The rate of disintegration of a radioactive substance is `……………………………` to the number of atoms present at that time .

Answer

To solve the question, we need to understand the relationship between the rate of disintegration of a radioactive substance and the number of atoms present at a given time. Here’s a step-by-step breakdown:

### Step 1: Understand Radioactive Disintegration
Radioactive disintegration refers to the process by which unstable atomic nuclei lose energy by emitting radiation. This process is characterized by a certain rate, which is influenced by the number of radioactive atoms present.

### Step 2: Recognize the Kinetic Order
Radioactive decay follows first-order kinetics. This means that the rate of disintegration is proportional to the concentration of the radioactive substance.

### Step 3: Write the Rate Equation
The rate of disintegration can be expressed mathematically as:
\[ -\frac{dN}{dt} = \lambda N \]
Where:
- \(N\) is the number of radioactive atoms present at time \(t\),
- \(\lambda\) is the decay constant,
- \(-\frac{dN}{dt}\) represents the rate of disintegration.

### Step 4: Analyze the Relationship
From the equation, we can see that the rate of disintegration (\(-\frac{dN}{dt}\)) is directly proportional to the number of atoms (\(N\)). This indicates that as the number of radioactive atoms increases, the rate of disintegration also increases.

### Step 5: Conclusion
Thus, we can conclude that the rate of disintegration of a radioactive substance is **directly proportional** to the number of atoms present at that time.

### Final Answer
The rate of disintegration of a radioactive substance is **directly proportional** to the number of atoms present at that time.

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