Radioactive decay follows first-order kinetic. The mean life and half-life of nuclear decay process are `tau = 1// lambda` and `t_(1//2) = 0.693//lambda`. Therefore are a number of radioactive elements in nature, their abundance is directly proportional to half life. The amount remaining after `n` half lives of radioactive elements can be calculated using the relation:
`N = N_(0) ((1)/(2))^(n)`
Which `"is"//"are"` true about the decay cosntant?

Options

A. Unit of `lambda` is `"time"^(-1)`
B. `lambda` is independent of temperature
C. `lambda` depends on the initial amount of element taken.
D. `lambda` depends on the nature of radioactive element.

Answer

To determine which statements about the decay constant (λ) are true, let's analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding the Decay Constant (λ)**:
- The decay constant (λ) is a proportionality constant in the radioactive decay equation. It relates the rate of decay of a radioactive substance to the number of atoms present.

2. **Analyzing the Options**:
- **Option 1**: "The unit of λ is time inverse."
- The decay rate (dN/dt) is proportional to the number of atoms (N), leading to the equation:
\[ -\frac{dN}{dt} = \lambda N \]
Rearranging gives:
\[ \lambda = -\frac{1}{N} \frac{dN}{dt} \]
The unit of dN is atoms, and dt is time. Thus, λ has units of time\(^{-1}\).
- **Conclusion**: This statement is **true**.

- **Option 2**: "λ is independent of temperature."
- The decay constant is a characteristic of the radioactive element and does not change with temperature. Radioactive decay is a nuclear process and is not affected by external conditions such as temperature.
- **Conclusion**: This statement is **true**.

- **Option 3**: "λ depends on the initial amount of element taken."
- The decay constant (λ) is intrinsic to the radioactive isotope and does not depend on how much of the substance is present. It is a fixed value for a given isotope.
- **Conclusion**: This statement is **false**.

- **Option 4**: "λ depends on the nature of the radioactive element."
- Each radioactive element has a unique decay constant that reflects its stability and the nature of its decay process. Thus, λ is indeed dependent on the type of radioactive element.
- **Conclusion**: This statement is **true**.

3. **Final Assessment**:
- The true statements about the decay constant are:
- Option 1: True
- Option 2: True
- Option 3: False
- Option 4: True

### Summary of True Statements:
- The unit of λ is time inverse.
- λ is independent of temperature.
- λ depends on the nature of the radioactive element.