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.
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$^{-1}$.
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!
