The probability disintegration per second of a nucleus in a given radio-active sample

Options

A. Increases in proportion to life time lived by the nucleus
B. Decreases with the life time lived
C. Is independent of life time lived
D. Depends on the total number of identical nuclei present in the sample

Answer

To solve the question regarding the probability of disintegration per second of a nucleus in a given radioactive sample, we can follow these steps:

### Step 1: Understand the concept of decay constant
The decay constant (λ) is a key parameter in radioactive decay, representing the probability per unit time that a nucleus will decay. It is defined as the ratio of the number of disintegrations per second (activity) to the number of nuclei present.

### Step 2: Write the relationship between decay and number of nuclei
The relationship can be expressed mathematically as:
\[ -\frac{dN}{dt} = \lambda N \]
Here, \(N\) is the number of radioactive nuclei at time \(t\), and \(-\frac{dN}{dt}\) represents the rate of decay (disintegration) of the nuclei.

### Step 3: Identify the parameters
- \(N\): Total number of radioactive nuclei present in the sample.
- \(\lambda\): Decay constant, which gives the probability of decay per nucleus per second.

### Step 4: Analyze the implications of the equation
From the equation, we can see that the rate of disintegration (or decay) is directly proportional to the number of nuclei present. As time progresses, the number of nuclei decreases, which in turn affects the rate of decay.

### Step 5: Conclusion
The probability of disintegration per second of a nucleus in a given radioactive sample is represented by the decay constant \(\lambda\). Thus, the correct options related to this concept would be those that mention the decay constant and its relationship with the number of nuclei.

### Final Answer
The probability disintegration per second of a nucleus in a given radioactive sample is given by the decay constant \(\lambda\).