Two radioactive substance `X` and `Y` initially contain equal number of nuclei. `X` has a half life of `1` hour and `Y` has a half-life of `2` hours. After 2 hours the ratio of activity of `X` to that of `y` is

Options

A. `1:4`
B. `1:2`
C. `1:1`
D. `2:1`

Explanation

To solve the problem of finding the ratio of activity of two radioactive substances \(X\) and \(Y\) after 2 hours, we can follow these steps:

### Step 1: Understand the given data
- Both substances \(X\) and \(Y\) initially contain equal number of nuclei, denoted as \(N_0\).
- The half-life of substance \(X\) is 1 hour.
- The half-life of substance \(Y\) is 2 hours.

### Step 2: Define the formula for activity
The activity \(A\) of a radioactive substance is given by the formula:
\[ A = A_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \]
where:
- \(A_0\) is the initial activity,
- \(t\) is the time elapsed,
- \(t_{1/2}\) is the half-life of the substance.

### Step 3: Calculate the activity of substance \(X\) after 2 hours
For substance \(X\):
- Initial activity \(A_{0X} = N_0\) (since the initial number of nuclei is the same).
- Half-life \(t_{1/2X} = 1\) hour.
- Time \(t = 2\) hours.

Using the formula:
\[ A_X = A_{0X} \left(\frac{1}{2}\right)^{\frac{2}{1}} = N_0 \left(\frac{1}{2}\right)^{2} = N_0 \cdot \frac{1}{4} = \frac{N_0}{4} \]

### Step 4: Calculate the activity of substance \(Y\) after 2 hours
For substance \(Y\):
- Initial activity \(A_{0Y} = N_0\).
- Half-life \(t_{1/2Y} = 2\) hours.
- Time \(t = 2\) hours.

Using the formula:
\[ A_Y = A_{0Y} \left(\frac{1}{2}\right)^{\frac{2}{2}} = N_0 \left(\frac{1}{2}\right)^{1} = N_0 \cdot \frac{1}{2} = \frac{N_0}{2} \]

### Step 5: Find the ratio of activities \(A_X\) to \(A_Y\)
Now, we can find the ratio of the activities:
\[ \text{Ratio} = \frac{A_X}{A_Y} = \frac{\frac{N_0}{4}}{\frac{N_0}{2}} = \frac{1/4}{1/2} = \frac{1}{4} \cdot \frac{2}{1} = \frac{1}{2} \]

### Step 6: Conclusion
The ratio of the activity of \(X\) to that of \(Y\) after 2 hours is:
\[ \text{Ratio} = 1 : 2 \]

### Final Answer
Thus, the final answer is \(1 : 2\).

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