`90%` of the active nuclei present in a radioactive sample are found to remain undecyayed after `1` day. The precentage of undecayed nuclei left after two days will be

Options

A. 0.81
B. 0.19
C. 0.2
D. 1

Answer

To solve the problem, we need to determine the percentage of undecayed nuclei left after two days, given that 90% of the active nuclei remain undecayed after one day. We can follow these steps:

### Step 1: Understand the initial condition
Let the initial number of active nuclei at time \( t = 0 \) be \( N_0 \). After one day, 90% of these nuclei remain undecayed. Therefore, the number of undecayed nuclei after one day is:
\[ N_1 = 0.9 N_0 \]

### Step 2: Use the decay formula
The number of undecayed nuclei at any time \( t \) can be expressed using the exponential decay formula:
\[ N_t = N_0 e^{-\lambda t} \]
where \( \lambda \) is the decay constant.

### Step 3: Set up the equation for one day
For \( t = 1 \) day, we can write:
\[ N_1 = N_0 e^{-\lambda \cdot 1} \]
Substituting the value of \( N_1 \) from Step 1:
\[ 0.9 N_0 = N_0 e^{-\lambda} \]

### Step 4: Simplify the equation
Dividing both sides by \( N_0 \) (assuming \( N_0 \neq 0 \)):
\[ 0.9 = e^{-\lambda} \]

### Step 5: Find the expression for \( e^{-\lambda} \)
Taking the natural logarithm of both sides, we can express \( \lambda \):
\[ -\lambda = \ln(0.9) \]
Thus,
\[ \lambda = -\ln(0.9) \]

### Step 6: Calculate the number of undecayed nuclei after two days
Now, we need to find the number of undecayed nuclei after two days (\( t = 2 \)):
\[ N_2 = N_0 e^{-\lambda \cdot 2} \]
Substituting \( \lambda \):
\[ N_2 = N_0 e^{-2(-\ln(0.9))} = N_0 e^{2\ln(0.9)} = N_0 (e^{\ln(0.9)})^2 = N_0 (0.9^2) \]

### Step 7: Calculate \( 0.9^2 \)
Calculating \( 0.9^2 \):
\[ 0.9^2 = 0.81 \]
Thus, the number of undecayed nuclei after two days is:
\[ N_2 = 0.81 N_0 \]

### Step 8: Find the percentage of undecayed nuclei
To find the percentage of undecayed nuclei left after two days, we can express it as:
\[ \text{Percentage undecayed} = \left( \frac{N_2}{N_0} \right) \times 100 = 0.81 \times 100 = 81\% \]

### Final Answer
The percentage of undecayed nuclei left after two days is **81%**.

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