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A sample of radioactive material is obtained from a very old rock. The activity of the rock over a period of time is monitored, and lnA is plotted as a function of t such as in Figure (b). The slope of the line has a value of -6.1 ×10^−8y^−1.   FInd the half-life in years.

A sample of radioactive material is obtained from a very old rock. The activity of the rock over a period of time is monitored, and lnA is plotted as a function of t such as in Figure (b). The slope of the line has a value of -6.1 ×10^−8y^−1.   FInd the half-life in years.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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A sample of radioactive material is obtained from a very old rock. The activity of the rock over a period of time is monitored, and lnA is plotted as a function of t such as in Figure (b). The slope of the line has a value of -6.1 ×10^−8y^−1.

 

FInd the half-life in years. 

**Problem 6:** The equation for the activity of a radioactive substance as a function of time is given by \[ A = A_0 e^{-\lambda t}, \] and is plotted in Figure (a). If we take the natural log of both sides of this equation, the result is the following: \[ \ln A = -\lambda t + \ln A_0. \] Notice that our equation now has the form of an equation of a line, \( y = mx + b \), with \( y = \ln A \) and a slope of \(- \lambda \). This line is plotted in Figure (b). **Figure Descriptions:** - **Figure (a):** - Graph of \( A = A_0 e^{-\lambda t} \). - The y-axis represents \( A \) (activity), and the x-axis represents \( t \) (time). - The curve shows an exponential decay, starting from \( A_0 \) and decreasing over time. - **Figure (b):** - Graph of \( \ln A = -\lambda t + \ln A_0 \). - The y-axis represents \( \ln A \), and the x-axis represents \( t \). - The line has a negative slope of \(-\lambda\), indicating a linear relationship when using the natural logarithm transformation.
Transcribed Image Text:**Problem 6:** The equation for the activity of a radioactive substance as a function of time is given by \[ A = A_0 e^{-\lambda t}, \] and is plotted in Figure (a). If we take the natural log of both sides of this equation, the result is the following: \[ \ln A = -\lambda t + \ln A_0. \] Notice that our equation now has the form of an equation of a line, \( y = mx + b \), with \( y = \ln A \) and a slope of \(- \lambda \). This line is plotted in Figure (b). **Figure Descriptions:** - **Figure (a):** - Graph of \( A = A_0 e^{-\lambda t} \). - The y-axis represents \( A \) (activity), and the x-axis represents \( t \) (time). - The curve shows an exponential decay, starting from \( A_0 \) and decreasing over time. - **Figure (b):** - Graph of \( \ln A = -\lambda t + \ln A_0 \). - The y-axis represents \( \ln A \), and the x-axis represents \( t \). - The line has a negative slope of \(-\lambda\), indicating a linear relationship when using the natural logarithm transformation.
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