The time of decay for the nuclear reaction is given by `t = 5 t_(1//2)`. The relation between average life `tau` and time of decay `(t)` is given by
a. `3 tau "In" 2` b. `4 tau "In" 2` c. `5 tau "In" 2` d. `6 tau "In" 2`

Options

A. a. `3 tau "In" 2`
B. b. `4 tau "In" 2`
C. c. `5 tau "In" 2`
D. d. `6 tau "In" 2`

Explanation

To solve the problem, we need to relate the time of decay \( t \) to the average life \( \tau \) using the given information. Let's break it down step by step.

### Step 1: Understand the relationship between average life and half-life
The average life \( \tau \) of a radioactive substance is related to its half-life \( t_{1/2} \) by the formula:
\[ \tau = \frac{t_{1/2}}{\ln 2} \]
This means that the average life is the half-life divided by the natural logarithm of 2.

### Step 2: Use the given time of decay
According to the problem, the time of decay \( t \) is given as:
\[ t = 5 t_{1/2} \]
This means that the time of decay is five times the half-life.

### Step 3: Substitute the half-life in terms of average life
From the relationship established in Step 1, we can express \( t_{1/2} \) in terms of \( \tau \):
\[ t_{1/2} = \tau \cdot \ln 2 \]
Now, substituting this expression for \( t_{1/2} \) into the equation for \( t \):
\[ t = 5 t_{1/2} = 5 (\tau \cdot \ln 2) \]
Thus, we can write:
\[ t = 5 \tau \ln 2 \]

### Step 4: Identify the relationship between \( t \) and \( \tau \)
From the equation derived in Step 3, we can see that:
\[ t = 5 \tau \ln 2 \]
This indicates that the time of decay \( t \) is directly proportional to the average life \( \tau \) multiplied by \( 5 \ln 2 \).

### Step 5: Match with the given options
Now, we need to identify which of the provided options matches our derived equation. The options are:
- a. \( 3 \tau \ln 2 \)
- b. \( 4 \tau \ln 2 \)
- c. \( 5 \tau \ln 2 \)
- d. \( 6 \tau \ln 2 \)

From our derivation, we see that the correct answer is:
\[ \text{Option c: } t = 5 \tau \ln 2 \]

### Final Answer
The correct option is **c. \( 5 \tau \ln 2 \)**.

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