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