weird half life problem

[Oh_Gee]

from GS-9


i thought that since there would be three lives, you would just double 7 g three times to get original mass which is what i usually do for these problems. where did i go wrong

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[Cawolf]

The original material is not losing mass through alpha decay, it is decaying into a different element.

 

[Oh_Gee]

The original material is not losing mass through alpha decay, it is decaying into a different element.

isn't that what usually happens in these processes? how would i calculate this differently? the solution doesn't make sense to me

 

[Shirafune]

As the answer says, beta decay is when a neutron is turned into a proton and a beta particle is emitted. Thus, there is no appreciable mass lost through beta decay. Instead, beta decay reduces the number of neutrons and increases the atom's atomic number (changing elements).

The problem gives you the amount of Pb (the product of process) after 3 half-lives (9.3 minutes/3.1 minutes per half life). Therefore, the total mass of Pb should equal the sum of the original sample's mass converted to Pb through beta decay. Since three half lives passed, the sum must be (1/2 + 1/4 + 1/8) * X = 7, where X is the original sample's mass. The fractions come from powers of two, aka what number half life each decay represents.

 

[pacer]

Can I not work backwards from the 7 grams of product given. If 3 half-lives have passed, working backwards gives me 7 -> 14 -> 28 -> 56. Since beta particles don't have weight, we started off with 56 grams??

What is wrong with this method?

 

[Shirafune]

Can I not work backwards from the 7 grams of product given. If 3 half-lives have passed, working backwards gives me 7 -> 14 -> 28 -> 56. Since beta particles don't have weight, we started off with 56 grams??

What is wrong with this method?

You cannot work backwards from 7 grams. Your method assumes that 7 grams is the mass of the original sample after 3 half-lives. I assume this is the logic you are using. The question gives you amount of product after 3 half-lives and asks you for the mass of the original sample. You are not given the remaining mass of the original sample so your method cannot be applied here. The total mass of the product is the sum of the product produced through beta-decay with each half-life.

 

[pacer]

Can I write out the reaction as follows:

Reactants -> product + beta negative particle

product + beta negative particle = 7 grams

Products = 4 grams

4 -> 8-> 16->32 grams??

 

[Shirafune]

Can I write out the reaction as follows:

Reactants -> product + beta negative particle

product + beta negative particle = 7 grams

Products = 4 grams

4 -> 8-> 16->32 grams??

No. Like I said before, the total mass of the reactant element and the product element are going to be approximately equal because the beta particle has negligible mass. Thus, the process is something closer to this:

Initial: Reactant -> No Product

Less Reactant -> Some Product

Even less Reactant -> More Product

Final: Almost Zero Reactant -> Product

As you can see, beta decay can be simplified to mass of one element being "converted" to another element. At any given time, we can calculate the ratio between the reactant and product. Consequently, we can calculate the total mass of the system, which is equivalent to the mass of the original sample.