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 find the rate constant for the radioactive decay of the sample, we can follow these steps:

### Step 1: Understand the decay process
The problem states that a radioactive sample decays to half of its initial concentration in 6.93 minutes. This means that the half-life (t₁/₂) of the radioactive substance is 6.93 minutes.

### Step 2: Use the half-life formula for first-order reactions
For first-order reactions, the relationship between the half-life (t₁/₂) and the rate constant (k) is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]

### Step 3: Rearrange the formula to solve for k
We can rearrange the formula to find the rate constant (k):
\[ k = \frac{0.693}{t_{1/2}} \]

### Step 4: Substitute the half-life into the equation
Now, we substitute the half-life (6.93 minutes) into the equation:
\[ k = \frac{0.693}{6.93 \text{ minutes}} \]

### Step 5: Calculate the rate constant
Now we perform the calculation:
\[ k = \frac{0.693}{6.93} \approx 0.09994 \text{ min}^{-1} \]

### Step 6: Round the value
Rounding off the value gives us:
\[ k \approx 0.10 \text{ min}^{-1} \]

### Final Answer
The rate constant for the reaction is approximately \( 0.10 \text{ min}^{-1} \).

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