A radioactive sample decays to half of its initial concentration in 6.93 minutes. If it further decays another half in next 6.93 minutes, then the rate constant for the reaction is:

Options

A. `0.10 min^-1`
B. `.01 min^-1`
C. `1.0 min^-1`
D. `.001 min^-1`

Answer

To solve the problem, we need to determine the rate constant (k) for a radioactive decay process given that the half-life (t_half) is 6.93 minutes.

### Step-by-Step Solution:

1. **Understanding Half-Life**:
The half-life of a radioactive substance is the time taken for half of the substance to decay. In this case, we are given that the half-life (t_half) is 6.93 minutes.

2. **Identifying the Reaction Order**:
Since the problem involves radioactive decay, we can conclude that the reaction follows first-order kinetics. For first-order reactions, the half-life is constant and does not depend on the initial concentration.

3. **Using the Half-Life Formula**:
The relationship between the rate constant (k) and the half-life (t_half) for a first-order reaction is given by the formula:
\[ k = \frac{0.693}{t_{half}} \]

4. **Substituting the Half-Life**:
Now, we can substitute the value of t_half into the formula:
\[ k = \frac{0.693}{6.93 \text{ minutes}} \]

5. **Calculating the Rate Constant**:
Performing the calculation:
\[ k = \frac{0.693}{6.93} \approx 0.1 \text{ minute}^{-1} \]

### Final Answer:
The rate constant for the reaction is approximately \( k \approx 0.1 \text{ minute}^{-1} \).

---