LP:wrs. — AUTOCATALYTIC DECOMPOSITION OF SILVER OXIDE. 727 



dx 

 where — measures the velocity, x is the fraction of oxide decomposed, 



and therefore proportional to the amount of silver; 1 — x is the fraction 

 remaining unchanged, and therefore proportional to the amount of silver 

 oxide. A" is a constant. 



An inspection of the equation shows that it would be represented by a 

 rising and falling curve, with a maximum of velocity at the point where 

 just one half the silver oxide is decomposed. In order to compare the 

 assumptions we are making with our experimental results we must obtain 

 the velocity as a function of t (time) rather than of x. From equation 1, 

 by integration. 



In = Kt + C, 



1 — X 



where hi stands for a natural logarithm and C is the integration con- 

 stant. If we count t from the point of maximum velocity, then at that 

 point Kt = 0. Also since 



X 



X = (1 — x), In =; 0, 



^ \ — X 



and therefore C=Q. Therefore our equation stands 



or 'y* p^* 



In = Kt, or = e^\ ov x = , (2) 



1 — X 1 — X l+« 



differentiating this equation 



dx Ke' 



dt (1 + e^'f 



(3) 



This is the equation sought, and in order to plot its curve on a suitable 



scale for comparison with those of Figure 3, we must choose in this case 



also such a unit of time as to make the maximum velocity equal to unity. 



dx 

 At this point, then, -— = 1, but as previously shown x=^\, and 1 — a; = A ; 



whence by equation 1 , \ =. K • }, • \, and K = 4, whence equation 3 

 becomes 



dx _ 4 e^' 



"^ "" (1 + e^'f 



The continuous curve shown in Figure 3 is obtained by plotting this 

 equation. Its great similarity to the experimental curves is obvious. 



