38. The calculation of the mechanical effect, in any case, which might
always be effected in the manner described in § 37 (with the proper
modification for fractions of degrees, when necessary), is much
simplified by the use of Table II., where the first number of Table I.,
the sum of the first and second, the sum of the first three, the sum of
the first four, and so on, are successively exhibited. The sums thus
tabulated are the values of the integrals
∫_{0}^1 μ_dt_, ∫_{0}^2 μ_dt_, ∫_{0}^3 μ_dt_, ... ∫_{0}^{231} μ_dt_;
and, if we denote ∫_{0}^t μ_dt_ by the letter _M_, Table II. may be
regarded as a table of the value of _M_.
_To find the amount of mechanical effect due to a unit of heat
descending from a body at a temperature S to a body at T, if these
numbers be integers, we have merely to subtract the value of M, for the
number T, from the value for the number S, given in Table II._
TABLE I.[56]
MEAN VALUES OF Μ FOR THE SUCCESSIVE DEGREES OF THE AIR-THERMOMETER FROM
0° TO 230°.
───────────────────────────────────┬───────────────────────────────────
° │ μ
───────────────────────────────────┼───────────────────────────────────
1│ 4.960
2│ 4.946
3│ 4.932
4│ 4.918
5│ 4.905
6│ 4.892
7│ 4.878
8│ 4.865
9│ 4.852
10│ 4.839
11│ 4.826
12│ 4.812
13│ 4.799
14│ 4.786
15│ 4.773
16│ 4.760
17│ 4.747
18│ 4.735
19│ 4.722
20│ 4.709
21│ 4.697
22│ 4.684
23│ 4.672
24│ 4.659
25│ 4.646
26│ 4.634
27│ 4.621
28│ 4.609
29│ 4.596
30│ 4.584
31│ 4.572
32│ 4.559
33│ 4.547
34│ 4.535
35│ 4.522
36│ 4.510
37│ 4.498
38│ 4.486
39│ 4.474
40│ 4.462
41│ 4.450
42│ 4.438
43│ 4.426
44│ 4.414
45│ 4.402
46│ 4.390
47│ 4.378
48│ 4.366
49│ 4.355
50│ 4.343
51│ 4.331
52│ 4.319
53│ 4.308
54│ 4.296
55│ 4.285
56│ 4.273
57│ 4.262
58│ 4.250
59│ 4.239
60│ 4.227
61│ 4.216
62│ 4.205
63│ 4.194
64│ 4.183
65│ 4.172
66│ 4.161
67│ 4.150
68│ 4.140
69│ 4.129
70│ 4.119
71│ 4.109
72│ 4.098
73│ 4.088
74│ 4.078
75│ 4.067
76│ 4.057
77│ 4.047
78│ 4.037
79│ 4.028
80│ 4.018
81│ 4.009
82│ 3.999
83│ 3.990
84│ 3.980
85│ 3.971
86│ 3.961
87│ 3.952
88│ 3.943
89│ 3.934
90│ 3.925
91│ 3.916
92│ 3.907
93│ 3.898
94│ 3.889
95│ 3.880
96│ 3.871
97│ 3.863
98│ 3.854
99│ 3.845
100│ 3.837
101│ 3.829
102│ 3.820
103│ 3.812
104│ 3.804
105│ 3.796
106│ 3.788
107│ 3.780
108│ 3.772
109│ 3.764
110│ 3.757
111│ 3.749
112│ 3.741
113│ 3.734
114│ 3.726
115│ 3.719
116│ 3.712
117│ 3.704
118│ 3.697
119│ 3.689
120│ 3.682
121│ 3.675
122│ 3.668
123│ 3.661
124│ 3.654
125│ 3.647
126│ 3.640
127│ 3.633
128│ 3.627
129│ 3.620
130│ 3.614
131│ 3.607
132│ 3.601
133│ 3.594
134│ 3.586
135│ 3.579
136│ 3.573
137│ 3.567
138│ 3.561
139│ 3.555
140│ 3.549
141│ 3.543
142│ 3.537
143│ 3.531
144│ 3.525
145│ 3.519
146│ 3.513
147│ 3.507
148│ 3.501
149│ 3.495
150│ 3.490
151│ 3.484
152│ 3.479
153│ 3.473
154│ 3.468
155│ 3.462
156│ 3.457
157│ 3.451
158│ 3.446
159│ 3.440
160│ 3.435
161│ 3.430
162│ 3.424
163│ 3.419
164│ 3.414
165│ 3.409
166│ 3.404
167│ 3.399
168│ 3.394
169│ 3.389
170│ 3.384
171│ 3.380
172│ 3.375
173│ 3.370
174│ 3.365
175│ 3.361
176│ 3.356
177│ 3.351
178│ 3.346
179│ 3.342
180│ 3.337
181│ 3.332
182│ 3.328
183│ 3.323
184│ 3.318
185│ 3.314
186│ 3.309
187│ 3.304
188│ 3.300
189│ 3.295
190│ 3.291
191│ 3.287
192│ 3.282
193│ 3.278
194│ 3.274
195│ 3.269
196│ 3.265
197│ 3.261
198│ 3.257
199│ 3.253
200│ 3.249
201│ 3.245
202│ 3.241
203│ 3.237
204│ 3.233
205│ 3.229
206│ 3.225
207│ 3.221
208│ 3.217
209│ 3.213
210│ 3.210
211│ 3.206
212│ 3.202
213│ 3.198
214│ 3.195
215│ 3.191
216│ 3.188
217│ 3.184
218│ 3.180
219│ 3.177
220│ 3.173
221│ 3.169
222│ 3.165
223│ 3.162
224│ 3.158
225│ 3.155
226│ 3.151
227│ 3.148
228│ 3.144
229│ 3.141
230│ 3.137
231│ 3.134
───────────────────────────────────┴───────────────────────────────────
TABLE II.
MECHANICAL EFFECT IN FOOT-POUNDS DUE TO A THERMIC UNIT CENTIGRADE,
PASSING FROM A BODY, AT ANY TEMPERATURE LESS THAN 230° TO A BODY AT 0°.
───────────────────────────────────┬───────────────────────────────────
Superior Limit of Temperature. │ Mechanical Effect.
───────────────────────────────────┼───────────────────────────────────
° │ Ft.-Pounds.
│
1│ 4.960
2│ 9.906
3│ 14.838
4│ 19.756
5│ 24.661
6│ 29.553
7│ 34.431
8│ 39.296
9│ 44.148
10│ 48.987
11│ 53.813
12│ 58.625
13│ 63.424
14│ 68.210
15│ 72.983
16│ 77.743
17│ 82.490
18│ 87.225
19│ 91.947
20│ 96.656
21│ 101.353
22│ 106.037
23│ 110.709
24│ 115.368
25│ 120.014
26│ 124.648
27│ 129.269
28│ 133.878
29│ 138.474
30│ 143.058
31│ 147.630
32│ 152.189
33│ 156.736
34│ 161.271
35│ 165.793
36│ 170.303
37│ 174.801
38│ 179.287
39│ 183.761
40│ 188.223
41│ 192.673
42│ 197.111
43│ 201.537
44│ 205.951
45│ 210.353
46│ 214.743
47│ 219.121
48│ 223.487
49│ 227.842
50│ 232.185
51│ 236.516
52│ 240.835
53│ 245.143
54│ 249.439
55│ 253.724
56│ 257.997
57│ 262.259
58│ 266.509
59│ 270.748
60│ 274.975
61│ 279.191
62│ 283.396
63│ 287.590
64│ 291.773
65│ 295.945
66│ 300.106
67│ 304.256
68│ 308.396
69│ 312.525
70│ 316.644
71│ 320.752
72│ 324.851
73│ 328.939
74│ 333.017
75│ 337.084
76│ 341.141
77│ 345.188
78│ 349.225
79│ 353.253
80│ 357.271
81│ 361.280
82│ 365.279
83│ 369.269
84│ 373.249
85│ 377.220
86│ 381.181
87│ 385.133
88│ 389.076
89│ 393.010
90│ 396.935
91│ 400.851
92│ 404.758
93│ 408.656
94│ 412.545
95│ 416.425
96│ 420.296
97│ 424.159
98│ 428.013
99│ 431.858
100│ 435.695
101│ 439.524
102│ 443.344
103│ 447.156
104│ 450.960
105│ 454.756
106│ 458.544
107│ 462.324
108│ 466.096
109│ 469.860
110│ 473.617
111│ 477.366
112│ 481.107
113│ 484.841
114│ 488.567
115│ 492.286
116│ 495.998
117│ 499.702
118│ 503.399
119│ 507.088
120│ 510.770
121│ 514.445
122│ 518.113
123│ 521.174
124│ 525.428
125│ 529.075
126│ 532.715
127│ 536.348
128│ 539.975
129│ 543.595
130│ 547.209
131│ 550.816
132│ 554.417
133│ 558.051
134│ 561.597
135│ 565.176
136│ 568.749
137│ 572.316
138│ 575.877
139│ 579.432
140│ 582.981
141│ 586.524
142│ 590.061
143│ 593.592
144│ 597.117
145│ 600.636
146│ 604.099
147│ 607.656
148│ 611.157
149│ 614.652
150│ 618.142
151│ 621.626
152│ 625.105
153│ 628.578
154│ 632.046
155│ 635.508
156│ 638.965
157│ 642.416
158│ 645.862
159│ 649.302
160│ 652.737
161│ 656.167
162│ 659.591
163│ 663.010
164│ 666.424
165│ 669.833
166│ 673.237
167│ 676.636
168│ 680.030
169│ 683.419
170│ 686.803
171│ 690.183
172│ 693.558
173│ 696.928
174│ 700.293
175│ 703.654
176│ 707.010
177│ 710.361
178│ 713.707
179│ 717.049
180│ 720.386
181│ 723.718
182│ 727.046
183│ 730.369
184│ 733.687
185│ 737.001
186│ 740.310
187│ 743.614
188│ 746.914
189│ 750.209
190│ 753.500
191│ 756.787
192│ 760.069
193│ 763.347
194│ 766.621
195│ 769.890
196│ 773.155
197│ 776.416
198│ 779.673
199│ 782.926
200│ 786.175
201│ 789.420
202│ 792.661
203│ 795.898
204│ 799.131
205│ 802.360
206│ 805.585
207│ 808.806
208│ 812.023
209│ 815.236
210│ 818.446
211│ 821.652
212│ 824.854
213│ 828.052
214│ 831.247
215│ 834.438
216│ 837.626
217│ 840.810
218│ 843.990
219│ 847.167
220│ 850.340
221│ 853.509
222│ 856.674
223│ 859.836
224│ 862.994
225│ 866.149
226│ 869.300
227│ 872.448
228│ 875.592
229│ 878.733
230│ 881.870
231│ 885.004
───────────────────────────────────┴───────────────────────────────────
_Note on the curves described in Clapeyron’s graphical method of
exhibiting Carnot’s Theory of the Steam-Engine._
39. At any instant when the temperature of the water and vapor is _t_,
during the fourth operation (see above, § 16, and suppose, for the sake
of simplicity, that at the beginning of the first and at the end of the
fourth operation the piston is absolutely in contact with the surface of
the water), the latent heat of the vapor must be precisely equal to the
amount of heat that would be necessary to raise the temperature of the
whole mass, if in the liquid state, from _t_ to _S_.[57] Hence, if _v′_
denote the volume of the vapor, _c_ the mean capacity for heat of a
pound of water between the temperatures _S_ and _t_, and _W_ the weight
of the entire mass, in pounds, we have
_kv′_ = _c_(_S_ − _t_)_W_.
Again, the circumstances during the second operation are such that the
mass of liquid and vapor possesses _H_ units of heat more than during
the fourth; and consequently, at the instant of the second operation,
when the temperature is _t_, the volume _v_ of the vapor will exceed
_v′_ by an amount of which the latent heat is _H_, so that we have
_v_ = _v′_ + (_H_)/(_k_).
40. Now, at any instant, the volume between the piston and its primitive
position is less than the actual volume of vapor by the volume of the
water evaporated. Hence, if _x_ and _x′_ denote the abscissæ of the
curve at the instants of the second and fourth operations respectively,
when the temperature is _t_, we have
_x_ = _v_ − σ_v_, _x′_ = _v′_ − σ_v′_,
and, therefore, by the preceding equations,
_x_ = (1 − σ)/(_k_){_H_ + _c_(_S_ − _t_)_W_}, (_a_)
_x′_ = (1 − σ)/(_k_)_c_(_S_ − _t_)_W_. (_b_)
These equations, along with _y_ = _y′_ = _p_, (_c_)
enable us to calculate, from the data supplied by Regnault, the abscissa
and ordinate for each of the curves described above (§ 17) corresponding
to any assumed temperature _t_. After the explanations of §§ 33, 34, 35,
36, it is only necessary to add that _c_ is a quantity of which the
value is very nearly unity, and would be exactly so were the capacity of
water for heat the same at every temperature as it is between 0° and 1°;
and that the value of _c_(_S_ − _t_), for any assigned values of _S_ and
_t_, is found, by subtracting the number corresponding to _t_ from the
number corresponding to _s_, in the column headed “_Nombre des unités de
chaleur abandonnées par un kilogramme d’eau en descendant de T° à 0°_,”
of the last table (at the end of the tenth memoir) of Regnault’s work.
By giving _S_ the value 230°, and by substituting successively 220, 210,
200, etc., for _t_, values for _x_, _y_, _x′_, _y′_, have been found,
which are exhibited in the table opposite.
─────────────┬─────────────────┬────────────────────────┬─────────────
Temperatures.│ Volumes to be │ Volumes from the │Pressures of
│described by the │ primitive position of │ saturated
│ piston, to │ the piston to those │ steam, in
│ complete the │occupied at instants of │pounds on the
│fourth operation.│ the second operation. │square foot.
_t_ │ _x′_ │ _x_ │_y_ = _y′_ =
│ │ │ _p_
─────────────┼─────────────────┼────────────────────────┼─────────────
0°│ 1269. _W_ │_x′_ + 5.409._H_ │ 12.832
10│ 639.6. _W_ │_x′_ + 2.847._H_ │ 25.567
20│ 337.3. _W_ │_x′_ + 1.571._H_ │ 48.514
30│ 185.5. _W_ │_x′_ + .9062._H_ │ 88.007
40│ 105.9. _W_ │_x′_ + .5442._H_ │ 153.167
50│ 62.62. _W_ │_x′_ + .3392._H_ │ 256.595
60│ 38.19. _W_ │_x′_ + .2188._H_ │ 415.070
70│ 21.94. _W_ │_x′_ + .1456._H_ │ 650.240
80│ 15.38. _W_ │_x′_ + .09962._H_ │ 989.318
90│ 10.09. _W_ │_x′_ + .06994._H_ │ 1465.80
100│ 6.744. _W_ │_x′_ + .05026._H_ │ 2120.11
110│ 4.578. _W_ │_x′_ + .03688._H_ │ 2999.87
120│ 3.141. _W_ │_x′_ + .02758._H_ │ 4160.10
130│ 2.176. _W_ │_x′_ + .02098._H_ │ 5663.70
140│ 1.519. _W_ │_x′_ + .01625._H_ │ 7581.15
150│ 1.058. _W_ │_x′_ + .01271._H_ │ 9990.26
160│ 0.7369. _W_ │_x′_ + .01010._H_ │ 12976.2
170│ 0.5085. _W_ │_x′_ + .008116._H_ │ 16630.7
180│ 0.3454. _W_ │_x′_ + .006592._H_ │ 21051.5
190│ 0.2267. _W_ │_x′_ + .005406._H_ │ 26341.5
200│ 0.1409. _W_ │_x′_ + .004472._H_ │ 32607.7
210│ 0.0784. _W_ │_x′_ + .003729._H_ │ 39960.7
220│ 0.3310. _W_ │_x′_ + .003130._H_ │ 48512.4
230│ 0 │_x′_ + .002643._H_ │ 58376.6
─────────────┴─────────────────┴────────────────────────┴─────────────
_Appendix._
(Read April 30, 1849.)
41. In p. 30 some conclusions drawn by Carnot from his general reasoning
were noticed; according to which it appears, that if the value of μ for
any temperature is known, certain information may be derived with
reference to the saturated vapor of any liquid whatever, and, with
reference to any gaseous mass, without the necessity of experimenting
upon the specific medium considered. Nothing in the whole range of
Natural Philosophy is more remarkable than the establishment of general
laws by such a process of reasoning. We have seen, however, that doubt
may exist with reference to the truth of the axiom on which the entire
theory is founded, and it therefore becomes more than a matter of mere
curiosity to put the inferences deduced from it to the test of
experience. The importance of doing so was clearly appreciated by
Carnot; and, with such data as he had from the researches of various
experimenters, he tried his conclusions. Some very remarkable
propositions which he derives from his theory coincide with Dulong and
Petit’s subsequently discovered experimental laws with reference to the
heat developed by the compression of a gas; and the experimental
verification is therefore in this case (so far as its accuracy could be
depended upon) decisive. In other respects, the data from experiment
were insufficient, although, so far as they were available as tests,
they were confirmatory of the theory.
42. The recent researches of Regnault add immensely to the experimental
data available for this object, by giving us the means of determining
with considerable accuracy the values of μ within a very wide range of
temperature, and so affording a trustworthy standard for the comparison
of isolated results at different temperatures, derived from observations
in various branches of physical science.
In the first section of this Appendix the theory is tested, and shown to
be confirmed by the comparison of the values of μ found above, with
those obtained by Carnot and Clapeyron from the observations of various
experimenters on air, and the vapors of different liquids. In the second
and third sections some striking confirmations of the theory arising
from observations by Dulong, on the specific heat of gases, and from Mr.
Joule’s experiments on the heat developed by the compression of air, are
pointed out; and in conclusion, the actual methods of obtaining
mechanical effect from heat are briefly examined with reference to their
economy.
I. _On the values of μ derived by Carnot and Clapeyron from observations
on Air, and on the Vapors of various liquids._
43. In Carnot’s work, pp. 80–82, the mean value of μ between 0° and 1°
is derived from the experiments of Delaroche and Bérard on the specific
heat of gases, by a process approximately equivalent to the calculation
of the value of (_Ep_{0}v_{0}_)/(_vdq_/_dv_) for the temperature ½°.
There are also, in the same work, determinations of the values of μ from
observations on the vapors of alcohol and water; but a table given in M.
Clapeyron’s paper, of the values of μ derived from the data supplied by
various experiments with reference to the vapors of ether, alcohol,
water, and oil of turpentine, at the respective boiling-points of these
liquids, affords us the means of comparison through a more extensive
range of temperature. In the cases of alcohol and water, these results
ought of course to agree with those of Carnot. There are, however,
slight discrepancies which must be owing to the uncertainty of the
experimental data.[58] In the opposite table, Carnot’s results with
reference to air, and Clapeyron’s results with reference to the four
different liquids, are exhibited, and compared with the values of μ
which have been given above (Table I.) for the same temperatures, as
derived from Regnault’s observations on the vapor of water.
────────────┬──────────────┬──────────────┬──────────────┬─────────────
│ │ │ Values of μ │
│ │ │ deduced from │
Names of the│ │ │ Regnault’s │
Media. │Temperatures. │ Values of μ. │Observations. │Differences.
────────────┼──────────────┼──────────────┼──────────────┼─────────────
│ ° │ (Carnot) │ │
Air │ 0.5│ 4.377│ 4.960│ .383
Sulphuric │ (Boil. pt.)│ (Clapeyron)│ │
Ether │ 35.5│ 4.478│ 4.510│ .032
Alcohol │ 78.8│ 3.963│ 4.030│ .071
Water │ 100│ 3.658│ 3.837│ .179
Essence of │ │ │ │
Turpentine│ 156.8│ 3.530│ 3.449│ −.081
────────────┴──────────────┴──────────────┴──────────────┴─────────────
44. It may be observed that the discrepancies between the results
founded on the experimental data supplied by the different observers
with reference to water at the boiling-point, are greater than those
which are presented between the results deduced from any of the other
liquids, and water at the other temperatures; and we may therefore feel
perfectly confident that the verification is complete to the extent of
accuracy of the observations.[59] The considerable discrepancy presented
by Carnot’s result deduced from experiments on air, is not to be
wondered at when we consider the very uncertain nature of his data.
45. The fact of the gradual decrease of μ through a very extensive range
of temperature, being indicated both by Regnault’s continuous series of
experiments and by the very varied experiment on different media, and in
different branches of Physical Science, must be considered as a striking
verification of the theory.