Half life of a radio-active substance is `20` minutes. The time between `20 %` and `80 %` decay will be

Options

A. 20 min
B. 40 min
C. 30 min
D. 25 min

Explanation

To solve the problem of finding the time between 20% and 80% decay of a radioactive substance with a half-life of 20 minutes, we can follow these steps:

### Step 1: Understand the half-life concept
The half-life (T_half) of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. In this case, T_half = 20 minutes.

### Step 2: Determine the decay fractions
- 80% decay means that 20% of the original substance remains.
- 20% decay means that 80% of the original substance remains.

### Step 3: Set up the equations
Using the formula for radioactive decay:
\[ N = N_0 \left(\frac{1}{2}\right)^{\frac{T}{T_{half}}} \]

For 80% decay:
\[ \frac{N}{N_0} = 0.20 \]
This can be expressed as:
\[ 0.20 = \left(\frac{1}{2}\right)^{\frac{T_2}{20}} \]

For 20% decay:
\[ \frac{N}{N_0} = 0.80 \]
This can be expressed as:
\[ 0.80 = \left(\frac{1}{2}\right)^{\frac{T_1}{20}} \]

### Step 4: Solve for T1 and T2
Taking logarithm for both equations:
1. For 80% decay:
\[ \log(0.20) = \frac{T_2}{20} \log\left(\frac{1}{2}\right) \]
Rearranging gives:
\[ T_2 = 20 \cdot \frac{\log(0.20)}{\log(0.5)} \]

2. For 20% decay:
\[ \log(0.80) = \frac{T_1}{20} \log\left(\frac{1}{2}\right) \]
Rearranging gives:
\[ T_1 = 20 \cdot \frac{\log(0.80)}{\log(0.5)} \]

### Step 5: Calculate T1 and T2
Using logarithmic values:
- \( \log(0.20) \approx -0.699 \)
- \( \log(0.80) \approx -0.097 \)
- \( \log(0.5) \approx -0.301 \)

Calculating T2:
\[ T_2 = 20 \cdot \frac{-0.699}{-0.301} \approx 20 \cdot 2.32 \approx 46.4 \text{ minutes} \]

Calculating T1:
\[ T_1 = 20 \cdot \frac{-0.097}{-0.301} \approx 20 \cdot 0.322 \approx 6.44 \text{ minutes} \]

### Step 6: Find the time difference
Now, we find the time between T2 and T1:
\[ T_2 - T_1 = 46.4 - 6.44 \approx 39.96 \text{ minutes} \]

### Step 7: Round the result
Rounding gives us approximately 40 minutes.

### Final Answer
The time between 20% and 80% decay is approximately **40 minutes**.

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