There are `10^(10)` radioactive nuclei in a given radioactive element. Its half-life time is 1 minute . How many nuclei will remain after 30 seconds ?
`(sqrt(2)=1.414)`

Options

A. `2xx10^(10)`
B. `4xx10^(10)`
C. `10^(5)`
D. `7xx10^(9)`

Answer

To solve the problem of how many radioactive nuclei will remain after 30 seconds, we can follow these steps:

### Step 1: Understand the given data
- Initial number of nuclei, \( N_0 = 10^{10} \)
- Half-life, \( t_{1/2} = 1 \text{ minute} = 60 \text{ seconds} \)
- Time elapsed, \( t = 30 \text{ seconds} \)

### Step 2: Calculate the decay constant (\( \lambda \))
The decay constant \( \lambda \) can be calculated using the formula:
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
Substituting the half-life:
\[ \lambda = \frac{\ln(2)}{60 \text{ seconds}} \]

### Step 3: Calculate \( \lambda t \)
Now we need to calculate \( \lambda t \):
\[ \lambda t = \lambda \times 30 \text{ seconds} = \frac{\ln(2)}{60} \times 30 = \frac{1}{2} \ln(2) \]

### Step 4: Use the radioactive decay formula
The number of remaining nuclei \( N \) after time \( t \) is given by:
\[ N = N_0 e^{-\lambda t} \]
Substituting the values we have:
\[ N = 10^{10} e^{-\frac{1}{2} \ln(2)} \]

### Step 5: Simplify the expression
Using the property of exponents \( e^{\ln(x)} = x \):
\[ e^{-\frac{1}{2} \ln(2)} = \frac{1}{\sqrt{2}} \]
Thus, we can rewrite the equation as:
\[ N = 10^{10} \cdot \frac{1}{\sqrt{2}} \]

### Step 6: Calculate the remaining nuclei
Now substituting the value of \( \sqrt{2} \):
\[ N = 10^{10} \cdot \frac{1}{1.414} \approx 10^{10} \cdot 0.707 \]
Calculating this gives:
\[ N \approx 7.07 \times 10^{9} \]

### Step 7: Final answer
Since the problem asks for the number of nuclei remaining after 30 seconds, we can round this to:
\[ N \approx 7 \times 10^{9} \]

### Summary
After 30 seconds, approximately \( 7 \times 10^{9} \) radioactive nuclei will remain.

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