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Question

To determine the half- life of radioactive element, a student plots graph of lnopen vertical bar fraction numerator d N blank left parenthesis t right parenthesis over denominator d t end fraction close vertical bar v e r s u s t. Here fraction numerator d N blank left parenthesis t right parenthesis over denominator d t end fraction is the rate of radioactive decay at time t. If the number of radioactive nuclei of this element decreases by a factor of p after 4.16 yr, the value of p is

  1. 8    
  2. 7    
  3. 4    
  4. 8.5    

The correct answer is: 8


    open vertical bar fraction numerator d N over denominator d t end fraction close vertical bar= |Activity of radioactive substance|
    equals blank lambda N blank equals blank lambda N subscript 0 end subscript e to the power of negative lambda t end exponent blank left parenthesis because blank N blank equals blank N subscript 0 end subscript e to the power of negative lambda t end exponent right parenthesis
    Taking log both sides
    l n open vertical bar fraction numerator d N over denominator d t end fraction close vertical bar equals blank l n left parenthesis blank lambda N subscript 0 end subscript right parenthesis minus lambda t
    Hence, lnopen vertical bar fraction numerator d N over denominator d t end fraction close vertical bar v e r s u st graph is a straight line with slope-lambda.
    From the graph we can see that,
    lambda equals fraction numerator 1 over denominator 2 end fraction equals 0.5 blank y r to the power of negative 1 end exponent
    Now applying the equation
    N blank equals blank N subscript 0 end subscript e to the power of negative lambda t end exponent equals N subscript 0 end subscript e to the power of negative 0.5 blank cross times blank 4.16 end exponent
    equals N subscript 0 end subscript e to the power of negative 2.08 end exponent blank equals blank 0.125 blank N subscript 0 end subscript
    equals blank fraction numerator N subscript 0 end subscript over denominator 8 end fraction

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