zenterix
- 774
- 84
- Homework Statement
- Technetium has not been found in nature. It can be obtained readily as a product of uranium fission in nuclear power plants, however, and is produced in quantities of many kilograms per year. MIT Chemistry Professor Alan Davison pioneered the use of techneticum in the diagnosis of heart disease. Calculate the total activity (in disintegrations per second) caused by the decay of 0.5 microgram of ##\mathrm{^{99m}Tc}## (an excited nuclear state of ##^{99}Tc##), which has a half-life of 6.0 hours.
- Relevant Equations
- It is not clear to me at all what the problem is asking. The total activity is a negative exponential. That is, the rate of decay of the technetium is going down exponentially fast. 0.5mcg represents a specific number of nuclei that decay. How quickly this number of nuclei decays depends on how much of the sample it is in it represents.
This is a problem from this problem set from MIT OCW.,
Here is my reasoning about the problem, even though I don't reach any conclusion since I am not sure what is being asked.
The decay rate of the number of nuclei of technetium in our sample is
$$\frac{dN}{dt}=-k_rN=\text{activity}=A$$
$$\implies N=N_0e^{-k_rt}$$
$$\implies A=A_0e^{-k_rt}$$
where ##A_0=-k_rN_0##, ##N_0## is the initial number of nuclei, and ##A_0## is the initial activity.
The half-life is
$$t_{1/2}=-\frac{\ln{2}}{k_r}=6\text{h}$$
so
$$k_r=\frac{\ln{2}}{21600}\mathrm{s^{-1}}$$
What I don't understand is what the problem is asking exactly.
The problem statement says that 0.5mcg of technetium in an excited state decays.
Here is my reasoning about the problem, even though I don't reach any conclusion since I am not sure what is being asked.
The decay rate of the number of nuclei of technetium in our sample is
$$\frac{dN}{dt}=-k_rN=\text{activity}=A$$
$$\implies N=N_0e^{-k_rt}$$
$$\implies A=A_0e^{-k_rt}$$
where ##A_0=-k_rN_0##, ##N_0## is the initial number of nuclei, and ##A_0## is the initial activity.
The half-life is
$$t_{1/2}=-\frac{\ln{2}}{k_r}=6\text{h}$$
so
$$k_r=\frac{\ln{2}}{21600}\mathrm{s^{-1}}$$
What I don't understand is what the problem is asking exactly.
The problem statement says that 0.5mcg of technetium in an excited state decays.