
calories of 20 deg. C. The heat-loss in this example is large, nearly
4.5% of the total supply, owing to the small flow and the large rise
of temperature, but this correction was greatly reduced in subsequent
observations on the specific heat of water by the same method. In the
case of mercury the liquid itself can be utilized to conduct the
electric current. In the case of water or other liquids it is
necessary to employ a platinum wire stretched along the tube as
heating conductor. This introduces additional difficulties of
construction, but does not otherwise affect the method. The absolute
value of the specific heat deduced necessarily depends on the absolute
values of the electrical standards employed in the investigation. But
for the determination of relative values of specific heats in terms of
a standard liquid, or of the variations of specific heat of a liquid,
the method depends only on the constancy of the standards, which can
be readily and accurately tested. The absolute value of the E.M.F. of
the Clark cells employed was determined with a special form of
electrodynamometer (Callendar, _Phil. Trans._ A. 313, p. 81), and
found to be 1.4334 volts, assuming the ohm to be correct. Assuming
this value, the result found by this method for the specific heat of
water at 20 deg. C. agrees with that of Rowland within the probable
limits of error.
S 15. _Variation of Specific Heat of Water._--The question of the
variation of the specific heat of water has a peculiar interest and
importance in connexion with the choice of a thermal unit. Many of the
uncertainties in the reduction of older experiments, such as those of
Regnault, arise from uncertainty in regard to the unit in terms of
which they are expressed, which again depends on the scale of the
particular thermometer employed in the investigation. The first
experiments of any value were those of Regnault in 1847 on the
specific heat of water between 110 deg. C. and 192 deg. C. They were
conducted on a very large scale by the method of mixture, but showed
discrepancies of the order of 0.5%, and the calculated results in many
cases do not agree with the data. This may be due merely to deficient
explanation of details of tabulation. We may probably take the
tabulated values as showing correctly the rate of variation between
110 deg. and 190 deg. C., but the values in terms of any particular
thermal unit must remain uncertain to at least 0.5% owing to the
uncertainties of the thermometry. Regnault himself adopted the
formula,
s = 1 + 0.00004t + 0.0000009t^2 (Regnault), (3)
for the specific heat _s_ at any temperature _t_ C. in terms of the
specific heat at 0 deg. C. taken as the standard. This formula has
since been very generally applied over the whole range 0 deg. to 200
deg. C., but the experiments could not in reality give any information
with regard to the specific heat at temperatures below 100 deg. C. The
linear formula proposed by J. Bosscha from an independent reduction of
Regnault's experiments is probably within the limits of accuracy
between 100 deg. and 200 deg. C., so far as the mean rate of variation
is concerned, but the absolute values require reduction. It may be
written--
s = S_100 + .00023(t - 100) (Bosscha-Regnault) (4).
The work of L. Pfaundler and H. Platter, of G.A. Hirn, of J.C. Jamin
and Amaury, and of many other experimentalists who succeeded Regnault,
appeared to indicate much larger rates of increase than he had found,
but there can be little doubt that the discrepancies of their results,
which often exceeded 5%, were due to lack of appreciation of the
difficulties of calorimetric measurements. The work of Rowland by the
mechanical method was the first in which due attention was paid to the
thermometry and to the reduction of the results to the absolute scale
of temperature. The agreement of his corrected results with those of
Griffiths by a very different method, left very little doubt with
regard to the rate of diminution of the specific heat of water at 20
deg. C. The work of A. Bartoli and E. Stracciati by the method of
mixture between 0 deg. and 30 deg. C., though their curve is otherwise
similar to Rowland's, had appeared to indicate a minimum at 20 deg.
C., followed by a rapid rise. This lowering of the minimum was
probably due to some constant errors inherent in their method of
experiment. The more recent work of Ludin, 1895, under the direction
of Prof. J. Pernet, extended from 0 deg. to 100 deg. C., and appears
to have attained as high a degree of excellence as it is possible to
reach by the employment of mercury thermometers in conjunction with
the method of mixture. His results, exhibited in fig. 6, show a
minimum at 25 deg. C., and a maximum at 87 deg. C., the values being
.9935 and 1.0075 respectively in terms of the mean specific heat
between 0 deg. and 100 deg. C. He paid great attention to the
thermometry, and the discrepancies of individual measurements at any
one point nowhere exceed 0.3%, but he did not vary the conditions of
the experiments materially, and it does not appear that the well-known
constant errors of the method could have been completely eliminated by
the devices which he adopted. The rapid rise from 25 deg. to 75 deg.
may be due to radiation error from the hot water supply, and the
subsequent fall of the curve to the inevitable loss of heat by
evaporation of the boiling water on its way to the calorimeter. It
must be observed, however, that there is another grave difficulty in
the accurate determination of the specific heat of water near 100 deg.
C. by this method, namely, that the quantity actually observed is not
the specific heat _at_ the higher temperature _t_, but the _mean
specific heat_ over the range 18 deg. to _t_. The specific heat itself
can be deduced only by differentiating the curve of observation, which
greatly increases the uncertainty. The peculiar advantage of the
electric method of Callendar and Barnes, already referred to, is that
the specific heat itself is determined over a range of 8 deg. to 10
deg. at each point, by adding accurately measured quantities of heat
to the water at the desired temperature in an isothermal enclosure,
under perfectly steady conditions, without any possibility of
evaporation or loss of heat in transference. These experiments, which
have been extended by Barnes over the whole range 0 deg. to 100 deg.,
agree very well with Rowland and Griffiths in the rate of variation at
20 deg. C., but show a rather flat minimum of specific heat in the
neighbourhood of 38 deg. to 40 deg. C. At higher points the rate of
variation is very similar to that of Regnault's curve, but taking the
specific heat at 20 deg. as the standard of reference, the actual
values are nearly 0.56% less than Regnault's. It appears probable that
his values for higher temperatures may be adopted with this reduction,
which is further confirmed by the results of Reynolds and Moorby, and
by those of Ludin. According to the electric method, the whole range
of variation of the specific heat between 10 deg. and 80 deg. is only
0.5%. Comparatively simple formulae, therefore, suffice for its
expression to 1 in 10,000, which is beyond the limits of accuracy of
the observations. It is more convenient in practice to use a few
simple formulae, than to attempt to represent the whole range by a
single complicated expression:--
Below 20 deg. C. s = 0.9982 + 0.0000045(t - 40)^2 - 0.0000005(t - 20)^3.
From 20 deg. to 60 deg., s = 0.9982 + 0.0000045(t - 40)^2 (5).
/ s = 0.9944 + .00004t + 0.0000009t^2
Above 60 deg. to 200 deg. < (Regnault corrd.)
\ s = 1.000 + 0.00022(t - 60), (Bosscha corrd.)
The addition of the cubic term below 20 deg. is intended to represent
the somewhat more rapid change near the freezing-point. This effect is
probably due, as suggested by Rowland, to the presence of a certain
proportion of ice molecules in the liquid, which is also no doubt the
cause of the anomalous expansion. Above 60 deg. C. Regnault's formula
is adopted, the absolute values being simply diminished by a constant
quantity 0.0056 to allow for the probable errors of his thermometry.
Above 100 deg. C., and for approximate work generally, the simpler
formula of Bosscha, similarly corrected, is probably adequate.
The following table of values, calculated from these formulae, is
taken from the _Brit. Assoc. Report_, 1899, with a slight modification
to allow for the increase in the specific heat below 20 deg. C. This
was estimated in 1899 as being equivalent to the addition of the
constant quantity 0.20 to the values of the total heat h of the liquid
as reckoned by the parabolic formula (5). This quantity is now, as the
result of further experiments, added to the values of h, and also
represented in the formula for the specific heat itself by the cubic
term.