September 1, 1922] 



SCIENCE 



259 



seven unknowns, the value of each of the un- 

 knowns may readily be found in terms of any 

 one of the unknowns, a numerical value being 

 then assigned to the latter such that fractional 

 coefficients will disappear. For example, from 

 the sis equations given above it follows that : 



a ^ a e =. 2a 



11a 

 'Y 

 llo £r — 4a 



~2~ 



11a 



& = 



2 



/ = 



and calling a = 2, the ehemical equation is : 

 2Ag^AsO^ + llZn -|- IIH^SO^ =z 

 2AsH +* 4Ag 4- llZnSO^"+ 8H^O. 



NUMBER OF EQUATIONS AND NUMBER OP 

 UNKNOVSTNS 



In applying the algebraic method there may 

 be written as many equations as there are ele- 

 ments concerned, and obviously there will be 

 as many coefficients as there are compounds. 

 Since the relation between number of equations 

 and number of unknowns determines the appli- 

 cability of the algebraic method, one is led to 

 inquire into this matter with respect to the 

 chemical equations ordinarily encountered. A 

 random selection of fifty equations from an 

 inorganic chemistry text-book reveals the fol- 

 lowing : 



If a; =: number of elements concerned 

 y =: number of compounds concerned 

 Then: 



For 4 of the equations : 2^ := j/ -|- 1 (Case I) 

 For 17 of the equations: x =z y (Case II) 



For 28 of the equations : x^=y — 1 (Case III) 

 For 1 of the equations : x^y — 2 (Case IV) 



Since x independent equations fix every pos- 

 sible ratio between x -\-l unknowns, it is evi- 

 dent that in Cases I, II and III above the 

 number of independent equations written will 

 be one less than the number of unknowns, 

 although the procedure of the algebraic method 

 yields actually two additional (dependent) 

 equations in Case I and one additional equa- 

 tion in Case II. 



Case IV offers a curious condition, for here 

 it is evident that the ratios between all the 

 unknowns can not be fixed, and there may be 

 found an infinite number of sets of coefficients 



which will balance the equation. The following 

 equation is an example : 



dK^SO^ -f- eMnSO^*+ fKO + gO^. 



The fact that this equation may be balanced 

 in an infinite number of ways has no signifi- 

 cance chemically, since the valence changes of 

 manganese and oxygen settle the matter and 

 there is only one set of coefficients which per- 

 mit the equation to represent the chemical facts 

 involved. In this case the valence changes 

 involved require that 5a ^= 2c and this adds an 

 additional equation to those required by stoichi- 

 ometric considerations, and the problem of 

 finding the coefficients falls then under Case III 

 above. There are chemical equations, however, 

 which can be balanced in an infinite number of 

 ways having due regard for valence require- 

 ments. In the equation : 



eK^SO^ -f /Cr, (SO^)*^ + gFeJSO^) + hB.fi + tS 



there are nine unknown coefficient and seven 

 equations. Valence changes of chromium, iron 

 and sulphur require that 6a ~ c -\- 2d, but this 

 equation is included in those required by 

 stoichiometric ratios and the problem remains 

 indeterminate. There will be 3a — 1 true solu- 

 tions for every value given a. 



NUMBER OP ELEMENTS AND NUMBER OP COM- 

 POUNDS IN A CHEMICAJJ REACTION 



From a ehemical standpoint, the fact that, 

 with but few exceptions, x elements enter into 

 reactions involving either a; or a; -f- 1 com- 

 pounds may strike one as curious. Closer ex- 

 amination of this point, however, does not 

 appear to reveal anything in the way of a law 

 of nature, but indicates that the rule arises 

 from certain limited values in the equations 

 ordinarily used in inorganic chemistry, as is 

 shown below : 



Let X = Number of elements 

 y -j^ Number of compounds 

 =z Total number of elementary symbols 



appearing 

 r = Unnecessary repetition of symbols 



(i. e., in excess of 2a;). 

 fc = Average number of elements per coni- 



c 

 pound (i. e., -) 

 2/ 



