THE M X T H E X . 1 T I C A L EXPRESSIOS O F THE P E R I O D I C LXW BY S. H. HARRIS
By a close comparison between a table of the atomic weights in numerical order and F. P. \‘enable’s table of tlie elements, we arrive at some interesting relations. T h e table of elements was taken from Paul C. Freer’s work on Inorganic Chemistry and Yenable’s table is inserted, containing the calculated atomic weights of several iinknown elements as derived by formulae obtained by the author froin a study of the table. T h e atomic weights are themselves only approximately determined at present and we could not reasonably expect to find a law which is exactly fulfilled in all cases. \Ye can, however, find a law thst will hold good approximately, and use it to calculate atomic weights of iinknown elements with a fair degree of exactness. T h e series Li, Re, R, C, N , 0, F are cailed bridge elements and the series Xa, Mg, h l , Si, P, S , C1 are called typical eleillen ts. Let us for a iiioinent consider tlie typical and bridge elements alone thus :
Li -- Be - B - C - K- 0 - F
I
I
Na-- Mg-A1
I
I
I
I
I
- Si - P - S - C1.
From this we observe this relation log Li : log Be = (Ea)’ : (Mg)’
.
TVorking this out, using logarithms to base IO, we obtain in verification 499. j 6 =: 506.99. R‘e do not obtain exact equality as any inherent error in atomic weights when squared is considerably increased.
I i/
L lr$
-,E:
x -r - -v
2 . log Be : log B =- Jlg : -41 verification 2 j . j 6 -- 25.3
3 log B : log C
== 1
A1 : 1 Si
j . 5 5 == j.60
4. log C : log S =- 1'Si : 1 P 6.00 = 6 . I I F -P : s 36.68 == 3 7 . 3 2
5 . log A- : log 0
6. log 0 : log F 1S' : C1' I j 1 7 . 4 9 -- 1309.33 wide discrepancy. Error greatly multiplied in squaring, became numbers are large. On collecting these proportions in a table we obtain tlie following very remarkable series : (
.; I_
log L,i : log Be = K a Z : hIg2 2 . log Be : log B = 1Ig : A1 3. log B : log C = 1/X1 : 1 Si 4. l o C~ : log N = 1 Si , v P j . log h' : log 0 = P : S 6. log 0 : log F = S' : C1' I.
From this table we see an approximate relationship betn.eeii the bridge elements and the typical elements. Approximate law T h e ratios of the logarithms of the atomic weights of bridge elements and typical elements, which stand next to them in the table, decrease in a certain geometrical progression as we pass from the left or positive side of the table toward the central carbon atom and increases as we pass from the carbon atom toward the right or negative eiid of the table. Relations between every other or alternate bridge element Consider the first two ratios in the previous table. From them we obtain tlie following : (Ij
logBe=
Mg'.log Li Sa'
(2)
log B e =
Mg.log B
A-
or
hlg” log Li
NaY
-
_-
l f g .log B A1
,
Dividing both sides by Mg‘ and inatiipulatiiig we obtain
In a similar manner we may combine all these six ratios and obtain f i \ ~new ones tabulated as follows :
i
’
i
log log log 4. log 3. j. log I.
2.
Li : log B = Na’ : hlg.Al Be : log C = 1 Xl.hIg : A41.1’Si B : l o g N = 1’Al : 1 P c : log o P1,”si : 1 /F.s X : log F = P.S : C1’
Here the logarithmic ratio decreases as we pass towards center and increases in same way as we go to negative end. Typical elements and periods T h e following ratios have been observed between the typical elements and periods which underlie tlirin : Summary
log S a : log Mg := K : Ca A1 =: 1 Ca : 1 Si -== +’SC : it P = Ge: f S =: As : 1 log S : log C1 := Se : Br log M g : log log AI : log log Si : log log P : log
Verifications
54.46 = 5.1..19 9.18 = 9 . 0 j 5.19 j.13 6.12 :- 6.21
Sc Ti As Se
~~
13.24 = 13.04
120.34 = 122.46
Typical elements and second group Sumtriary
( log Na : log S I g log SIg : log AI
I
log log log 1 log
iI
Rb : Sr : 1 Sr : 1
Verifications I
: :
/ ~ t
A1 : log Si ~ 15 Yt::i3 Zr SI : log P =: 6 S n : T? Sb = I Sb : 1 Te P : log S S : log C1 == Te : I
19.28 = I 18.16 13.08 = 13.39
6.43 =
6.49
7.18 =;
7.33
16.67 = 16.48 190.92 x: 193.77
Comparing these proportions to those already given, we obtain :
( K Ca I Sc 1 Ge As Se
i
: Ca = R b : S r
Sc : Ti
Sr : Y t -= Ut : Zr : As - Sn : Sh : Se = Sb : T e : Br ~- T e : I
:
z
Verifications
Sminiary
187.14 = 184.14 log Na : log Mg =- Cs : Ba 16.29 = 16.77 2 . log M g : log A1 = 1 Ba : /La 3. log A1 : log Si = f La : f / C e 7.51 7.44 8.62 8.61 4. log Si : log P = f P b : 1” Bi x = 212.58 j . log P : log S 1 1 Bi : 1 x 6. log S : log C1 = x :2’ 3’ = 218.94 Calculating atomic weights of unknown elements I.
\
By a n examination of Yenable’s table we see that the eleiiieiit lying just below Te in sixth group is unknown. T o calculate its atomic weight log P : log S
1’Bi : v’x
x = 212.5s
= calculated atomic weight
T o calculate the element following iodine in seventh group log
s
:
log c1 = x : y but x : 212.58 4’ = 218.94
These two unknown elements fill the g a p between Ri 208.9 and T h =- 232.6. By comparing the last two sets of ratios, we obtain Cs : Ba = Rb Ba : La -= Sr La : Ce = Ut Pb : Bi =- Sn Bi : x =- Sb .z- : 1’ =- Te
: Sr : Ut :
Zr
: Sb :
Te
:I
&S.H. H a v
582
i 5
T o obtain the atomic weight of the unknown element following La in third group, we have log A1 : log Si = solving, x :=
+ x-: i3 Th 22 I
, 4j
For the unknown following Ea in second group log Mg : log XI = 1 X: 1 .z = 208.23
__ 221.
$5
Element following Cs in first group log Na : log M g = x
:
208.23 Verifications == 88.12
88.92 log M g : log A1 = 1 Zn : 1 Ga 3. log A1 : log Si = G a : Ger { 4. log Si : log P = 1” Ti : $ l‘ 5 . log P : log S = k’V : b’Cr 2.
= 11.j6 j.96 = j.96 5.10 : j.41 10.75
