For the reaction of bromocresol green and bleach, I was supposed to conduct an experiment to find the order of the reaction with respect to bleach. To do this, I used a fixed amount of bromocresol green and varied the amount of bleach in each trial. I recorded the amount of time it took for each reaction to finish.
To find the order, I was told that if the order = x, $\frac{time_1}{time_2} = (\frac{[bleach]_2}{[bleach]_1})^x$. However, I'm having trouble understanding why this is true. For example, say we begin with $[bleach]_1=y$ and $[bleach]_2=2y$. If $\frac{time_1}{time_2} = 2^x$, this means that the rate of reaction 2 is always $2^x$ times faster than the rate of reaction 1.
In the beginning, this is definitely true, since rate = $k[bleach]^x$. Initial $rate_1 = ky^x$ and initial $rate_2=k(2y)^x$. But at a time $dt$ later, $[bleach]_1=y-ky^x\cdot dt$ and $[bleach]_2=2y-k(2y)^x\cdot dt=2y-2^xky^x\cdot dt$. At this time, $\frac{rate_2}{rate_1} = (\frac{2y-2^xky^x\cdot dt}{y-ky^x\cdot dt})^x$, which does not equal $2^x$ unless $x=1$.
Thus, I'm not understanding why $\frac{time_1}{time_2} = (\frac{[bleach]_2}{[bleach]_1})^x$ is true for x other than 1. Any clarifications would be much appreciated.