Chemistry Calculating total activity from decay of this sample of technetium

AI Thread Summary
The discussion focuses on calculating the activity of a technetium-99m sample undergoing decay. The decay rate is expressed as A = -dN/dt, with the initial activity A0 derived from the initial number of nuclei N0. Participants clarify that the problem specifically asks for the number of disintegrations per second for 0.5 micrograms of excited technetium. Steps to solve include calculating the number of nuclei in the sample, finding the decay constant, and applying the activity formula correctly. The conversation emphasizes the importance of understanding the problem's requirements and correcting any misconceptions about the decay process.
zenterix
Messages
774
Reaction score
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.
 
Physics news on Phys.org
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.
 
@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.
 
Steve4Physics said:
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.
 
Thread 'Confusion regarding a chemical kinetics problem'
TL;DR Summary: cannot find out error in solution proposed. [![question with rate laws][1]][1] Now the rate law for the reaction (i.e reaction rate) can be written as: $$ R= k[N_2O_5] $$ my main question is, WHAT is this reaction equal to? what I mean here is, whether $$k[N_2O_5]= -d[N_2O_5]/dt$$ or is it $$k[N_2O_5]= -1/2 \frac{d}{dt} [N_2O_5] $$ ? The latter seems to be more apt, as the reaction rate must be -1/2 (disappearance rate of N2O5), which adheres to the stoichiometry of the...
I don't get how to argue it. i can prove: evolution is the ability to adapt, whether it's progression or regression from some point of view, so if evolution is not constant then animal generations couldn`t stay alive for a big amount of time because when climate is changing this generations die. but they dont. so evolution is constant. but its not an argument, right? how to fing arguments when i only prove it.. analytically, i guess it called that (this is indirectly related to biology, im...
Back
Top