Calculate frequency dose from different light intensities


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> The quantum yield for homolytic O-O cleavage of (CH$\_3$)$\_3$COOC(CH$\_3$)$\_3$ at $266$ nm is $0.21$. In one experiment, a pulsed laser of this wavelength was used to induce the formation of t-butoxyl radical from (CH$\_3$)$\_3$COOC(CH$\_3$)$\_3$. The experiment is to study the kinetics of recombination (dimerization) of phenoxyl radicals. These are formed by the t-butoxyl radical abstracting a hydrogen atom from phenol in solution (inert solvent). To get good statistics, the laser is allowed to shoot light pulses with a certain frequency (ie with a certain time between the pulses) and then follow the disappearance of phenoxyl radical by measuring its absorbance as a function of the time after each pulse. Then the results from each pulse are superimposed to obtain a statistically reliable mean value. You can regulate both pulse frequency and energy per pulse. To obtain a good mean value, at least $95$% of the phenoxyl radicals from the previous pulse must have been consumed before the next light pulse hits the solution. **Calculate the maximum pulse frequency that can be used if the incident light intensity is $10^{-7}$, $10^{-5}$ and $10^{-3}$ J cm$^{-2}$, respectively.**
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> The illuminated area is $0.5$ cm$^2$ and the volume is $0.5$ mL. The absorbance at $266$ nm (which is entirely attributed to (CH$\_3$)$\_3$COOC(CH$\_3$)$\_3$) is $0.14$. We can assume that the length of the laser pulse is negligibly short in this context and that the formation of phenoxyl radical from the reaction between t-butoxyl radical and phenol is quantitative and immediate. The rate constant for recombination of phenoxyl radicals is $3.5 \*10^7$ M$^{-1}$ s$^{-1}$.
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I am completely lost when it comes to this problem. To begin with I have never heard of pulse frequency nor can I find any equation related to it. The only equation I can find with the frequency of radiation is: $T = I\*t\*h\*v$, where $v$ is the frequency and $T$ is the radiation dose (which I don't really know how to calculate).


I would be truly thankful if someone could push me in the direction of how to solve this. I have never encountered a similar problem and there is so much information given that I don't know what is useful and what is not.


Thank you!


**EDIT: Attempted answer**


**$1$. Calculate the absorbed energy, in J.**


E = I$\_0$ \* A = $10^{-7}Jcm$^{-2}$ \* $0.5$cm$^2$ = $5\*10^{-8}$J


**$2$. Calculate the energy of one 266 nm photon, in J.**


E = hc/λ = $\frac{6.62\*10^{-34}Js \* 3 \* 10^8m/s}{266\*10^{-9}m}$ = $7.46\*10^{-19}$ J


**3. Divide to get number of photons absorbed.**


Here I am not sure which energy should be in the numerator and which in the denominator and depending on it I either get a very high number ($6.675\*10^{10}$) or a very small number ($1.49\*10^{-11}$).


**4. Multiply by quantum efficiency to get number of effective photons.**


Continuing with both gives me: $0.21$ \* $6.675\*10^{10}$ = $1.4\*10^{10}$


and, $0.21$ \* $1.49\*10^{-11}$ = $3.13\*10^{-12}$


**5. Multiply by 2 to get number of radicals.**


$3.13\*10^{-12}$ \* $2$ = $6.258\*10^{-12}$


$1.4\*10^{10}$ \* $2$ = $2.8\*10^{10}$


**6. Divide by volume and Avogadro’s number to get molar concentration of radicals.**


[R] = $\frac{6.258\*10^{-12}}{5\*10^{-4}L \* 6.022\*10^{23}mol^{-1}}$ = $2.08\*10^{-32}$ mol/L


[R] = $\frac{2.8\*10^{10}}{5\*10^{-4}L \* 6.022\*10^{23}mol^{-1}}$ = $9.3\*10^{-11}$ mol/L


**7. Use the second order integrated rate expression to find the time required for 95% of the radicals to recombine.**


Here I am also a bit unsure if what I am doing is correct. So we have a second order reaction so the rate equation is:


$\frac{1}{a-x}-\frac{1}{a}$ = k$\_2$t


We have that at least $95$% has to be consumed so $a-x$ is the $5$% that's left.


$a-x$ = $2.08\*10^{-32}$ mol/L \* $0.05$ = $1.04\*10^{-33}$ mol/L


$a-x$ = $9.3\*10^{-11}$ mol/L \* $0.05$ = $4.65\*10^{-12}$ mol/L


We now solve for $t$:


t =$\frac{\frac{1}{1.04\*10^{-33}}-\frac{1}{2.08\*10^{-32}}}{3.5\*10^{7}}$ = $2.61\*10^{25}$s


t = $\frac{\frac{1}{4.65\*10^{-12}}-\frac{1}{9.3\*10^{-11}}}{3.5\*10^{7}}$ = $5837$ s


**8. Take the reciprocal to find the max laser repetition rate.**


max laser repetition rate = $\frac{1}{t}$ = $\frac{1}{2.61\*10^{25}}$ = $3.83\*10^{-26}$ s$^{-1}$


max laser repetition rate = $\frac{1}{t}$ = $\frac{1}{5837}$ = $1.7\*10^{-4}$s$^{-1}$


Which is not the same answer that @Ed V stated that he got. Did I understand the help given in the comments correctly or where am I going wrong?


Thank you!


