If 100 grams of Au-198 decays to 6.25 grams in 10.8 days, what is the half-life of Au-198?
The Correct Answer and Explanation is :
The correct answer is: 2.7 days.
To find the half-life of Au-198 given that 100 grams decays to 6.25 grams in 10.8 days, we can use the concept of half-life in radioactive decay.
Formula for Half-Life
The amount of a radioactive substance remaining after a certain time can be calculated using the formula:
[
N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}}
]
Where:
- ( N ) is the remaining quantity of the substance.
- ( N_0 ) is the initial quantity.
- ( t ) is the time elapsed.
- ( t_{1/2} ) is the half-life.
Given Values
- Initial amount (( N_0 )) = 100 grams
- Remaining amount (( N )) = 6.25 grams
- Time elapsed (( t )) = 10.8 days
Steps to Solve
- Set up the equation: [
6.25 = 100 \left( \frac{1}{2} \right)^{\frac{10.8}{t_{1/2}}}
] - Divide both sides by 100: [
0.0625 = \left( \frac{1}{2} \right)^{\frac{10.8}{t_{1/2}}}
] - Recognize that 0.0625 is ( \left( \frac{1}{2} \right)^4 ): [
\left( \frac{1}{2} \right)^4 = \left( \frac{1}{2} \right)^{\frac{10.8}{t_{1/2}}}
] - Set the exponents equal to each other: [
4 = \frac{10.8}{t_{1/2}}
] - Solve for ( t_{1/2} ): [
t_{1/2} = \frac{10.8}{4} = 2.7 \text{ days}
]
Conclusion
The half-life of Au-198 is 2.7 days.
Explanation
Radioactive decay follows an exponential pattern, where the quantity of a radioactive substance decreases over time at a rate proportional to its current amount. The half-life is a critical concept that defines the time required for half of the radioactive atoms in a sample to decay.
In this example, starting with 100 grams of Au-198, it takes 10.8 days for the quantity to reduce to 6.25 grams, demonstrating that four half-lives have occurred. Each half-life results in the remaining quantity being halved, confirming that the decay process is consistent with the observed amount remaining. Understanding half-lives is vital in various fields, including medicine and nuclear physics, where precise knowledge of decay rates informs safety protocols and treatment planning.