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The disintegration rate of a certain rad...

The disintegration rate of a certain radioactive sample at any instant is 4750 disintegrations per minute. Five minutes later the rate becomes 2700 per minute. Calculate
(a) decay constant and (b) half-life of the sample

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The correct Answer is:
To solve the problem, we will follow these steps: ### Step 1: Understand the relationship between disintegration rates and decay constant The disintegration rate of a radioactive sample decreases exponentially over time. The relationship can be expressed as: \[ R(t) = R_0 e^{-\lambda t} \] where: - \( R(t) \) is the disintegration rate at time \( t \), - \( R_0 \) is the initial disintegration rate, - \( \lambda \) is the decay constant, - \( t \) is the time elapsed. ### Step 2: Identify the given values From the problem, we have: - Initial disintegration rate, \( R_0 = 4750 \) disintegrations per minute, - Disintegration rate after 5 minutes, \( R(5) = 2700 \) disintegrations per minute, - Time elapsed, \( t = 5 \) minutes. ### Step 3: Set up the equation using the disintegration rate formula Using the values in the formula, we can write: \[ 2700 = 4750 e^{-\lambda \cdot 5} \] ### Step 4: Rearrange the equation to solve for the decay constant \( \lambda \) First, divide both sides by 4750: \[ \frac{2700}{4750} = e^{-5\lambda} \] Now take the natural logarithm of both sides: \[ \ln\left(\frac{2700}{4750}\right) = -5\lambda \] ### Step 5: Solve for \( \lambda \) Rearranging gives: \[ \lambda = -\frac{1}{5} \ln\left(\frac{2700}{4750}\right) \] ### Step 6: Calculate \( \lambda \) Now, calculate the value: 1. Calculate \( \frac{2700}{4750} \): \[ \frac{2700}{4750} \approx 0.5684 \] 2. Take the natural logarithm: \[ \ln(0.5684) \approx -0.566 \] 3. Substitute into the equation for \( \lambda \): \[ \lambda = -\frac{1}{5} \times (-0.566) \approx 0.1132 \text{ min}^{-1} \] ### Step 7: Calculate the half-life \( t_{1/2} \) The half-life is given by the formula: \[ t_{1/2} = \frac{\ln(2)}{\lambda} \] ### Step 8: Substitute \( \lambda \) to find \( t_{1/2} \) 1. Calculate \( \ln(2) \): \[ \ln(2) \approx 0.693 \] 2. Substitute \( \lambda \): \[ t_{1/2} = \frac{0.693}{0.1132} \approx 6.12 \text{ minutes} \] ### Final Answers (a) The decay constant \( \lambda \approx 0.1132 \text{ min}^{-1} \) (b) The half-life \( t_{1/2} \approx 6.12 \text{ minutes} \)

To solve the problem, we will follow these steps: ### Step 1: Understand the relationship between disintegration rates and decay constant The disintegration rate of a radioactive sample decreases exponentially over time. The relationship can be expressed as: \[ R(t) = R_0 e^{-\lambda t} \] where: - \( R(t) \) is the disintegration rate at time \( t \), - \( R_0 \) is the initial disintegration rate, ...
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Knowledge Check

  • Half Life of a certain radioactive substance is 69.3 days. Its disintegration constant is:

    A
    `0.010 "day"^(-1)`
    B
    `0.100 "day"^(-1)`
    C
    `0.001 "day"^(-1)`
    D
    `1.00 "day"^(-1)`
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