Exponential Decay and Half-Life

Exponential Decay Graph

• N: Number of radioactive nuclei as a function of time (T).
• Equation: ( N = N_0 e^{-\lambda T} )
  • ( N_0 ): Initial number of nuclei
  • ( \lambda ): Decay constant (can also be called K)

Initial Number of Nuclei

• At ( T = 0 ):
  • ( N = N_0 e^{-\lambda \cdot 0} = N_0 )
  • ( N_0 ) represents the initial number of radioactive nuclei.

Half-Life

• Half-life: Time when the number of radioactive nuclei is half its initial value.
• At half-life:
  • ( N = \frac{N_0}{2} )
• Equation for half-life:
  • ( \ln(\frac{1}{2}) = -0.693 )
  • ( -\lambda T_{1/2} = -0.693 )
  • Solving for Half-life: ( T_{1/2} = \frac{0.693}{\lambda} )
• Solving for Decay Constant: ( \lambda = \frac{0.693}{T_{1/2}} )_

Semi-log Plots

Overview

• Another method to analyze exponential decay data.
• Conversion to linear form:
  • Start with ( N = N_0 e^{-\lambda T} )
  • Divide by ( N_0 ): ( \frac{N}{N_0} = e^{-\lambda T} )
  • Take the natural log: ( \ln(\frac{N}{N_0}) = -\lambda T )
  • Rearrange to linear form: ( \ln(N) = -\lambda T + \ln(N_0) )

Interpretation

• Resembles the straight-line equation ( Y = MX + B )
  • Y: ( \ln(N) )
  • M: (-\lambda)
  • X: Time (T)
  • B: ( \ln(N_0) ) (Vertical intercept)
• Graphing:
  • Y-axis: ( \ln(N) )
  • X-axis: Time (T)
  • Vertical intercept: ( \ln(N_0) )
  • Slope: (-\lambda)
• Uses:
  • Determining the decay constant from the slope.
  • Calculating half-life using the decay constant.