Does Radioactive decay cause a loss of mass?

[Dave1982]

I would like to note first of all that this is NOT a homework question. This is a question that appeared on a science lab assignment that has ALREADY been graded. Also, I am providing the “correct” answer, so I am clearly not looking for help with my homework.

Here’s the problem as it appears on the lab worksheet:

100 grams of Thorium-234 (half life=24 days) and 100 grams on Americium-241 (half life=432 years) are placed together in a sealed lead container. The contents and the container weigh 500 grams. The container is stored in a vault for 1 year. When the lead container is removed from the vault, how much does the container and its contents weigh?

OK, so in one year, the Thorium goes through about 15.2 half lives, and the Americium goes through 1/432 of a half life. Therefore about 0.004 grams of Thorium and about 99.999 grams of Americium remain. According to my instructor that is correct, so therefore the container weighs 400.003 grams.

However, I believe that this is a trick question. I had a discussion with my instructor about this problem, and she said she’d talk with the lab coordinator about it (the coordinator wrote the lab manual and all the problems).

My contention is that regardless of how much of the original contents of the container decay, the container will still weigh 500 grams. True, the point of the question is to actually calculate the decay, but considering the phrasing of the question, I think the right answer is 500 grams, not 400.003 grams.

Considering that the container is both sealed and made of lead, any particle radiation should be contained in the box, and the by products of the decay of thorium and americium (whatever they may be) should also be contained. Therefore “the container and its contents” should weigh 500 grams.

Any and all feedback is welcome.

[Q.E.D]

At least some of the energy released will be converted to heat which will eventually leak away, thereby carrying energy–and thus, per E=mc[sup]2[/sup], mass.

[Polycarp]

Dave, your answer is not quite right, but closer than the lab instructor’s answer.

When the Th-234 (forgive my not using subscripts) breaks down (assuming alpha decay), it turns into Ra-230 and He-4. The Ra-230 will then break down (I don’t have a table of isotopes handy, but I don’t recall that being a long-lived nuclide) to Rn-226 and He-4, possibly with beta decays taking it back up the table (increasing atomic number while maintaining effectively steady atomic mass). What you will end up with is the sum of the surviving Th-234, Ra-230, Rn-226 (and any beta-decay products), and so on, plus of course the Americium and what it decays into (Np-237, which has a long enough half-life that its products will be negligible).

The sealed container will retain the He-4 (and the Rn-226), so it’s appropriate to keep its mass in the calculation.

But what you do not take into account is that radioactive decay towards iron (i.e., down the periodic table from the actinides) is exothermic – it produces energy at the expense of mass. The amount is very little, and may not cause serious modification in your numbers…but it will not weigh precisely 500 grams after the year.

[Urban_Ranger]

It will weight less.

For one thing, lead is a conductor, and will remove heat and excess electrons from inside the container. Besides, the question did not tell you how thick the container is, thus you cannot know if it can stop all gamma radiation.

[WereOtter]

[nitpick on] Once mass is converted to heat energy, it doesn’t contribute to the weight, so it doesn’t matter how much of the heat escapes from the container.[nitpick off]

[Desmostylus]

*Originally posted by WereOtter *
**[nitpick on] Once mass is converted to heat energy, it doesn’t contribute to the weight, so it doesn’t matter how much of the heat escapes from the container.[nitpick off] **

Sorry, but your nitpick is wrong.

You recognise that the nuclear binding energy can be detected as mass. So far so good. It doesn’t make any difference to the overall mass of the system if the binding energy is converted to another form of energy. Until the energy leaks out.

There’s nothing special about nuclear binding energy that sets it apart from other forms of energy, in term of having a mass equivalent. A charged NiCd battery weighs more than a discharged battery. A moving electron weighs more than one at rest. Hot lead weighs more than cold lead. And so on.

[Armilla]

The question is really pretty poorly posed. You can’t determine the answer without making outrageous assumptions.

Since it’s a sealed box question I’ll assume that matter can’t leak out (if it could then you’d have to factor gas diffusion (in and out) etc, all depending on the thickness of the lead etc), so we’re left with energy as the only way out. As other posters have shown above this is subject to the E=mc[sup]2[/sup] rule.

Some back of the envelope calculations show that if the box sat there for the year radiating a constant 1000 watts (as heat, light etc) then it would weigh about 0.0000004 kilograms less. The grease from a fingerprint probably weighs more.

Of course it wouldn’t radiate anything like 1000 watts. At the end of the experiment the box would probably weigh exactly the same down to the limit of measurable accuracy (and some way beyond).

[Desmostylus]

OK. Let’s have an answer. If I stuff up any of the following calculations or get any of the data wrong, please feel free to point out the errors and correct them.

Ignore the Am[sub]231[/sub], and assume that all of the Th[sub]234[/sub] goes to Fe.

Th[sub]234[/sub] has as excess mass of 4.06x10[sup]7[/sup] eV, or 6.51x10[sup]-12[/sup] J.

100 g is 0.427 moles, or 2.57x10[sup]23[/sup] atoms.

6.51x10[sup]-12[/sup] J times 2.57x10[sup]23[/sup] is 1.67x10[sup]12[/sup] J.

So 1.67x10[sup]12[/sup] J are released by the Th[sub]234[/sub].

The mass equivalent of that is 1.67x10[sup]12[/sup]/9x10[sup]16[/sup] = 1.86x10[sup]-5[/sup] kg = 0.0186 g.

Armilla assumed an average energy loss rate of 1 kW.

If my calculations above are correct so far (and given the assumptions I made), the average energy loss rate would be 1.67x10[sup]12[/sup]/(365x24x60x60) = 53.1 kW.

[Shade]

Have we decided that the energy loss is negligible yet? It never occured to me to do otherwise in this sort of problem.

My next thought was to wonder if any beta particles would get through the lead, but I’m guessing not.

Dave: I think your answer is correct. Well, if all the other questions take mass/energy losses into account then maybe its incomplete, but your prof. is certainly wrong. In retrospect I can see why he asks that question, but the wording is pretty clearly supportive of your answer.

[Evil_Captor]

Well, when my teeth decay, they lose mass. And THEY’RE not even radioactive, that I know of.

pa-dum!

Thanks, folks, I’ll be here all week.

[swansont]

*Originally posted by Desmostylus *
Ignore the Am[sub]231[/sub], and assume that all of the Th[sub]234[/sub] goes to Fe.

Your point is valid, but it would be more reasonable to assume you stop at the appropriate stable isotope of Pb. That would be Pb-206, if I traced the chain properly, and would release 47.438 MeV of energy per parent, assuming most likely mode of decay. Of course, this is an upper bound since some of the half-lives are very long.

[Ring]

*Originally posted by Desmostylus *
A moving electron weighs more than one at rest.

Unless you expand on it I’m going to have to disagree with this statement. A moving electron taken as a single object cannot weigh more than a stationary one because all of its kinetic energy can be transformed away.

On the other hand if a moving electron is part of a system’s internal energy ( the system has a zero momentum frame) then I agree that the system will weigh more.

[Achernar]

Ring, it took me a while to get a handle on what you mean by mass, but I think I got it. That’s not what he said.

He said it weighs more.

[Achernar]

Sorry, I hit Submit too soon.

I think that in this case, saying it “weighs” more is appropriate. By saying it’s moving, there’s a preferred frame of reference implied, and we would assume that any weighing is done in that frame.

[swansont]

*Originally posted by Ring *
Unless you expand on it I’m going to have to disagree with this statement. A moving electron taken as a single object cannot weigh more than a stationary one because all of its kinetic energy can be transformed away.

On the other hand if a moving electron is part of a system’s internal energy ( the system has a zero momentum frame) then I agree that the system will weigh more.

Actually, if the electron is part of a bound system, the sytem will weigh less. There is potential energy to consider as well; the PE is negative and in greater magnitude than the KE for a bound system.

Your first statement is somewhat vague - “transforming away” KE sounds a little like you expect energy to be the same in different frames of reference. One usually accounts for rest energy and KE seperately, so the argument is not quite according to Hoyle, but from a total energy perspective, the electron will weigh more, if measure in the same frame as the stationary one.

[Ring]

*Originally posted by swansont *
Actually, if the electron is part of a bound system, the sytem will weigh less. There is potential energy to consider as well; the PE is negative and in greater magnitude than the KE for a bound system.

Yes that’s true. The mass of a free electron plus the mass of a free proton is higher than the mass of an hydrogen atom. But that’s not what I was saying or at least trying to say. If you have an electron bouncing around in a box the box/electron system will have a higher mass than if the electron is stationary in the box.

As far as the second part of your post here’s what Steve Carlip a Ph.D. physicist at UC Davis has to say:

You have two cans, each full of ball bearings. In one of the cans, the ball bearings are at rest; in the other, they are bouncing around rapidly. Both cans are initially at rest relative to you. You measure the relative acceleration of the cans in each other’s gravitational field. You will find that they accelerate more rapidly than if the ball bearings in can number two were at rest.

Only in the last case can you say that ``a moving body weighs more than the same body at rest.’’ A clearer way to put it is that a body with internal motion, or with constituents that have kinetic energy relative to the body’s center of mass, weighs more than a body without such internal motion.

You can check Carlip’s credentials at http://www.physics.ucdavis.edu/Text/Carlip.html

[Ring]

As far as energy being transformed away; Chronos explains it better than I can.

Now, mass is that part of the energy of a system that can’t be transformed away: In other words, when you go to the zero-momentum frame, all of the energy that’s left is the mass. Two parallel photons (or a single photon, for that matter) don’t have a zero-momentum frame, and as your frame moves faster, the energy of the system becomes arbitrarily close to zero. If the photons are moving away from each other, however, then there is a zero momentum reference frame, and the energy in that frame is the mass. If you go into any other reference frame, the total energy you observe for the photons can only be greater. If you had a large number of photons radiating outward from some central point (so you have spherical symmetry), then an object outside the photon shell would orbit exactly as though you had a chunk of rock there instead, with the same energy.

http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=111712&highlight=energy+mass+transformed