# Given a general reaction $5x+2y\rightarrow w+3z$, the instantaneous rate of reaction is :

$\begin{array}{1 1}(a)\;\large\frac{1}{5}\frac{d[x]}{dt}=\frac{1}{2}\frac{d[y]}{dt}=-\frac{d[w]}{dt}=-\frac{d[z]}{dt}\\(b)\;-5\large\frac{d[x]}{dt}=\frac{-2d[y]}{dt}=\frac{d[w]}{dt}=\frac{3d[z]}{dt}\\(c)\;-\large\frac{1}{5}\frac{d[x]}{dt}=-\frac{1}{2}\frac{d[y]}{dt}=\frac{d[w]}{dt}=\frac{1}{3}\frac{d[z]}{dt}\\(d)\large\frac{5d[x]}{dt}=\frac{2d[y]}{dt}=-\frac{d[w]}{dt}=-\frac{3d[z]}{dt}\end{array}$

Answer: Rate = $\;-\large\frac{1}{5}\frac{d[x]}{dt}=-\frac{1}{2}\frac{d[y]}{dt}=\frac{d[w]}{dt}=\frac{1}{3}\frac{d[z]}{dt}$
For any reaction $ax+by \rightarrow cw+dz$, Rate of reaction=$-\large\frac{1}{a}\frac{dx}{dt}=-\large\frac{1}{b}\frac{dy}{dt}=\large\frac{1}{c}\frac{dw}{dt}=\large\frac{1}{d}\frac{dz}{dt}$
This represents the rate of disappearance of x and y (hence the negative sign) and the rate of appearance of w and z (hence the positive signs).
Therefore, the Rate $= \;-\large\frac{1}{5}\frac{d[x]}{dt}=-\frac{1}{2}\frac{d[y]}{dt}=\frac{d[w]}{dt}=\frac{1}{3}\frac{d[z]}{dt}$
edited Jul 26, 2014