Radioactivity & Specific Heat Capacity Question

Magda|A380

1. The problem statement, all variables and given/known data
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)

2. Relevant 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

3. 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

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 :)