How to convert rate to probability?

A "rate" here is an exponential rate. This means that you wait an "exponential length of time" until a collision/conversion event occurs. This is a random length of time that is governed by the exponential probability distribution with rate parameter $w_1$. The probability distribution function is $f_X(x)=w_1 e^{-w_1 x}$.

For a randomly selected molecule in the solution, the probability it will be converted over the next $t$ units of time is $P(X\leq t)$ where $X$ is an exponential random variable with parameter $w_1$. We calculate like this:

$P(X\leq t)=\int_0^t w_1 e^{-w_1 x} dx = \left[ -e^{-w_1 x} \right]_0^t=1-e^{-w_1 t}$

If $X$ is an exponential random variable with rate $\lambda$ in appropriate units, then its probability density function is $f_X(t) = \lambda e^{-\lambda t},$ for $t > 0.$ Its cumulative distribution function is $P(X \le t) = 1 - e^{-\lambda t}.$