If the decay constant of a radioactive substance is 5.3 x 10^-3 per year, then what is its half-life?
Understand the Problem
The question is asking for the calculation of the half-life of a radioactive substance based on its decay constant. Specifically, it provides the decay constant value and asks for the corresponding half-life value using the relationship between them.
Answer
The half-life is approximately $130.56$ years.
Answer for screen readers
The half-life is approximately $130.56$ years.
Steps to Solve
- Identify the decay constant and formula for half-life
The decay constant ($\lambda$) is given as $5.3 \times 10^{-3}$ per year. The formula to calculate the half-life ($t_{1/2}$) is:
$$ t_{1/2} = \frac{\ln(2)}{\lambda} $$
where $\ln(2)$ is the natural logarithm of 2.
- Calculate the half-life
Substituting the value of $\lambda$ into the formula:
$$ t_{1/2} = \frac{\ln(2)}{5.3 \times 10^{-3}} $$
- Evaluate $\ln(2)$
The natural logarithm of 2 is approximately $0.693$.
- Perform the calculation
Now substitute $\ln(2)$ into the equation:
$$ t_{1/2} = \frac{0.693}{5.3 \times 10^{-3}} $$
- Final calculation
Calculate the value:
$$ t_{1/2} = \frac{0.693}{0.0053} $$
After calculation, you will find the half-life.
The half-life is approximately $130.56$ years.
More Information
The half-life of a radioactive substance is the time it takes for half of the material to decay. This is an important concept in nuclear physics and radiochemistry, as it helps understand the stability of different isotopes over time.
Tips
- Mistaking the decay constant units can lead to incorrect calculations. Always ensure the decay constant is in compatible units.
- Neglecting to round or keep track of significant figures in final answers.
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