The rate of decay per second of a radioactive sample

Options

A. proportional to the life time lived by the nucleus
B. decreases with the life time lived
C. is independent of the life time lived
D. depends upon the total number of radioactive nuclei present in the sample

Explanation

To solve the question regarding the rate of decay per second of a radioactive sample, we can follow these steps:

### Step 1: Understand the Concept of Radioactive Decay
Radioactive decay is a random process by which unstable atomic nuclei lose energy by emitting radiation. The rate of decay is often expressed in terms of the number of decays per unit time.

### Step 2: Use the Decay Formula
The rate of decay (R) can be expressed using the formula:
\[ R = \frac{0.693}{t_{1/2}} \times N \]
Where:
- \( R \) = rate of decay (decays per second)
- \( t_{1/2} \) = half-life of the radioactive substance
- \( N \) = number of radioactive nuclei present

### Step 3: Analyze the Relationship
From the formula, we can see:
- The rate of decay is **inversely proportional** to the half-life (\( t_{1/2} \)). This means that as the half-life increases, the rate of decay decreases.
- The rate of decay is **directly proportional** to the number of radioactive nuclei (\( N \)). This means that as the number of nuclei increases, the rate of decay increases.

### Step 4: Conclusion
Based on the analysis:
- If the lifetime of the atom increases (i.e., the half-life increases), the rate of decay decreases.
- The rate of decay also depends on the total number of radioactive nuclei present in the sample.

Thus, the correct options based on the relationships derived are:
- The rate decreases with the lifetime lived (increased half-life).
- The rate depends upon the total number of radioactive nuclei present in the sample.

### Final Answer
The correct options are:
- **B**: Rate decreases with the lifetime lived.
- **D**: Rate depends upon the total number of radioactive nuclei present in the sample.

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