Radioactive decay Problem

**apple12** · September 11th 2008, 07:15 PM

A physicst a Geiger counter to measure the decay of a radioactive sample of bismuth 212 over a period of time, he then obtained this:
Time (min) | 0 | 20 | 40| 60| 80 |100|120|140|160|180|200|
Counts per min. |702|582|423|320|298|209|164|154|124| 81 |79|

What is the half-life of this isotope? (Please explain to me how the answer was worked out, I'm not very good at half-life)
The bismuth decays into thallium by emitting an alpha particle, and this particle is 6.64 x 10^(-27) kg. What is the momentum of the alpha particle? And What is the KE of the rocoiling Thaillium nucleus? Also, What is the total energy released during the decay of 1 mole of bismuth 212?

**TKHunny** · September 11th 2008, 08:00 PM

Does that make you less than half-good at life?

You know already that you need an exponential model. This is a good place to start. Do you know how to fit an exponantial model to a give data set?

One possible model is $I(t) = 681.82 \cdot e^{-0.011 \cdot t}$ Hint: It's a least-squares fit. Can you do that?

After that, you find the value of ' $t_{0}$ ' such that $I(t_{0}) = (1/2) \cdot I(0)$ .

Note: Quit beating yourself up. How can you not be good at it? Practice and GET good at it.

**topsquark** · September 12th 2008, 07:22 AM

A little more explanation might be useful. Using TKHunny's terminology
$I(t) = I_0e^{-at}$

Take the natural log of both sides:
$ln(I) = -at + ln(I_0)$

So if you make a plot of the data where the x axis is time and the y axis is ln(I) your data should form a line. So do a linear regression on t vs. ln(I). The slope is your a value from which you can work out the half-life and the intercept is your $ln(I_0)$ . (Notice that your slope will be a negative number and that the slope is equal to -a. That get's rid of the negative sign.)

-Dan

**apple12** · September 12th 2008, 11:32 PM

Okay Thanks! I've done that part already.

Can you teach me how to do this part?: The bismuth decays into thallium by emitting an alpha particle, and this particle is 6.64 x 10^(-27) kg. What is the momentum of the alpha particle? And What is the KE of the rocoiling Thaillium nucleus? Also, What is the total energy released during the decay of 1 mole of bismuth 212?

**CaptainBlack** · September 13th 2008, 12:52 AM

Originally Posted by apple12

Okay Thanks! I've done that part already.

Can you teach me how to do this part?: The bismuth decays into thallium by emitting an alpha particle, and this particle is 6.64 x 10^(-27) kg. What is the momentum of the alpha particle? And What is the KE of the rocoiling Thaillium nucleus? Also, What is the total energy released during the decay of 1 mole of bismuth 212?

Don't you need to know the energy of the alpha to answer this?

RonL

**mr fantastic** · September 13th 2008, 01:23 AM

Originally Posted by apple12

Okay Thanks! I've done that part already.

Can you teach me how to do this part?: The bismuth decays into thallium by emitting an alpha particle, and this particle is 6.64 x 10^(-27) kg. What is the momentum of the alpha particle? And What is the KE of the rocoiling Thaillium nucleus? Also, What is the total energy released during the decay of 1 mole of bismuth 212?

Originally Posted by CaptainBlack

Don't you need to know the energy of the alpha to answer this?

RonL

You (the OP not CB) should be getting taught all this in class. If you know the KE of the alpha particle you should be able to calculate the momentum using the usual formula that connects KE with p.

Knowing the momentum of the alpha particle you can use equations (1) and (2) below to calculate the KE of the product Thallium nucleus.

You know the energy released from one atom of Bismuth. How many atoms in 1 mole of Bismuth ....

I can speculate at the level you're studying at so I'll outline a very simplistic (ie. non-relativistic) general approach that might be useful to you. topsquawk can elabourate/correct it if and how he sees fit.

Let $Q_{\alpha}$ represent the total energy released in the process of alpha-particle disintegration. This energy consists of the kinetic energy of the alpha particle and the kinetic energy of the product nucleus. It comes from the difference in mass between the parent nucleus and the product nuclei.

The value of $Q_{\alpha}$ is readily calculated in terms of the kinetic energy $E_{\alpha}$ of the alpha particle using the principles of conservation of energy amd conservation of momentum.

Suppose that the mass of the parent nucleus is $m_1$ and assume it to be at rest. When it emits an alpha particle of mass $m_{\alpha}$ and velocity $v_{\alpha}$ the residual nucleus of mass $m_2$ will recoil with velocity $v_2$ such that:

Conservation of momentum: $m_2 v_2 = m_{\alpha} v_{\alpha}$ .... (1)

Note also that:

$Q_{\alpha} = \frac{1}{2} m_2 v_2^2 + \frac{1}{2} m_{\alpha} v_{\alpha}^2$ .... (2)

Eliminate $v_2$ from equations (1) and (2):

$Q_{\alpha} = \frac{1}{2} \frac{m_{\alpha}}{m_2} m_{\alpha} v_{\alpha}^2 + \frac{1}{2} m_{\alpha} v_{\alpha}^2 = \frac{1}{2} m_{\alpha} v_{\alpha}^2 \left( \frac{m_{\alpha}}{m_2} + 1\right) = E_{\alpha} \left( \frac{m_{\alpha}}{m_2} + 1\right)$ .

To a very close approximation the ratio of masses can be replaced with the ratio of mass numbers: $\frac{m_{\alpha}}{m_2} = \frac{4}{A - 4}$

where A is the mass number of the parent atom.

Hence you have the very simple formula: $Q_{\alpha} = \left( \frac{4}{A - 4}\right) E_{\alpha}$

**mr fantastic** · September 13th 2008, 03:16 AM

Originally Posted by apple12

Okay Thanks! I've done that part already.

Can you teach me how to do this part?: The bismuth decays into thallium by emitting an alpha particle, and this particle is 6.64 x 10^(-27) kg. What is the momentum of the alpha particle? And What is the KE of the rocoiling Thaillium nucleus? Also, What is the total energy released during the decay of 1 mole of bismuth 212?

It would help to know where else the questions in this thread have been asked - it might save unnecessary expenditure of time. Eg: Help! Radio Active Decay, Dead urgent!? - Yahoo! Answers

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