What is the disintegration constant of radon if the number of its atoms diminishes by `18%` in 24 h? `["Take ln "(0.82)~~-0.2]`

Options

A. `2.3xx10^(-3)s^(-1)`
B. `2.3xx10^(-4)s^(-1)`
C. `2.3xx10^(-5)s^(-1)`
D. `2.3xx10^(-6)s^(-1)`

Answer

To find the disintegration constant (λ) of radon when the number of its atoms diminishes by 18% in 24 hours, we can follow these steps:

### Step 1: Understand the Problem
We know that 18% of the atoms have decayed, which means 82% of the atoms remain after 24 hours.

### Step 2: Write the Relationship
The relationship between the number of atoms remaining (N) and the initial number of atoms (N0) can be expressed using the formula:
\[ N = N_0 e^{-\lambda t} \]

### Step 3: Set Up the Equation
Since 82% of the atoms remain, we can express this as:
\[ \frac{N}{N_0} = 0.82 \]
Thus, we can rewrite our equation as:
\[ 0.82 = e^{-\lambda t} \]

### Step 4: Substitute the Time
We know the time (t) is 24 hours. To convert this into seconds:
\[ t = 24 \text{ hours} = 24 \times 60 \times 60 = 86400 \text{ seconds} \]

### Step 5: Substitute into the Equation
Now, substituting the value of t into the equation:
\[ 0.82 = e^{-\lambda \cdot 86400} \]

### Step 6: Take the Natural Logarithm
To solve for λ, we take the natural logarithm of both sides:
\[ \ln(0.82) = -\lambda \cdot 86400 \]

### Step 7: Solve for λ
We know from the problem that:
\[ \ln(0.82) \approx -0.2 \]
Substituting this value into the equation gives:
\[ -0.2 = -\lambda \cdot 86400 \]
Now, solving for λ:
\[ \lambda = \frac{0.2}{86400} \]

### Step 8: Calculate λ
Calculating the value:
\[ \lambda \approx \frac{0.2}{86400} \approx 2.31 \times 10^{-6} \text{ s}^{-1} \]

### Final Answer
Thus, the disintegration constant (λ) of radon is approximately:
\[ \lambda \approx 2.31 \times 10^{-6} \text{ s}^{-1} \]

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