91Ó°ÊÓ

The decay rate is a measure of how quickly a substance reduces over time. For krypton-85, the decay rate is given as 6.3% per year. This means every year, 6.3% of the krypton-85 atoms will decay into other substances. To understand this concept, consider a starting amount of krypton-85. After one year, only 93.7% of the original amount remains. By the end of the second year, 93.7% of the remaining 93.7% will be left, and so on. Knowing the decay rate helps in predicting how long it will take for the substance to diminish to a specific quantity.

Exponential decay describes the process where the quantity of a substance decreases at a rate proportional to its current value. The formula for exponential decay is:
\ N(t) = N_0 \times (1 - r)^t,
where:

• \( N(t) \) is the remaining quantity after time \( t \)
• \( N_0 \) is the initial quantity
• \( r \) is the decay rate
• \( t \) is the time

In our example, the initial quantity \( N_0 \) is reduced by a factor of 1 - 0.063 each year. This factor represents the remaining percentage of the substance after one time unit (one year). Because the decay is exponential, the amount of krypton-85 will never reach zero, but it will get very close over a long duration. Every time unit (year in this case), the quantity of krypton-85 reduces by 6.3%.

The natural logarithm (ln) is a mathematical function often used in exponential decay calculations. It works as the inverse of the exponential function. In this problem, we use the natural logarithm to transform the equation
\ (1 - 0.063)^{T_{1/2}} = 0.5
into a form where we can solve for the half-life \( T_{1/2} \). Taking the natural logarithm on both sides allows us to deal with the exponent more easily because:
\ \text{ln} ((1 - 0.063)^{T_{1/2}}) = \text{ln}(0.5)
The property of logarithms that says \( \text{ln}(a^b) = b \times \text{ln}(a) \) simplifies our equation:
\ T_{1/2} \times \text{ln}(0.937) = \text{ln}(0.5)
By solving this equation, we find the half-life \( T_{1/2} \) of krypton-85. Calculations using the natural logarithm can often be done with scientific calculators for exact results.