24.106 CP - Chemistry: Molecular Nature Of Matter And Change

Question

Technetium-99m is a metastable nuclide used in numerous cancer diagnostic and treatment programs. It is prepared just before use because it decays rapidly through y  emission:

 ${}^{99 m}T{\to }^{99}Tc+\gamma$

Use the data below to determine 

(a) The half-life of  ${}^{99 m}Tc$; 

(b) The percentage of the isotope that is lost if it takes 2.0 h to prepare and administer the dose.

Time(h)	y emission (photons / s)
0	5000
4	3150
8	2000
12	1250
16	788
20	495

Step-by-Step Solution

Answer

(a) The half-life of   ${}^{99 m}Tc$5.99 hours

(b) 21%  of isotope that is lost if 2.0 h  is taken to prepare and administer the dosage

Step 1: Find the decay constant

Consider the given information:

Initial amount,  $N_0=5000$photons/s 

Amount of decay after time  $t,N_t=495photons/s$

Time, t = 20h                 

The rate constant, k, can be calculated using the following equation:

 $$\begin{gathered} \left(\frac{N_t}{N_0}\right)=-kt \\ \ln \left(\frac{495photons/s}{5000photons/s}\right)=-k(20 h) \\ -2.3126=-k(20 h) \\ k=0.11563 h^{-1} \end{gathered}$$

Step 2: Find the half-life of T 99   m c

(a)

The ${}^{99 m}Tc$ 

half-life is calculated as follows:

 $$\begin{gathered} t_{1/2}=\frac{\ln 2}{k} \\ =\frac{0.693}{0.11563 h^{-1}} \\ =5.99 h \end{gathered}$$

Therefore, the required value is  5.99h.

Step 3: Find the percentage of the isotope

(b)The following equation can be used to calculate the remaining percentage of isotope:

 $$\begin{gathered} \left(\frac{N_t}{N_0}\right)=-kt \\ \ln \left(\frac{N_t}{N_0}\right)=-\left(0.11563 h^{-1}\right)(2.0 h) \\ =-0.23126 \\ \left(\frac{N_t}{N_0}\right)=0.7935 \\ \left(\frac{N_t}{N_0}\right)\times 100\%=79.35\% \end{gathered}$$

The percentage of isotope lost if  2.0 h is used to prepare and administer the dosage is displayed below. Fraction lost during preparation  =100%-79.35%

$$\begin{gathered} =20.65\% \\ \approx 21\% \end{gathered}$$

                                                                                 

Therefore,  21% of isotope is lost if  2.0 h is taken to prepare and administer the dosage.