80% of the radioactive nuclei present in a sample is found to remain undecayed after one day. The percentage of undecayed nuclei left after two days will be
a. 64 b. 20 c. 46 d. 80

Options

A. a. 64
B. b. 20
C. c. 46
D. d. 80

Explanation

To solve the problem step by step, we will use the concept of radioactive decay and the first-order kinetics of the decay process.

### Step 1: Understand the problem
We know that 80% of the radioactive nuclei remain undecayed after one day. This means that 20% of the nuclei have decayed. We need to find out the percentage of undecayed nuclei left after two days.

### Step 2: Define initial conditions
Let's assume the initial number of radioactive nuclei (N₀) is 100. After one day, the number of undecayed nuclei (N₁) is:
\[ N_1 = 80 \text{ (since 80% remain)} \]

### Step 3: Calculate the decay constant (λ)
Using the first-order decay formula:
\[ \lambda = \frac{2.303}{t} \log \left(\frac{N_0}{N_f}\right) \]
For the first day (t = 1 day):
\[ \lambda = \frac{2.303}{1} \log \left(\frac{100}{80}\right) \]

### Step 4: Simplify the decay constant
Calculating the logarithm:
\[ \log \left(\frac{100}{80}\right) = \log(1.25) \]
Using a calculator, we find:
\[ \log(1.25) \approx 0.09691 \]
Thus,
\[ \lambda \approx 2.303 \times 0.09691 \approx 0.223 \]

### Step 5: Calculate undecayed nuclei after two days
Now, we want to find the number of undecayed nuclei after two days (N₂). We will use the same decay constant (λ) for the second day:
\[ N_2 = N_0 e^{-\lambda t} \]
For t = 2 days:
\[ N_2 = 100 e^{-0.223 \times 2} \]
Calculating the exponent:
\[ e^{-0.446} \approx 0.640 \]
Thus,
\[ N_2 \approx 100 \times 0.640 \approx 64 \]

### Step 6: Calculate the percentage of undecayed nuclei
The percentage of undecayed nuclei after two days is:
\[ \text{Percentage undecayed} = \frac{N_2}{N_0} \times 100 = \frac{64}{100} \times 100 = 64\% \]

### Conclusion
The percentage of undecayed nuclei left after two days is **64%**. Therefore, the correct answer is **a. 64**.

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