Radioactive decays - given power value

[mahoteacher]

Homework Statement

The masses of the following atoms are:

238 U → 238.050786(2)
234 Th→ 234.043598(5)
4 He → 4.002603

a) Calculate the energy available in the decay of 238U.

b) Calculate the decay rate of 238U

238U half time = 4.468*10^9yrs

c) If a block of uranium produces 1mW of power, how many disintegrations per second is this and what is the activity in Curies. Neglect other decays other than the one above.

d) Calculate the number of uranium atoms and the mass of the block.

- This is a question for an upper level medical physics class, but it seems to be relatively easy. So I am not sure if I am putting this in the right place.

Homework Equations

Mass calculation of radioactive decay

1 amu = 1.66 * 10^ -27 [kg]

t½ = ln2 / λ

A = λ*N = ΔN/ Δt

1 Ci = 3.7* 10^10 [s^-1]

The Attempt at a Solution

[/B]
a)
Δ amu = 238.050786(2) - 234.043598(5) - 4.002603 [amu] = 4.58*10^-3 [amu]

Δ mass = Δ amu * 1. 66*10^-27 [kg]
Δ mass = 4.58*10^-3 * 1.66 *10^-27 = 7.6106 *10^-30 [kg]

E = Δ mass * c^2
E = 7.6106 *10^-30 * 3*10^8 = 2.28 * 10^-21 J

b)

Δ t = 4.468*10^9 * 365 * 24 *60 * 60 = 1.409028* 10^17
Δ N = 6.022*10^23 atoms/ Δ amu = 7.9126* 10^-8

decay rate = ΔN/ Δt = 5.615 * 10^ -25

c)

disintegration per second = A = λN and 1 Ci = 3.7 * 10^10 [s^-1]

But how does power play a role? Does power affect the number of atoms? If yes how?
Assuming power has nothing to do with N, I am approaching the problem like this:

t½ = ln2/ λ
λ = 4.91932 * 10^ -18 [s^-1]
N = 6.022 * 10^ 23/ 238

A = λN = 12447 Bq = 3.36 * 10^ -7 [Ci]

d) since I don't know how power is related, I don't really know :(

 

[mfb]

E = Δ mass * c^2
E = 7.6106 *10^-30 * 3*10^8 = 2.28 * 10^-21 J

Where did the square of the speed of light go?

Δ N = 6.022*10^23 atoms/ Δ amu = 7.9126* 10^-8

What did you calculate here? Working with units would make it easier to follow.

decay rate = ΔN/ Δt = 5.615 * 10^ -25

That is not correct. You can directly get the decay rate from the half-life without other quantities involved.

But how does power play a role? Does power affect the number of atoms? If yes how?

If every apple has a mass of 0.2 kilogram and I give you 5 kg per second ("power"), how many apples per second do you get?
Power does not "affect" the number of atoms but you need power to figure out how many decays happen per second.

 

[mahoteacher]

Thank you for correcting me my mistakes, and helping me to move the thread to the right place!

E = Δ mass * c^2
E = 7.6106 *10^-30 *( 3*10^8)^2 = 6.84*10^-13 J

Δ N = 6.022*10^23 [atoms/mol] / Δ amu [g/mol] = 7.9126* 10^-8 [atoms/g]

Regarding the decay rate, I thought it means activity, which is A = λN. How can I get directly from half life? Can you give me some pointers?
t½ = ln2/ λ ==> λ = 4.91932 * 10^ -18 [s^-1]

For power, does that mean for question c)
Disintegration per second = A = 1*10^-3 [s^-1] ?

Thank you so much!

 

[mfb]

Δ N = 6.022*10^23 [atoms/mol] / Δ amu [g/mol] = 7.9126* 10^-8 [atoms/g]

That would give 12 tons per atom (the inverse value). Very heavy atoms...

Decay rate: Hmm, could be a matter of definition. If you used it as decays per second per gram, then you need the number of atoms per gram, sure.

For power, does that mean for question c)
Disintegration per second = A = 1*10^-3 [s^-1] ?

No. Where does that value come from?

 

[mahoteacher]

That would give 12 tons per atom (the inverse value). Very heavy atoms...

Oh I see that doesn't make sense now...

Decay rate: Hmm, could be a matter of definition. If you used it as decays per second per gram, then you need the number of atoms per gram, sure.

So am I correct to think that for decay rate A = λN
t½ = ln2/ λ => λ = 4.91932 * 10^ -18 [s^-1]
Δ N = 6.022*10^23 [atoms/mol] / 238 [g / mol] = 2.530*10^21 [atoms/g]

A = 12447 [s^-1] = 1.24*10^4 [s^-1]

No. Where does that value come from?

That value comes from the power value 1 mW = 1*10^-3 W

This question seems really easy, but I haven't done physics in a while and couldn't really wrap my head around.
Thanks so much for your help!

 

[mfb]

A = 12447 [s^-1] = 1.24*10^4 [s^-1]

Another value that comes out of the blue...

That value comes from the power value 1 mW = 1*10^-3 W

W is not the same as 1/s.

All the quantities you need can be derived by combining given values in a way that the units match. If the units do not match, the answer is certainly wrong.