Many unstable nuclie can decay spontaneously to a nucleus of lower mass but differnet combination of nucleons. The process of spontaneous emission of radiation is called radioactivity substance.
Radioactive decay is a statistical process. Radiaactivity is independent of all external conditions. The number of decays per unit time or decay rate is called activity. Activity exponentially decrease with time. Mean lifetime is always greater than half-life time.
n number of `alpha`-particels per second are being emitted by B atoms of a radioactive element. The half-life of element will be

Options

A. `(n)/(N) s`
B. `(N)/(n )s`
C. `0693 (N)/(n) s`
D. `0.693 (n)/(N) s`

Explanation

To find the half-life of a radioactive element based on the number of alpha particles emitted per second, we can follow these steps:

### Step 1: Understand the relationship between activity and decay constant
The activity \( A \) of a radioactive substance is defined as the number of decays per unit time, which can be expressed as:
\[ A = \lambda N \]
where:
- \( A \) is the activity (number of decays per second),
- \( \lambda \) is the decay constant,
- \( N \) is the number of radioactive nuclei present.

### Step 2: Relate activity to the number of emitted alpha particles
In this problem, we are given that \( n \) number of alpha particles are emitted per second, which means:
\[ A = n \]
Thus, we can write:
\[ n = \lambda N \]

### Step 3: Express the decay constant in terms of half-life
The decay constant \( \lambda \) is related to the half-life \( t_{1/2} \) of the radioactive element by the formula:
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
where \( \ln 2 \) is approximately 0.693.

### Step 4: Substitute the decay constant into the activity equation
Substituting for \( \lambda \) in the activity equation gives:
\[ n = \frac{\ln 2}{t_{1/2}} N \]

### Step 5: Solve for half-life
Rearranging the equation to solve for the half-life \( t_{1/2} \):
\[ t_{1/2} = \frac{N \ln 2}{n} \]

### Step 6: Conclusion
Thus, the half-life of the element can be expressed as:
\[ t_{1/2} = \frac{N \cdot 0.693}{n} \]