Calculating total activity from decay of this sample of technetium

[zenterix]

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.

 

[mjc123]

The question is quite specific; it asks for the number of disintegrations per second (this is the definition of "activity" it gives you; don't try to work with any other) due to the decay of 0.5 ug of excited Tc. Be careful with signs; the activity is positive (it equals -dN/dt, as N goes down by 1 for each disintegration). Numerically, what you've done so far looks good, now you need to calculate N₀ and dN/dt.

"How quickly this number of nuclei decays depends on how much of the sample it is in it represents."
This statement is untrue.

 

[Steve4Physics]

@zenterix, if you haven’t yet solved the problem, I’d like to add this.

You are being asked to find the rate of decay (activity) of a sample of Tc-99m at the moment when the remaining amount is 0.5$\mu$g.

Step 1. Calculate the number, $N$, of Tc-99m atoms (nuclei really) in 0.5$\mu$g of Tc-99m. You will need the mass of 1 atom; if you are not given the value then you need to calculate an approximate value for yourself. (Hint: what is the significance of '99'?)

Step 2. Find the decay constant (which you call $k_r$) - you’ve already done this.

Step 3. Use $-\frac{dN}{dt}=k_rN=\text{activity}=A$. Note I’ve corrected the signs.

Maybe it's also worth noting that original question is inaccurate/misleading. Tc-99m was discovered as “a product of uranium fission in nuclear power plants.”. That’s not how it is “readily obtained” nowadays.

 

[Borek]

Step 1. Calculate the number, $N$, of Tc-99m atoms (nuclei really) in 0.5$\mu$g of Tc-99m. You will need the mass of 1 atom; if you are not given the value then you need to calculate an approximate value for yourself. (Hint: what is the significance of '99'?)

Funny thing. As much as it is a perfectly correct way of doing things, my first instinct is to convert to moles using molar mass, then multiply by Avogadro number. Yes, in a way it is a bit of going around, at the same time it is always easier to follow paths you are used to.