A container is filled with a radioactive substance for which the half-life is` 2 days` . A week later, when the container is opened, it contains `5 g` of the substance. Approximately how many grams of the substance were initally placed in the container ?

Options

A. `40`
B. `60`
C. `80`
D. `100`

Explanation

To find the initial mass of the radioactive substance in the container, we can use the concept of half-life and the decay formula. Here’s the step-by-step solution:

### Step 1: Understand the half-life
The half-life of the substance is given as 2 days. This means that every 2 days, half of the remaining substance decays.

### Step 2: Determine the time elapsed
A week is equivalent to 7 days. We need to find out how many half-lives have passed in this time period:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{7 \text{ days}}{2 \text{ days}} = 3.5 \]
This means that 3.5 half-lives have passed in 7 days.

### Step 3: Calculate the remaining mass after 3.5 half-lives
The remaining mass of a radioactive substance after \( n \) half-lives can be calculated using the formula:
\[ M = M_0 \left(\frac{1}{2}\right)^n \]
where \( M_0 \) is the initial mass and \( n \) is the number of half-lives.

For 3.5 half-lives:
\[ M = M_0 \left(\frac{1}{2}\right)^{3.5} \]

### Step 4: Substitute the known values
We know that after 3.5 half-lives, the remaining mass \( M \) is 5 grams:
\[ 5 = M_0 \left(\frac{1}{2}\right)^{3.5} \]

### Step 5: Calculate \( \left(\frac{1}{2}\right)^{3.5} \)
Calculating \( \left(\frac{1}{2}\right)^{3.5} \):
\[ \left(\frac{1}{2}\right)^{3.5} = \frac{1}{2^{3.5}} = \frac{1}{\sqrt{2^7}} = \frac{1}{\sqrt{128}} \approx 0.0884 \]

### Step 6: Solve for \( M_0 \)
Now, we can rearrange the equation to find \( M_0 \):
\[ M_0 = \frac{5}{\left(\frac{1}{2}\right)^{3.5}} \approx \frac{5}{0.0884} \approx 56.5 \text{ grams} \]

### Step 7: Round to the nearest option
Since we are looking for an approximate value, we can round this to 60 grams.

### Conclusion
Thus, the initial mass of the substance placed in the container was approximately **60 grams**.

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