To calculate the time required for 75% of a 0.010 M solution of hydrogen peroxide to decompose, we can use the first-order reaction equation:ln [A]t / [A]0 = -ktwhere [A]t is the concentration of hydrogen peroxide at time t, [A]0 is the initial concentration of hydrogen peroxide, k is the rate constant, and t is the time.Since we want to find the time required for 75% decomposition, we can set [A]t to be 25% of the initial concentration:[A]t = 0.25 * [A]0Now we can plug in the values and solve for t:ln 0.25 * 0.010 M / 0.010 M = - 4.6 x 10^-4 s^-1 * tln 0.25 = - 4.6 x 10^-4 s^-1 * tt = ln 0.25 / - 4.6 x 10^-4 s^-1 t 3012.6 sSo, it takes approximately 3012.6 seconds for 75% of the 0.010 M solution of hydrogen peroxide to decompose at 25C.