A radioactive sample has `N_0` active at `t = 0` . If the rate of disintegration at any time is `R` and the number of atoms is `N`, them the ratio `R//N` varies with time as.

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Explanation

To solve the problem step by step, we need to analyze the relationship between the rate of disintegration \( R \) and the number of radioactive atoms \( N \) over time.

### Step 1: Understand the relationship between \( R \) and \( N \)
The rate of disintegration \( R \) is defined as the number of disintegrations per unit time. According to the principles of radioactive decay, the rate of disintegration can be expressed as:
\[ R = -\frac{dN}{dt} \]
This equation states that the rate of disintegration is equal to the negative rate of change of the number of atoms \( N \).

### Step 2: Express \( N \) in terms of time
The number of radioactive atoms \( N \) at any time \( t \) can be expressed using the exponential decay formula:
\[ N(t) = N_0 e^{-\lambda t} \]
where \( N_0 \) is the initial number of atoms at \( t = 0 \) and \( \lambda \) is the decay constant.

### Step 3: Differentiate \( N(t) \) to find \( R \)
To find the rate of disintegration \( R \), we differentiate \( N(t) \):
\[ R = -\frac{dN}{dt} = -\frac{d}{dt}(N_0 e^{-\lambda t}) = N_0 \lambda e^{-\lambda t} \]

### Step 4: Find the ratio \( \frac{R}{N} \)
Now, we can find the ratio \( \frac{R}{N} \):
\[ \frac{R}{N} = \frac{N_0 \lambda e^{-\lambda t}}{N_0 e^{-\lambda t}} = \lambda \]
This shows that the ratio \( \frac{R}{N} \) is equal to the decay constant \( \lambda \).

### Step 5: Analyze the result
Since \( \lambda \) is a constant, we conclude that the ratio \( \frac{R}{N} \) does not vary with time. It remains constant as time progresses.

### Final Answer
Thus, the ratio \( \frac{R}{N} \) varies with time as a constant value \( \lambda \).

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