Calculate the uncertainty in the momentum of an electron if it is confined to a linear region of length `1xx10^(-10)`metre

Options

A. `5.37xx10^(-27)" kg "ms^(-1)`
B. `5.27xx10^(-27)" g "ms^(-1)`
C. `5.37xx10^(-25)" g "ms^(-1)`
D. `5.27xx10^(-25)" kg "ms^(-1)`

Answer

To calculate the uncertainty in the momentum of an electron confined to a linear region of length \(1 \times 10^{-10}\) meters, we will use Heisenberg's uncertainty principle, which states:

\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]

where:
- \(\Delta x\) is the uncertainty in position,
- \(\Delta p\) is the uncertainty in momentum,
- \(h\) is Planck's constant (\(6.63 \times 10^{-34} \, \text{Js}\)),
- \(\pi\) is approximately \(3.14\).

### Step 1: Identify the values
We know:
- \(\Delta x = 1 \times 10^{-10} \, \text{m}\)
- \(h = 6.63 \times 10^{-34} \, \text{Js}\)

### Step 2: Rearrange the uncertainty principle
We need to isolate \(\Delta p\):

\[ \Delta p \geq \frac{h}{4\pi \Delta x} \]

### Step 3: Substitute the values
Now, substitute the known values into the equation:

\[ \Delta p \geq \frac{6.63 \times 10^{-34}}{4 \times 3.14 \times (1 \times 10^{-10})} \]

### Step 4: Calculate the denominator
First, calculate \(4 \times 3.14\):

\[ 4 \times 3.14 = 12.56 \]

Now, multiply this by \(1 \times 10^{-10}\):

\[ 12.56 \times 10^{-10} = 1.256 \times 10^{-9} \]

### Step 5: Calculate \(\Delta p\)
Now substitute back into the equation:

\[ \Delta p \geq \frac{6.63 \times 10^{-34}}{1.256 \times 10^{-9}} \]

Calculating this gives:

\[ \Delta p \geq 5.27 \times 10^{-25} \, \text{kg m/s} \]

### Conclusion
The uncertainty in the momentum of the electron is:

\[ \Delta p \geq 5.27 \times 10^{-25} \, \text{kg m/s} \]