At time `t=0` , number of nuclei of a radioactive substance are 100. At `t=1` s these numbers become 90. Find the number of nuclei at `t=2 s`.

Explanation

To solve the problem step by step, we will use the radioactive decay law.

### Step 1: Understand the initial conditions
At time \( t = 0 \), the number of nuclei \( N_0 \) is given as 100. At time \( t = 1 \) second, the number of nuclei \( N \) becomes 90.

### Step 2: Use the radioactive decay formula
The formula for radioactive decay is given by:

\[ N = N_0 e^{-\lambda t} \]

Where:
- \( N \) is the number of nuclei at time \( t \)
- \( N_0 \) is the initial number of nuclei
- \( \lambda \) is the decay constant
- \( t \) is the time elapsed

### Step 3: Set up the equation for \( t = 1 \) second
Substituting the known values into the decay formula for \( t = 1 \) second:

\[ 90 = 100 e^{-\lambda \cdot 1} \]

### Step 4: Solve for \( e^{-\lambda} \)
Rearranging the equation gives:

\[ \frac{90}{100} = e^{-\lambda} \]

This simplifies to:

\[ e^{-\lambda} = 0.9 \]

### Step 5: Find the number of nuclei at \( t = 2 \) seconds
Now we need to find the number of nuclei at \( t = 2 \) seconds. Using the decay formula again:

\[ N = N_0 e^{-\lambda t} \]

Substituting \( t = 2 \) seconds:

\[ N = 100 e^{-\lambda \cdot 2} \]

### Step 6: Substitute \( e^{-\lambda} \) into the equation
From the previous step, we know that \( e^{-\lambda} = 0.9 \). Therefore:

\[ e^{-\lambda \cdot 2} = (e^{-\lambda})^2 = (0.9)^2 = 0.81 \]

### Step 7: Calculate \( N \)
Now substituting back into the equation:

\[ N = 100 \cdot 0.81 = 81 \]

### Conclusion
The number of nuclei at \( t = 2 \) seconds is **81**.

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