If the uncertainty in the position of an electron is `10^(-10)` m, then what be the value of uncertainty in its momentum in kg `m s^(-1)` ? `(h = 6.62 xx10^(-34) Js)`

Options

A. `0.52xx10^(-24)`
B. `1.01xx10^(-24)`
C. `1.09xx10^(-24)`
D. `1.07xx10^(-24)`

Answer

To find the uncertainty in momentum of an electron given the uncertainty in its position, we can use the Heisenberg 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, approximately \(6.62 \times 10^{-34} \, \text{Js}\).

Given:
- \(\Delta x = 10^{-10} \, \text{m}\)

We need to find \(\Delta p\).

### Step 1: Rearranging the Uncertainty Principle

From the uncertainty principle, we can rearrange the equation to solve for \(\Delta p\):

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

### Step 2: Plugging in the Values

Now we substitute the values into the equation:

\[ \Delta p \geq \frac{6.62 \times 10^{-34}}{4 \cdot \pi \cdot 10^{-10}} \]

### Step 3: Calculating the Denominator

First, calculate \(4 \cdot \pi\):

\[ 4 \cdot \pi \approx 4 \cdot 3.14 = 12.56 \]

So, the denominator becomes:

\[ 4 \cdot \pi \cdot 10^{-10} \approx 12.56 \times 10^{-10} \]

### Step 4: Performing the Division

Now, we can perform the division:

\[ \Delta p \geq \frac{6.62 \times 10^{-34}}{12.56 \times 10^{-10}} \]

Calculating this gives:

\[ \Delta p \geq \frac{6.62}{12.56} \times 10^{-34 + 10} = 0.526 \times 10^{-24} \]

### Step 5: Final Result

Thus, the uncertainty in momentum is approximately:

\[ \Delta p \geq 0.526 \times 10^{-24} \, \text{kg m/s} \]

This can be rounded to:

\[ \Delta p \approx 0.53 \times 10^{-24} \, \text{kg m/s} \]

### Summary

The uncertainty in the momentum of the electron is approximately \(0.53 \times 10^{-24} \, \text{kg m/s}\).

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