Radioactive decay. Given Ax, Ay and t½ for Y, get t½ for X. Not possible?

[catkin]

[SOLVED] Radioactive decay. Given Ax, Ay and t½ for Y, get t½ for X. Not possible?

Homework Statement

This is from Advanced Physics by Adams & Allday, spread 8.13 Question 1.

The activity of 20 g of element X is four times the activity of 10 g of element Y. Element Y has a half-life of 20,000 y. What is the half-life of X?

Homework Equations

$$A = \lambda N$$
$$\lambda t_{0.5} = 0.69$$

The Attempt at a Solution

Rewriting the first relevant equation in $t_{0.5}$, rather than λ, using the proportionality from the second relevant equation
$$A = 0.69 N / t_{0.5}$$

Considering 10g of both elements
$$A_{X} = 2A_{Y}$$

Expressing these activities in terms of the number of atoms in 10 g and half life
$$0.69 N_{X} / t_{0.5X} = 2 \times 0.69 N_{Y} / t_{0.5Y}$$
$$N_{X} / t_{0.5X} = 2N_{Y} / t_{0.5Y}$$
$$t_{0.5X} = (N_{X} / 2 N_{Y}) t_{0.5Y}$$

Substituting, using years as time units
$$t_{0.5X} = (N_{X} / 2 N_{Y}) {20000}$$

If the number of atoms in 10 g of element X were the same as the number of atoms in 10 g of element Y (there is no reason why it should be) then $t_{0.5X}$ would be 10,000 years (the answer the book gives).

4. Question Am I right in thinking there is not enough information in the question to answer it?

 

[rl.bhat]

10 g of x and y cannot have same number of atoms. Nx and Ny depend on their molicular weights.

 

[catkin]

Thanks rl.bhat

It doesn't answer my question though; is the problem soluble?

[Separate issue: what if the elements had the same atomic number? Say Th-234 and Pa-234? Wouldn't the number of atoms in 10 g be the same, at least to the number of significant figures the question implies?]

 

[rl.bhat]

In case of Th- 234 and Pa- 234 the problem is soluble.

 

[catkin]

Thanks again. That makes sense. Is the problem soluable as it is posed in the original question?

 

[Astronuc]

Certainly if X -> Y (or Y -> X) by beta decay, then the same mass would have approximately the same number of atoms within 1% or less.

The question becomes - "does X -> Y, or vice versa, i.e. do they represent sequential steps in a decay chain?"

The approach seems correct. The problem hinges on the assumption of N_x and N_y, which would be determined by the atomic masses.

 

[catkin]

Certainly if X -> Y (or Y -> X) by beta decay, then the same mass would have approximately the same number of atoms within 1% or less.

The question becomes - "does X -> Y, or vice versa, i.e. do they represent sequential steps in a decay chain?"

The approach seems correct. The problem hinges on the assumption of N_x and N_y, which would be determined by the atomic masses.

Thanks, astronuc

That helps understanding.

There's nothing in the question to indicate either any decay relationship or the atomic mass relationship between X and Y, though. Decay chains are introduced in a later "spread" in the textbook so should not be necessary for the solution.

I'm increasingly coming to think that the question as set is not soluable.

 

[Astronuc]

Raise this concern with the professor.

With mass and activity, one can get the specific activity, but one needs to know the atomic mass to obtain the number of atoms.

Since you obtained the answer given in the book with the assumption that the atomic mass of X and Y are roughly equal, that would seem to indicate an implicit assumption on the part of the author. If beta decay was involved (e.g. X -> Y), then that is a reasonable assumption.

 

[catkin]

Thanks Astronuc

That's enough to consider this one SOLVED