Nuclear HalfLife Problems

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Nuclear Half-Life Problems
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Background

• The amount of time required for ½ of a
  radioactive sample to transmutate (change into
  another substance).
• 1 half-life 1/2 remains
• 2 half-lives 1/4 remains
• 3 half-lives 1/8 remains
• 4 half-lives 1/16 remains
• The pattern is 1/2n where n of half-lives

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Background cont

• As long as you are working with whole half-lives,
  the math is relatively easy.
• The purpose of this lesson is to solve half-life
  problems that are not whole number or 1/2n
  fractions.

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Logarithms

• Logarithm (log) of a positive number is the
  exponent or power of a given base that is
  required to give that number.
• log (common log) uses base 10
• ln (natural log) uses base e (e 2.71828)
• e 1 1/1! 1/2! 1/3! ...
• e is the limit of (1 1/n)n as n tends to
  infinity

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Logarithms cont

• Since logarithms are exponents, all properties of
  exponents are also properties of logarithms
• The logarithm of the product of two numbers is
  the sum of their logarithms (log ab log a
  log b)
• The logarithm of the quotient of two numbers is
  the logarithm of the numerator minus the
  logarithm of the denominator (log a/b log a
  log b)

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Logarithms cont

• The log of the nth power of a number is n times
  the log of the number (log an n log a)
• ln N 2.303 log N (convenient conversion)
• Negative numbers do not have logarithms
• Logarithms of numbers greater than 1 are positive
  and numbers less than 1 are negative
• Logarithms of 0 are infinitely negative

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Logarithms cont

• Given the value of the logarithm of a number to a
  certain base, the process of finding the number
  is called taking the antilogarithm.
• 10x and ex raising a base to the power of the
  logarithm is synonymous with taking the
  antilogarithm.

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First Order Rate Equation

• The rate for a nuclear half-life is a first order
  equation and is proportional to the concentration
  of the reactant A and the methods of integral
  calculus gives us

AE -Conc. Sub. left AO - Orig. Conc. k -
rate constant t - time duration
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• To simplify

If t is half-life then AE ½ AO
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First Order Rate Equation cont

• Another useful equation is

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First Order Rate Equation cont
AE amount of substance remaining AO
original amount of substance t - elapsed
time t ½ - half-life of substance
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Graphing Calculators Rejoice!

• Your graphing calculator can handle all of this
  heavy math ? hurrah!!!
• Lets use the equation solver
• Press the MATH key
• Choose 0Solver
• Enter the equation solver. You may need to press
  the up key

• Enter the following equation
• eqn 0O0.5(T/H)-A
• Letter O represents orig. amt.
• Letter T represents time
• Letter H represents half-life
• Letter A represents remaining amt.
• Press enter

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Graphing Calculators Rejoice!

• Enter the values in the problem for each letter.
• Place the cursor on the unknown and press ALPHA
  SOLVE. That is the ALPHA and ENTER keys
  simultaneously.
• The answer should appear.
• Pretty slick isn't it?
• Happy calculating