Question

The quantum efficiency of fluorescence ${\varphi }_f$can be written as

${\varphi }_f=\frac{\tau }{{\tau }_0}$

where $\tau$is the observed lifetime of the excited state in the presence of a quenching agent and ${\tau }_0$is the natural lifetime in the absence of a quencher. The fluorescence radiant power F is given by Equation 15-7. This quantity is affected by collisional quenching because the lifetime t is influenced by collisional

quenching. Derive an equation to show that the $F-\tau$ratio is independent of collisional quenching and directly related to concentration. (From G. M. Hieftje and G. R. Haugen, Anal. Chim. Acta, 1981, 123, 255, DOI: 10.1016/S0003-2670(01)83178-0.)

Short answer

The expression which shows the $F-\tau$ ratio is independent of collisional quenching and directly related to concentration is $\frac{F}{\tau }=K_c$.

Step-by-step solution

Given Information

The expression which shows the independence of collision quenching and direct relation to concentration by$F-\tau$ ratio should be determined.

Explanation

The expression of quantum efficiency of fluorescence is:

${\varphi }_f=\frac{\tau }{{\tau }_0}$ ......... (I)

Here, the observed lifetime of the excited state in presence of quenching agent is $\tau$and the natural lifetime of the excited state in absence of quenching agent is ${\tau }_0$.

The expression of fluorescence radiant power is:

localid="1646804951974" $F=2.303{\varphi }_f{K}^{"}\epsilon bc{P}_0$ ...... (II)

Here, the constant is K" , the molar absorptivity is $\epsilon$, the concentration of fluorescence species is c, the length of the medium is b and the power of the beam incident on the solution is P₀.

Substitute $\frac{\tau }{{\tau }_0}$for ${\varphi }_f$in Equation (II).

$$\begin{gathered} F=2.303\left(\frac{\tau }{{\tau }_0}\right)K"\epsilon bc{P}_0 \\ \frac{F}{\tau }=2.303\left(\frac{1}{{\tau }_0}\right)K"\epsilon bc{P}_0 \end{gathered}$$

......... (III)

Here, the term $2.303\left(\frac{1}{{\tau }_0}\right)K"\epsilon bc{P}_0$in Equation (III) is a constant.

The expression of constant is:

$K=2.303\left(\frac{1}{{\tau }_0}\right)K"\epsilon b{P}_0$

Substitute K for $2.303\left(\frac{1}{{\tau }_0}\right)K"\epsilon b{P}_0$in Equation (III).

$\frac{F}{\tau }=Kc$ ........ (IV)

Here, the Equation (IV) shows that the ratio of fluorescence radiant power (F) and the observed lifetime of the excited state in presence of quenching agent $\left(\tau \right)$is directly related to concentration (c) , because the K is a constant. Equation (IV) also shows that the ratio of fluorescence radiant power (F) and the observed lifetime of the excited state in presence of quenching agent $\left(\tau \right)$is independent of collision quenching, because the ratio is related to concentration.