The activity of a nucleus is inversely proportional to its half of average life. Thus, shorter the half life of an element, greater is its radioactivity, i.e., greater the number of atomsd disintegrating per second. The relation between half life and average life is `t_(1//2) = (0.693)/(lambda) = tau xx 0.693`
or `tau = 1.44 t_(1//2)`
Mark the incorrect relation.

Options

A. `N_(0) = Ne^(lambda t)`
B. `tau = 1.44 t_(0.5)`
C. `N = N_(0) ((1)/(2))^(n)`
D. `t_(1//2) = 2.303 lambda "log" 2`

Explanation

To solve the question, we need to analyze the given relations concerning radioactivity and identify the incorrect one. Let's go through the options step-by-step.

### Step 1: Understand the Relationships
1. **Activity and Half-Life**: The activity of a radioactive nucleus is inversely proportional to its half-life. This means that the shorter the half-life, the greater the activity.
2. **Half-Life and Average Life**: The relationship between half-life (t_(1/2)) and average life (τ) is given by:
\[ t_{1/2} = \frac{0.693}{\lambda} = \tau \times 0.693 \]
or
\[ \tau = 1.44 \times t_{1/2} \]

### Step 2: Analyze Each Option
1. **Option A**: \( n_0 = n \times e^{\lambda t} \)
- This is a correct representation of the relationship in radioactive decay, where \( n_0 \) is the initial number of atoms, \( n \) is the remaining number of atoms, and \( \lambda \) is the decay constant.

2. **Option B**: \( t = 1.44 \times t_{1/2} \)
- This is also correct as it represents the relationship between average life (τ) and half-life (t_(1/2)).

3. **Option C**: \( n = n_0 \times \left( \frac{1}{2} \right)^n \)
- This is correct as it describes the remaining number of atoms after n half-lives.

4. **Option D**: \( t_{1/2} = 2.303 \lambda \log(2) \)
- This is incorrect. The correct formula for half-life in terms of the decay constant (λ) is:
\[ t_{1/2} = \frac{0.693}{\lambda} \]
The expression given in Option D is not valid as it incorrectly places the constants.

### Conclusion
The incorrect relation is **Option D**.