Radioactivity & Specific Heat Capacity Question

[Magda|A380]

Homework Statement

0.2g of a radium salt was separated from a ton of uranium ore. The radioactive radium nuclide Ra-226 decays by alpha-particle emission with a half-life of 1600 years. 1 year = 3.16x10⁷s.
The curie is defined as the number of disintegrations per second from 1.0g of Ra.

Show that:
a)i) the decay constant of the radium nuclide is 1.4x10-11 s^-1
ii) 1 curie equals 3.7x10¹⁰Bq
b) Show that the energy release in the decay of a single nucleus of Ra-226 by alpha-particle emission is 7.9x10^-13J.
nuclear mass of Ra-226 = 226.0254u
nuclear mass of Rn-222 = 222.0175
nuclear mass of He = 4.0026u
c)Estimate the time it would take a freshly made sample of radium of mass 0.2g to increase in temperature by 1^oC. Assume that 80% of the energy of the alpha particles is absorbed within the sample so that this is the energy which is heating the sample. Take the specific heat capacity of radium to equal 110Jkg^-1K^-1. Use the data from a) and b)

Homework Equations

For a)i) I used λt_1/2 = 0.693
For ii) I used A = λN (I used Avogadro's constant to find N of 1.0g of Ra-226.)
For b) I used E = Δmc²; 1u = 1.661x10^-27kg
For c) E=mcΔT

The Attempt at a Solution

Both part a) and b) were fine, but I'm having trouble with part c)
I was thinking of using E = mcΔT and having E = 0.8 x 7.9x10^-13, m = 2x10^-4kg, c=110; what value of ΔT should I be using? I thought it should be 1 because the temperature is being increased by 1^oC? However, that doesn't fit the equation. Also, how can I find the time from this? Should I equate E to Qt or should I be using a different equation?
Sorry this is so long; I included parts a) and b) as they might be needed to work out part c).
Any help would really be appreciated.
Thanks

 

[pcm]

find no. of nucleus decayed in time t.


the energy release in the decay of a single nucleus of Ra-226 by alpha-particle emission is 7.9x10-13J.

use this to calculate energy released in time t.
then, E=mcΔT.

 

[Magda|A380]

find no. of nucleus decayed in time t.


the energy release in the decay of a single nucleus of Ra-226 by alpha-particle emission is 7.9x10-13J.

use this to calculate energy released in time t.
then, E=mcΔT.

How would I go about finding the no. of nuclei decaying in time, t? I was thinking of using N = N₀e^-λt; but which values should I use for N₀ and N?

I worked out, using Avogadro's constant, that 0.2g of radium-226 has 5.3x10²⁰ nuclei. The question says that 80% of the energy of the alpha-particles heats the sample; so 0.8 x 7.9x10^-13 = 6.3x10^-13J. So I thought that to find the amount of energy required to heat 0.2g of radium, I'd have to multiply it by the number of nuclei in 0.2g of radium. I tried incorporating this into E=mcΔT, but I'm still not sure how to figure out the time from this
Thanks again

 

[Magda|A380]

Could anyone help with the above question? My exam is in less than a week...
Thanks :)

 

[asteriatic]

I would first clarify the question with the person who provided it. Specifically, I would ask for clarification on what is meant by "freshly made sample of radium." Is this referring to a sample that has just been separated from the uranium ore, or a sample that has been stored for a certain amount of time after separation? This is important because the temperature increase will depend on the amount of time the sample has been decaying.

Assuming the sample is freshly separated, here is a possible solution to part c):

1. Calculate the energy released by 0.2g of Ra-226 in 1 year (3.16x107 seconds). This can be done by multiplying the energy released by a single nucleus (7.9x10-13J) by the number of decays that occur in 1 year. This can be found by using the decay constant (1.4x10-11 s-1) and the initial number of nuclei (0.2g of Ra-226).

2. Use the equation E = mcΔT to find the temperature increase (ΔT) of the sample. You already have all the values needed for this equation.

3. Since the question asks for the time it would take for the sample to increase in temperature by 1oC, we can rearrange the equation to solve for time (t). So, t = ΔT / (mc). Plug in the values you have from step 2 to find the time it would take for the sample to increase in temperature by 1oC.

Note: Since the question states that 80% of the energy is absorbed by the sample, you may want to multiply the final answer by 0.8 to account for this.

I hope this helps!