CORRELATIOW OF ENTROPY .4ND PROBABILITT
1881
[CONTRIBUTION FROM THE EUREKA JUNIOR COLLEGE, SCIENCE DEPARTXENT]
CORRELATION OF ENTROPY AND PROBABILITY BY GEORGE A. LIXIIART Received M a y 3 , 1922
In a recent article’ two equations are given, one connecting entropy and probability, and the other, specific heats and temperature. These equations are ccio (2% s= log K cx-c
CcckTK c= ___ kTK
+1
In the article just cited the validity of Equation 2 is amply substantiated by a comparison of the calculated entropies for 18 substances with the values obtained for the same substances by a graphical method. For Equation 5, however, only one set of calculated values is given, namely, those for the specific heats of copper. The purpose of the present article is to show that this equation which is based upon the laws of entropy arid probability holds for the specific heats of all substances thus far obtained by thermo-electric methods. It will be seen that in all cases where check values are given (see silver chloride and aluminum) the deviations from the values calculated by means of Equation 5 are far less than the variations among the experimental values themselves. Most of these data have been obtained in Nernst’s Laboratory3 and are scattered all through the literature in fragments, each writer smoothing, readjusting and generally vitiating the experimental data to suit his particular empirical equations. In the present paper are given presumably only the actually “observed” values. These have been subjected to the treatment described in the article previously cited. The results with the 31 substances, with the exception of those for ice and silver iodide, leave nothing to be desired. The data for the specific heats of these two compounds seem to be quite erroneous. In conclusion it may be pointed out that from the results obtained with the aid of Equations 2 and 5 the values for Sp can in every case be found as follows: Calculate Su(298) by means of Equation 2, then with the aid Linhart, THISJOURNAL, 44, 140 (1922). Here, as in the article cited above, C denotes the average gram atomic heat a t any temperature, T ; C a equals 3 R, equals 5.900 cal.; k and K are constants. For convenience of computation and tabulation, log k is given instead of k and is denoted by K‘. Where the values for K and K’ are omitted in the present article they may be found in Table TI of the article cited. The data for Na, Mg, Ca, Cd, TiCL, CCla, SiC14, SnC14, HCOOH, CO(KHB)~ were obtained in the Chemical Laboratory of the University of California and have been pulAished in THISJOURNAL within the past 5 years. 1
1882
GEORGE A. LINHART
of two or three calculated Cv values, from the point where C p begins ~ Cp and Cv against log T on uniform definitely to exceed C V , sketch tracing paper or cloth, cut out the segment thus obtained and weigh it. Having previously determined the relation between a unit of entropy and a square unit of the tracing cloth, the difference between Sp and Sv is very accurately obtained, provided, of course, the substance does not undergo a modification in form or a change of state for the chosen temperature interval. TABLES Calcium
Sodium T
64.6 67.9 71.1 74.2 84.6 94.8 156.8 35.0 39.1 42.9 45.5 51.4 53.8 77 .O 100 .o
Cg (obs.)
Cv(ca1c.)
4.52 4.54 4.66 4.66 4.77 4.77 4.81 4.86 5.08 5.11 5.30 5.29 6.02 .. Silver 1.58 1.58 1.90 1.90 2.26 2.19 2.47 2.50 2.81 2.81 2.90 2.97 4.07 4.17 4.86 4.84
Diff.
-0.02 0.00 0.00 -0.05 -0.03 $0.01
*...
0.00 0.00 $0.07 -0.03 0.00 -0.07 -0.10 +0.02 ---a
200 .o 273 .O 331 .O 535.0 589.0 74.9 78.3 83.5 92.1 101.5 114.5 115.8 132.7 155.2 172.4
5.78 5.71 6.00 6.01 .. 6.46 . 6.64 Magnesium 2.90 2.90 3.03 3.08 3.33 3.32 3.61 3.68 3.99 4.04 4.47 4.41 4.44 4.44 4.77 4.80 5.08 4.96 5.31 5.31
.. .
..
$0 .07
.... .... .... ....
.oo
0 -0.05 $0.01 -0.05 -0.05 $0.06 0.00 -0.03 +0.12 0.00
T
80.6 92.7 94.9 103.4
Cp(obs.)
Cv(ca1c.)
Diff.
4.42 4.81 4.92 5.07
4.44 4.81 4.86 5.04
-0.02 0.00 $0.06 $0.03 -a
141.4 143.3 145.2 163.2 198.5 68.4 70.7 72.5 74.8 75.7 76.8 78.8 79.1 80.7 81.8 84.2 86.2 87.0 88.4 90.0 92.5 298.0
31.1 36.6
5.70 5.51 5.68 5.52 5.75 5.54 5.97 5.64 .. 6.36 Cadmium 4.98 5.00 5.15 5.20 5.29 5.25 5.25 5.36 5.36 5.40 5.45 5.44 5.51 5.32 5.52 5.56 5.51 5.49 5.60 5.65 5.65 5.71 5.69 5.63 5.70 5.67 5.74 5.73 5.78 5.75 5.78 5.73 .. 6.35 Mercury K = 2.07 K'= -2.8050 3.89 3.93 4.36 4.35
4-0.19 $0.16 $0.19 4-0.16
....
-0.02 $0.05 +0.04 -0.11 -0.04 $0.01 -0.19 +0.04 -0.08 $0.05 $0.06 -0.06 -0.03 +0.01 $0.03 -0.05
....
-0.04 +0.01
For the elements 4 This is due to an increase in atomic or molecular volume. and compounds here presented this phenomenon begins to appear a t about 100' A., except for Cu, AI, diamond, graphite, HCOOH and CO(NH2)2. For some of these substances the increase in volume does not seem to be appreciable even up to room temperature.
1883
CORRELATION OF ENTROPY AND PROBABILITY ---a
192.6 220.0 237.3 253.5 255.2 2SS.5 67.6 70.4 73.3 76.2
5.53 5.46 5.82 5.59 5.83 5.66 5.95 5.70 5.86 5.71 6.11 .. Calcium 3.93 3.89 4.02 4.03 4.16 4.16 4.28 4.28
$0.09 +0.23 f0.17 '1-0.28 $0.15
.... $0.04 -0.01 0.00 0.00
43.0 62.0 65.0 69.0
4.70 5.34 5.37 5.43
4.71 5.31 5.36 5.43
80.5 86.0 92.0 164.0 168.0 201.0 207.0
5.58 5.66 5.79 6.26 6.29 6.42 6.48
5.56 5.61 5.65
+0.02 SO.05 +0.14
..
*... ....
.. .. ..
-0.01 +0.03 +O.Ol 0.00
--
a
.... ....
a These lines indicate approximately where the increase in volume begins to be apprecFble and, with the exception of a few suhstances, this is approximately a t about 100:'A . Diamond Mercury
T
213.0 214.0 229.0 232.0 33.1 34.3 36.3 41.0 43.7 161.8 M.0
68.0 75.0 80.0 85.0 39.0 89.3 94.0
Cp(obs.)
Ca(ca1c.)
6.58 .. 6.58 .. 6.62 .. 6.70 .. Zinc 1.25 1.26 1.32 1.34 1.49 1.71 1.60 1.84 2.17 2.04 3.25 3.25 3.51 3.37 3.39 3.59 3.95 3.91 4.11 4.17 4.29 4.24 4.39 4.42 4.32 4.44 4 55 4.57
Diff.
T
.... .... .... ....
88 92 205 209 220 222 232 243 262 284 306 331 358 413 1169
-0.01 -0.02 $0.22 -0.24 t o . 13 0.00 +o. 14 0 .oo +o .04 +O .06 -0.05 -0.03 -0.12 -0.02 ---a
207.0 274.0 290.0 366.0 395.0 19.1 23.6 2; 2 33.5 37.1 41.9 40 6
53.4 62.4
5.83 5.70 5.90 5.83 6.03 .. 6.20 .. 6.28 .. Aluminum 0.07 0.06 0.11 0.11 0.16 0.17 0.30 0.30 0.40 0.40 0.60 0.55 0.90 0.85 1.11 1.02 1.Lj
1.45
$0.13 4-0.07
.... ....
.... +0.01 0.00 -0.01 0.00 0.00 4-0.05 +o .03 +o .09 t0.10
28.7 38.1 44.1 58.8 85.0 137 232 284 334 412 622 1095 1250 79.8 87.3 94.8 194.9
Co(obs.)
Cu(ca1c.)
Diff.
.... ....
(0.03) 0.064 (0.03) 0.073 0.62 0.62 0.00 0.66 0.65 $0.01 0.72 0.74 -0.02 0.76 0.75 +O.Ol 0.86 0.84 4-0.02 0.95 0.94 +O.Ol 1.14 1.12 +0.02 1.35 1.34 $0.01 1.58 1.57 $0.01 1.84 1.83 + O . O l 2.12 2.12 0.00 2.66 2.70 -0.04 (5.45) 5.60 (-0.15) Graphite 0.06 0.04 $0.02 0 .oo 0.07 0.07 0 .oo 0.10 0.10 0.14 -0.02 0.16 0.31 0 .oo 0.31 0 .oo 0.69 0.69 1.50 1.52 -0.02 1.92 1.98 -0.06 0.00 2.39 2.39 2.95 $0.09 3.04 4.03 -0.03 4.00 5.45 5.10 5.60 5.26 Tin (white) 4.68 4.64 4.87 4.88 5.07 5.07 6.20 ..
+0.35 +0.34
0.00 -0.01 0.00
....
1884
GEORGE A. LINHART T.4B LE S
Mercury Cfi(obs.) Ca(calc.) 2.08 2.03 2.36 2.32 0.25 0.27 0.33 0.34 2.41 2.53 2.52 2.68 2.62 2.79 3.97 4.53 5.32 5.60 5.82 5.82 6.10 .. 6.48 ..
T
73.4 79.1 32.4 35.1 83.0 86.0 88.3 137.0 235.0 331.0 433.0 655.0
Diff.
$0.05 $0.04 -0.02 -0.01 -0.12 -0.16 -0.17 -0.56 -0.28 0.00
.... ....
(Continued) T
197.2 205.2 248.4 256.4 264.3 273.0 288.1 79.8 87.3 94.8 194.9
197.2 205.2 248.4 256.4 264.3 273.0 288.1 23.0 28.3 36.8 38.1 85.5 90.2 200.0 290.0 332.0 409.0 22.7 25.9 27.5 28.3 29.9 57.0 69.0 83.0 93.0 138.0 198.0 235.0 297.0
Cs(ca1c.)
Diff.
7’
5.71 .. 5.75 .. 5.87 .. 5.88 .. 5.89 .. 5.90 .. 5.91 .. Lead 2.96 3.00 3.92 3.66 4.40 4.40 4.45 4.49 5.65 5.65 5.71 5.68 .. 6.13 6.33 .. 6.41 .. 6.61 Sulfur (rhombic) 0.86 0.96 0.99 0.99 1.05 1.04 1.08 1.08 1.14 1.14 2.05 2.06 2.37 2.29 2.70 2.70 2.91 2.93 3.63 3.63
.... .... ....
28.3 33.5 36.5 77.0
Cp(obs.)
..
4.72 4.93 5.47
4.22 4.47 4.77
Diff.
.... .... .... . . .
.... ....
.... 0.00
0.oo 0 .oo ....
Iodine
Tin (gray) (cont.) T
Diamond Cw(calc.1 6.23 .. 6.25 .. 6.36 .. 6.37 .. 6.38 .. 6.39 .. 6.40 .. Tin (gray) 3.80 3.80 4.07 4.07 4.30 4.30 5.66 ..
C$(obs.)
.... ....
.... . . I .
-0.04 $0.26 0.00 -0.04 0.00 $0.03
.... ....
.,..
.... $0.10 0.00 -0.01 0.00 0.00 $0.01 -0.08 0.00 $0.02 0.00
__-
$0.50 $0.46 $0.70
C p ( o b s . ) Cu(ca1c.)
3.78 3.97 4.17 5.38
3.54 3.97 4.17 5.38
Diff,
+0.24 0.00 0.00 0.00
25.0 25.5 28.0 67.5 69.0 81.4 83.4 138.0 235.0
5.92 5.84 $0.08 6.36 .. .... 6.64 .. .... Sodium chloride 0 . 3 0 -0.01 0.29 0.31 0.00 0.31 0.40 0.00 0.40 3.06 2.90 $0.16 3.13 3.00 $0.13 3.72 -0.18 3.54 3.81 -0.06 3.75 (3.87)? 5.29 .... (5,76)? 5.81 -0.05
22.8 26 9 30.1 33.7 39.0 48.3 52.8 57.6 63.2 70.0 76.6 86.0 137.0 235.0
Potassium chloride 0 57 $0.01 0.58 0.76 0.82 -0.06 0.98 1.01 -0.03 1.25 1 . 3 0 -0.05 1.83 1.70 $0.13 2.85 2.40 $0.45 2.72 $0.08 2.80 3.04 $0.02 3.06 3.37 -0.01 3.36 3.74 $0.05 3.79 4.11 4.04 +0.07 -0.04 4.36 4.40 5.25? 5.36 -0.11 5.892 5.79 $0.10
186.0 235.0 298.0
1885
CORRELATION O F ENTROPY AND PROBABILITY
Sulfiir imonocl.) 0 0 0 0 0 ‘260 102 0
2 2 2 2 2 2 3
IC14 0 200 0 201 0
4 92 5 00 4 81
83 87 87 ,49 01
75 82 90 90
95 97 15
75 84 84 88 93 03 15
0 00 -0 02 fO 06 $0 02 -0 02 -0 03 0 00
4 30 4 35 4 35
+O 62 +0 63 $0 4(i
2 2 2 2 2 3 3
CP(ohs.)
23 5 20.4 32.8 45.6 87.0 ll(i.0
1.49 1.72 2.40 3.63 4.87 5.17
1.49 1.75 2.28 3.18 4.72 5.16
207.5 330.0 403.0 4:K.O
5.90 6.51 6.80 6.86
5.66
$0.24
..
.... ....
..
....
..
DiR.
0.00 -0.03 $0.12 $0.45 +0.15 $0.01
--_
Thallium chloride
:23 1 :26 7 29.7 32.4 36.3 40.9 44.8 50.1 90.4 94.0 138.0 236.0 297.0
K = 1.90 K‘=-2.8838 1.89 2.02 2.32 2.40 2.69 2.69 2.95 2.94 3.22 3.25 3.53 3.59 3.79 3.83 4.03 4.11 5.20 5.20 5.25 5.25 5.62 5.60 6.02 .. 6.34 .. I50 1.74 2.18 2 54 4 20 4.40
1.58 1.79 2.01 2.43 4.20 4.40
. ..
..
Silver E:hloride 1.39 1.40 1.73 1.77 1.99 2.16 2.75 3.02 4.20 4.20 4.32 4.34 4.60 4.58 4.91 4.80
....
....
-0.01 -0.04 -0.17 -0.27 0.00 -0.02 $0.02 $0.11
I’
Cp(obs.)
Cu(ca1c.)
Diff.
99.3 4.54 4.54 0.00 194.6 6.19 5.45 .... Carbon tetrachloride K = 2.92 K’=-5.605 39.1 0 . 6 0 0.59 $0.01 40.7 0.65 0.67 -0.02 6 3 . 8 2.05 1.89 $0.16 79.6 2 . 8 1 2.80 +0.01 91.0 3.40 3.38 $0.02 95.0 3.57 3.56 $0.01 99.5 3.60 3.75 -0.15
-_-
-0.13 -0.08 0.00 fO.01 -0.03 -0.06 -0.04 -0.08 0.00 0.00 $0.02
.... *
I
.
.
Mercurous chloride 23.0 25.7 29.0 34 5 75.0 83 0
22 5 26.6 31.3 43.1 68.0 72.2 81.3 91.4
6.16 6.36 6.54
Titanium tetrachloride (cont.)
Silver chloride T
Co(ca1c.)
331.0 416.0 550.0
-0.02 -0.05 f0.17 $0.11 0.00 0.00
199.5 5.84 5.54 204.1 5.90 5.56 208.0 5.90 5.58 239.4 6.20 .. Silicon tetrachloride K = 1.89 K’=-3.3882 7 7 . 4 3.60 3.60 3.75 81.8 3.70 3.90 8 6 . 6 3.83 94.8 4.14 4.12 131.3 4.80 4.80 168.6
5.68 5.18 181.0 5.95 5.27 185.8 6 . 0 8 .. Tin tetrachloride K = 2.25 K‘=-3,8800 89.0 4.59 4.55 89.5 4.53 4.55 95.0 4.73 4.70
$0.30 $0.34 $0.32
....
0.00 -0.05 -0.07 $0.02 0.00
-_-
$0.50 -I-0.68
....
$0.04 -0.02 $0.03
GEORGE A . LINHART
TABLES(Concluded) Silver chloride
Titanium tetrachloirde (cont.) Diff.
T
CP(obs.1
$0.01 0.00 0.00
95.9
4.71
T
Cp(obs.)
84.0 86.0 89.0
4.43 4.46 4.52
4.42 4.46 4.52
198.0 326.0 331.0
5.69 6.12 6.24
5.44
$0.25
..
.... ....
Cw(ca1c.)
..
___
Titanium tetrachloride K = 1.79 Kt=-3.0725 4.26 0.00 86.7 4 . 2 6 4.40 0.00 92.8 4 . 4 0
161.2 200.5
87.7 106.5 205.5 330.0 405.0 430.0
71.0 73.3 76.6 77.7 82.0 86.0 89.0 90.0 90.3 94.0 176.3 180.0 184.5 196.0
CP(obs.)
4.52 4.97
Cu(ca1c.)
4.63 4.97
5.80 5.64 6.17 .. 6.32 .. 6.42 .. Formic acid K = 0.90 Kt=-2.1650 1.44 1.45 1.47 1.50 1.51 1.50 1.52 1.52 1.58 1.55 1.63 1.60 1.67 1.60 1.68 1.66 1.69 1.68 1.73 1.73 2.49 2.48 2.52 2.51 2.56 2.59 2.63 2.69
EUREKA, CALIFOENIA
4.72
5.89 5.51 .. 6.32 Lead chloride 0.71 0.72 1.04 1.03 1.39 1.17 1.63 1.63 3.52 3.49 3.82 3.91 4.55 4.46
15.6 19.8 24.0 27.0 54.9 61.5 54.4
Diff.
-0.01
--$0.38 .,..
+0.01 -0.01 -0.22 0.00 -0.03 $0.09 -0.09
Formic acid (cont.)
Lead chloride T
Co(ca1c.)
Diff.
-0.11 0.00
--4-0.16
T
Cp(0bs.j
Co(ca1c.)
Diff.
205.0
2.73
2.70
$0.03
237.0 243.2
3.02 3.12
2.89 2.92
86.4 90.0 90.0 90.3 96.5 97.0 104.0 107.5 128.1 198.5 199.5 201.4 204.7 208.2
Urea 1.79 1.75 1.76 1.78 1.83 1.78 1.82 1.79 1.89 1.84 1.85 1.83 1.91 1.87 1.94 1.94 2.13 2.11 2.57 2.54 2.55 2.55 2.56 2.54 2.58 2.59 2.58 2.59
223.9 244.2 274.0
2.83 2.87 3.21
....
....
....
$0.01 -0.03 -0.01 0.00 -0.03 -0.03 -0.07 -0.02 -0.01 0.00 -0.01 -0.01 $0.03 $0.06
2.67 2.76 2.88
--$0.13 $0.20 $0.04 -0.02 $0.05 $0.03 $0.05 -0.02 -0.04 0.00 $0.02 $0.03 0.00 -0.02 $0 01 $0.01 $0.16 +O.ll $0.33
