A radioactive sample at any instant has its disintegration rate `5000` disintegrations per minute After `5` minutes , the rate is `1250` disintegration per minute. Then , the decay constant (per minute)

Options

A. 0.8 In 2
B. 0.4 In 2
C. 0.2 In 2
D. 0.1 In 2

Answer

To find the decay constant (λ) of a radioactive sample, we can follow these steps:

### Step 1: Understand the relationship between disintegration rates and half-lives
The disintegration rate of a radioactive sample decreases over time. After each half-life, the disintegration rate is halved.

### Step 2: Identify the initial and final disintegration rates
Given:
- Initial disintegration rate (A₀) = 5000 disintegrations per minute
- Disintegration rate after 5 minutes (A) = 1250 disintegrations per minute

### Step 3: Determine the number of half-lives that have passed
From the initial and final disintegration rates, we can see that:
- After 1 half-life: A₀/2 = 5000/2 = 2500 disintegrations per minute
- After 2 half-lives: A₀/4 = 5000/4 = 1250 disintegrations per minute

Since the disintegration rate has decreased from 5000 to 1250 in 5 minutes, this indicates that 2 half-lives have passed.

### Step 4: Calculate the duration of one half-life
If 2 half-lives correspond to 5 minutes, then:
- 1 half-life (T₁/₂) = 5 minutes / 2 = 2.5 minutes

### Step 5: Use the relationship between half-life and decay constant
The relationship between half-life (T₁/₂) and decay constant (λ) is given by the formula:
\[ T_{1/2} = \frac{\ln(2)}{\lambda} \]
Rearranging this gives:
\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]

### Step 6: Substitute the half-life value into the equation
Substituting T₁/₂ = 2.5 minutes into the decay constant formula:
\[ \lambda = \frac{\ln(2)}{2.5} \]

### Step 7: Calculate the decay constant
Using the approximate value of ln(2) ≈ 0.693:
\[ \lambda = \frac{0.693}{2.5} \]
\[ \lambda ≈ 0.2772 \text{ per minute} \]

### Step 8: Final answer
Thus, the decay constant (λ) is approximately 0.2772 per minute.

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