Results 1 to 7 of 7

Thread: Radioactive decay Problem

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    9

    Radioactive decay Problem

    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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2
    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 $\displaystyle I(t) = 681.82 \cdot e^{-0.011 \cdot t}$ Hint: It's a least-squares fit. Can you do that?

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

    Note: Quit beating yourself up. How can you not be good at it? Practice and GET good at it.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    11,124
    Thanks
    720
    Awards
    1
    A little more explanation might be useful. Using TKHunny's terminology
    $\displaystyle I(t) = I_0e^{-at}$

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

    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 $\displaystyle ln(I_0)$. (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
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Sep 2008
    Posts
    9
    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?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    5
    Quote Originally Posted by apple12 View Post
    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
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    9
    Quote Originally Posted by apple12 View Post
    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?
    Quote Originally Posted by CaptainBlack View Post
    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 $\displaystyle Q_{\alpha}$ 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 $\displaystyle Q_{\alpha}$ is readily calculated in terms of the kinetic energy $\displaystyle E_{\alpha}$ of the alpha particle using the principles of conservation of energy amd conservation of momentum.

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

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

    Note also that:

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

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

    $\displaystyle 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: $\displaystyle \frac{m_{\alpha}}{m_2} = \frac{4}{A - 4}$

    where A is the mass number of the parent atom.

    Hence you have the very simple formula: $\displaystyle Q_{\alpha} = \left( \frac{4}{A - 4}\right) E_{\alpha}$
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    9
    Quote Originally Posted by apple12 View Post
    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
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Radioactive Decay Problem
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: Aug 26th 2011, 10:50 AM
  2. Radioactive Decay Problem?
    Posted in the Differential Equations Forum
    Replies: 6
    Last Post: Aug 25th 2010, 03:16 PM
  3. radioactive decay word problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 3rd 2010, 12:00 AM
  4. Radioactive Decay Problem
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: Apr 25th 2010, 04:32 PM
  5. Differntial Problem Radioactive decay
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Feb 24th 2009, 01:49 PM

Search tags for this page

Search Tags


/mathhelpforum @mathhelpforum