THE APPLICATION O F THE LAW O F MATHEMATICAL PROBABILITY TO THE BEHAVIOR OF GASES IN THEIR PRESSURE-VOLUME-TEMPERATURE RELATIONS' GEORGE A. LINHART
Department of Mathematics, Riverside Junior College, Riverside, California Received October 18, 193.9
The equation,
+ kzK)
2/ = y,lczK/(l
has been shown to hold for the course of a large variety of natural processes (1)) and it is of interest to find that it also applies to the behavior of a gas in its pressure-volume-temperature relations. I n the present case the equation assumes the form,
l/vp=
IcPK/(l
+ IcPK)V,
(1)
where V p denotes the volume at any pressure, P; V,, the ultimate molal volume of the gas; and k and K are constants, characteristic of the gas considered. V is expressed in standard units, i.e., the volume of a mole of gas at standard conditions is taken as unity. P is expressed in atmospheres. Equation 1 may also be written in the form, V,/(Vp
- V,>
=
kPK
(2)
and, at 1 atmosphere pressure may be written, (3)
V , / ( V , - V m ) = IC
Dividing equation 2 by equation 3, we obt,ain, (VI
-
vm>/(v,- v,) =
PK
(4)
In testing the constancy of K in equation 4, it was assumed for convenience of calculation that a t 1 atmosphere pressure Charles' Law is obeyed, i.e., 'VI = 'V0T/To,where V o = 1 and T o = 273"A. V , is given by the point of inflection on the curve obtained by plotting on a rather
' Read before the Mathematical Association of America, Southern California Section, San Diego Teachers College, March, 1932. 645
646
GEORGE A. LINHART
TABLE 1 Hydrogen gas
--
T P
___
1/
= 092.
v
K
T
= 15.5"C.
1IV
K
T = 99.3"C.
T
K
1/V
1lV
=
200.3"C K
atm.
1
100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000
93.5 136.0 175.7 213.2 248.1 280.9 311.8 340.9 368.6 394.8 318.9 442.7 465.1 487.1 507.6 532.2 544.9 563 .,8 579.7 610.9 642.0 670.7 698.3 724.6 749.3 772.5 794,9 816.3 837.2 857.3 876.4 894.5 911.1 927.6 948.2 959.7 975. e
0 999 0.999 0.999 0.999 0,998 0.998 0.997 0.997 0.996 0.996 0.995 0.995 0.994 0.994 0.994 0.995 0.992 0.993 0.992 0.991 0.991 0.990 0.990 0.991 0.991 0.991 0.992 0.992 0.993 0.993 0.995 0.995 '0.995 0.996 0.998 0.998 0.999
1500.C
Av. = 0.994
W
1.7334
1.9463
1
--
88.6 129.0 166.9 202,4 236.5 268.2 298,3 326.6 353.4 378,8 402.9 426.1 447.9 469.3 489.0 509.2 527.5 545.8 562.8 593.5 623,4 652.3 679.3 705.2 729.9 754.1 776.4 797,l 818.0 837.5 855.8 873.7 887.4 908.3 923.7 940.7 956.9 972.8 987.6
0.998 0.998 0.998 0,997 0.997 0.997 0.997 0,996 0.996 0.995 0.995 0.994 0.994 0,994 0,993 0.993 0.993 0.993 0,993 0.992 0.991 0.991 0.991 0.991 0.991 0.992 0.992 0.993 0.993 0.994 0.994 0.994 0.994 0.996 0.997 0.998 1.000 1,001 1.002
1500.0
Av. = 0.995
D ,5768
101.6 132.2 161.3 189.2 215.7 241,l 265.5 288.8 311.1 332.7 353.2 373.1 392.0 410.5 428.1 445.6 460.0 477,8
0.998 0.997 0.997 0.997 0.997 0.997 0.997 0.996 0.996 0.996 0.995 0.995 0.995 0.994 0.994 0.994 0.993 0,993
1500.0
Av. = 0,996
81.2 106.2 130.2 153.4 175.6 197.0 217.8 237,5 257.0 275.7 293.9 311.4 328.4 344.8 360.8 376,4
0,999 0.998 0,998 0,998 0.997 0.997 0.997 0,997 0,996 0.996 0.996 0.996 0.996 0.995 0.995 0,995
1500.0
Av. = 0.997
-
-
FIG. 1. HYDROQEN GAS A T 0°C.
FIG.2. OXYGENGAS AT 0°C. 647
648
GEORGE A. LINHART
TABLE 2 Oxygen gas P
T = 0°C.
v
11
T K
=
15.6"C.
K 1/v -
T I/
=
v
QQ.5"C. K
T = ISQ.5OC. K
1/v
alm.
1
1
100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850
107.9 164.2 218.8 268.4 311.7 348.6 380.4 408.2 432.5 453.7 472.5 489.7 505.3 519.5 532.2 543.2 555.6 566.3 576.4 594.5 611.6 627.4 642.3 655.3 667.8 679.3 690.3 700.8 710.2 719.4 728.0 736.4 744.3 752.2 759.6 766.9 774.0 791.3
1.043 1.058 1.068 1.075 1.079 1.080 1.080 1.os0 1.079 1.077 1.075 1.073 1.072 1.070 1.068 1.066 1.065 1.064 1.062 1.060 1.059 1.058 1.058 1.057 1.057 1.057 1.057 1.058 1.058 1.059 1.060 1.061 1.062 1.064 1.065 1.068 1.070 1.073
920.0
Av. = 1.066
900 950 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 m
0.9459 99.6 151.2 201.1 246.7 287.9 324.0 355.6 383.0 407.5 429.2 448.8 466.4 481.9 497.3 510.5 523.3 534.5 546.1 555.6 574.7 592.1 607.9 623.0 636.5 649.4 660.7 671.8 682.6 692.5 702.2 711.2 719.4 727.8 735.3 742.9 750.8 757.9 765.1 771.6 920.0
0.7329 1.036 1.049 1.058 1.064 1.068 1.070 1.071 1.071 1.070 1.069 1.068 1.066 1.065 1.064 1.062 1.061 1.060 1.059 1.057 1.055 1.053 1.052 1.052 1.051 1.051, 1.051 1.051 1.051 1.051 1.052 1.053 1.054 1.055 1.056 1.057 1.059 1.061 1.063 1.065
72.7 108.5 142.9 175.6 206.5 234.9 261.1 285.5 308.3 329.5 348.8 366.7 383.1 398.7 413.7 427.7 440.9 453.3 464.9
0.5778 1.016 1.022 1.027 1.031 1.033 1.035 1.036 1.037 1.038 1.038 1.038 1.038 1.038 1.037 1.037 1.037 1.037 1.036 1.036
-
-
920.0
Av. = 1.034
83.3 110.0 135.1 159.2 182.1 204.0 224.6 243.9 262.5 280.1 296.6 312.3 327.2 341.4 355.6 367.9 380.4
1.011 1.015 1.017 1.019 1.020 1.021 1.022 1.022 1.023 1.023 1.023 1.024 1.024 1.024 1.024 1.024 1.024
920.0
Av. = 1.021
Av. = 1.058
649
MATHEMATICAL PROBABILITY IN BEHAVIOR OF GASES
TABLE 3 Nitrogen gas T
P I/
T=
= 0°C.
v
K
1/
v
16.0"C.
K
T
= 99.5OC.
1/v
K
-
T
= 199.5"C.
11 v
K
alm.
1
100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000
1
100.9 148.5 192.5 230.9 264.1 292.9 318.3 340.1 359,7 377.1 393.2 407.8 421.2 433.5 445.0 455.6 465.3 474.6 483,3 499.1 513.9 527.3 539.7 551.4 562,3 573.1 583.3 592.3 601.1 609.4 617.3 624.6 631.5 637.7 643.9 649.8 655.7 665.8 667.1
0,9446 1.031 1.040 1.044 1.047 1.048 1.047 1.046 1.045 1.043 1.041 1.039 1.038 1.036 1.035 1.034 1.032 1.031 1.030 1.029 1.027 1.025 1.024 1.023 1.023 1.023 1.023 1.024 1.024 1.025 1.026 1.027 1.028 1.029 1.029 1.030 1.031 1.033 1.039 1.036
-
94.2 138.7 179,5 216.0 247.8 276.2 301 .O 322.8 342.7 360.4 376.4 391.2 404.9 417.2 428.8 439.8 449,6 459.1 468,2 485.0 500.0 514.1 527.1 539.4 550,l 560.5 570.8 579.9 588.6 597.0 605.0 612.7 619.8 626.6 633.3 639.4 645.4 651 .O 656.8
1.026 1.033 1.038 1.041 1.041 1.041 1.041 1.039 1.038 1.037 1.035 1.034 1.032 1.031 1.030 1.029 1.028 1.027 1.026 1.024 1.023 1.022 1.021 1.021 1.021 1.021 1.021 1.021 1.022 1.022 1.023 1.024 1.025 1.026 1.027 1.028 1.029 1.030 1.032
800.0
Av. = 1.029
Av. = 00
800.0
1.032
0,7329 103.5 134.3 162.6 188.6 212.6 234.5 254.9 273.6 291 .O 306.9 321.7 335.6 348,3 360.4 371.8 382.3 392.9
0.5780 1.015 1.018 1.015 1.020 1.021 1.020 1.020 1.020 1.020 1.019 1.019 1.018 1.017 1.017 1.016 1.015 1.014
80.6 104.9 127.6 148.9 168.9 187.6 205.1 221.5 237.1 251.7 265.5 278.6 291.1 303.0 314.1 324.8 334.8
Av. = 800.0
1.018
800.0
1.006 1.008 1.009 1.009 1.010 1.010 1.009 1.009 1.009 1.009 1.009 1.008 1.008 1.008 1.007 1.007 1.007
-
Av. = 1.007
650
GEORGE A . LINHART
TABLE 4 Air P
T
= 0°C.
T
= 15.7"C.
T
T
= BD.4"C.
K
K
K
1/
= 200.4"C.
v
K
atm.
I . 9453
1
1
100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000
102.8 152.4 198.0 238.3 273.4 303,3 329.4 352.5 373.1 391.7 408.2 423.2 437.1 449.4 461.3 472.6 482.9 492.6 502.0 515.7 531.1 545,3 558.2 570.1 581.2 591.5 601.7 610.9 620.0 628.7 636.9 645.0 652.1 659.4 666.7 673.6 680.5 687.3 694.0
1.035 1.044 1.050 1.053 1.054 1.054 1.053 1.052 1.050 1.049 1.047 1.045 1.044 1.042 1.041 1.040 1.039 1.038 1.037 1.033 1.031 1.031 1.030 1.029 1.028 1.029 1.029 1.030 1.030 1.031 1.033 1.034 1.035 1.037 1.039 1.041 1.043 1.046 1.049
822.0
Av. = 1.040
00
-
0.7329
95.6 141.8 184.3 222.0 255.6 285.7 311.6 334.3 354,4 373.1 390.2 405.8 420,O 432.5 444.5 455.8 466.4 476.0 485.4 502.0 517.3 531.9 545.3 557.6 569.2 579.7 590,O 599,5 608.6 613.9 625.8 633.7 641.2 648.5 655.7 662.3 668.9 675.4 682.1
1.029 1.038 1.043 1.045 1.047 1.048 1.047 1.046 1.044 1.043 1.042 1.041 1.040 1.038 1.037 1.035 1.035 1.033 1.033 1.031 1.029 1.028 1.028 1.028 1.027 1.027 1.027 1.028 1.028 1.026 1.030 1.031 1.032 1.033 1.035 1.036 1.038 1.040 1.043
822.0
Av. = 1.035
71.3 104.8 136.3 165.5 192.5 219.6 240.6 261.6 280.7 298.2 314.8 330.5 344.8 358.4 371.2 383.3 394.2 404.4 414.2
0.5766 1.014 1.018 1.020 1.022 1.023 1.024 1.025 1.025 1.024 1.023 1.023 1.023 1.022 1.022 1.021 1.021 1.020 1.020 1.019
81.4 106.0 129.3 151.O 171.5 190.8 209.0 226.1 242.3 257.5 272.0 285.5 298.4 310.7 322.4 333.3 344.5 353.6
1.009 1.010 1.011 1.012 1.012 1.012 1.012 1.012 1.013 1.012 1.012 1.012 1.012 1.012 1.012 1.011 1.011 1.010
822.0
Av. = 1.012
-
822.0
Av. = 1.022
65 1
MATHEMATICAL PROBABILITY I N BEHAVIOR O F GASES
large scale l / V p against log P, for a t this point, lcPR = 1. This may be verified by taking the second derivative of 1 / V with respect to log P , and
FIG.3. NITROGEN GAS A T 0°C.
6
FIG.4. AIR AT 0°C.
placing the resulting expression equal to zero. Hence, by equation 1, 1/2V, = 1 / V at inflection, which can be easily located from the perfect
652
’
GEORGE A. LINHART
symmetry of the curve. Only one curve for each gas is given, since they are all of the same general trend and of nearly equal accuracy. V , may also be obtained approximately from density measurements of the given substance in its solid state near the absolute zero of temperature, since V is practically independent of temperature and pressure. However, the inflection point method is preferable, for density measurements a t extremely low temperatures are not likely to be generally reliable. The gases chosen for this study are hydrogen, oxygen, nitrogen and air. As the most complete P-V-T data (2) available are those of Amagat, ranging from 0°C. to 200°C. and from 1 atmosphere pressure to 3000 atmospheres pressure, these data were used in the calculation of K . RESULTS O F CALCULATIONS
The data for hydrogen gas are those of tables 4 and 8 of the original article cited above; for oxygen gas, of tables 4 and 7; for nitrogen gas, of tables 5 and 9; and for air, of tables 5 and 10. TABLE 5 Variation of K with the absolute temperature: a summary T
OXYGEN
HYDROQEN
I
NITROQEN
1
AIR
~~~
273 290 373 473 m
‘ .
1.OB6 1.058 1.034 1.021 1.000
0.994 0.995 0.996 0.997 1.000 I
I
1.032 1,029 1.018 1.007 1.ooo
I
1.040 1.035 1.022 1.012 1.ooo I
DISCUSSION O F T H E CONSTANTS
The concordance of the constants in each of the sixteen sets of results is quite remarkable and justifies the assumption that a t 1 atmosphere pressure Charles’ Law is valid. It is also of considerable interest to note that with rise in temperature of the gas the constant, K , in every case approaches unity (see figure 5 ) . This phenomenon is in accord with the kinetic theory of gases, leading to the simple expression, PV = a constant. The conditions postulated by the kinetic theory can be fulfilled only a t high temperature and a t moderately low pressure, in which case equation 4,
(v,- V , > / ( V , - v,) = P K reduces to PV = a constant, since a t high temperatures and low pressures K approaches, in all cases, unity, and V , becomes negligibly small as compared with either VI or V p . It may be noted that K for hydrogen approaches unity from below, while in the other cases K approaches unity from above. This phenom-
MATHEMATICAL PROBABILITY IN BEHAVIOR OF GASES
653
enon is shown on figure 5, where K is plotted to four significant figures against the absolute temperature. In conclusion, it may be pointed out that equation 4 is an empirical one, in that y of the general equation is replaced by 1 / V without any theoretical reason for doing so. The general equation, however, does possess a theoretical foundation, since, as will be shown in a subsequent paper, it is based upon the law of mathematical probability. Furthermore, equation 4 is not intended either for extrapolation or interpolation
purposes, for, as stated in the first paragraph of this article, the writer is merely endeavoring to show that the equation,
describes the general trend of a vast variety of natural processes in the fields of chemistry, physics, botany, biology, bacteriology and sociology, and even in practical engineering; for example, the depreciation and the life expectancy of physical property. REFERENCES (1) LINHART: J. Phys. Chem. 36, 1908 (1932). (2) AMAGAT: Ann. chim. phys. 29, 68 (1893). Extensive work in this field is contemplated by Mitchells: Proc. Roy. SOC.London A1930, 127-258.
