Questions: A substance has a half-life of 4.044 minutes. If the initial amount of the substance was 140.8 grams, how many half-lives will have passed before the substance decays to 8.8 grams? half-lives What is the total time of decay? min

Solution Steps

The half-life of a substance is the time it takes for half of the substance to decay. In this problem, the half-life is given as 4.044 minutes.

To find the number of half-lives that have passed, we use the formula:

\[ N = \frac{\log(\frac{A_0}{A})}{\log(2)} \]

where \( A_0 \) is the initial amount, \( A \) is the final amount, and \( N \) is the number of half-lives.

Given:

• \( A_0 = 140.8 \) grams
• \( A = 8.8 \) grams

Substitute these values into the formula:

\[ N = \frac{\log(\frac{140.8}{8.8})}{\log(2)} \]

Calculate:

\[ N = \frac{\log(16)}{\log(2)} = \frac{4}{1} = 4 \]

The total time of decay is the number of half-lives multiplied by the half-life duration:

\[ \text{Total time} = N \times \text{half-life duration} = 4 \times 4.044 \, \text{minutes} \]

Calculate:

\[ \text{Total time} = 16.176 \, \text{minutes} \]

Final Answer