The position of a hydrogen atom `(m = 1.7 xx 10^(-27) kg)` is known within `2.0 xx 10^(-6) m`. What is the minimum uncertainty in the atom's velocity?
(a) `zero m//s`
(b) `0..011 m//s`
(c) `0.0085 m//s`
(d) `0.016 m//s`
(e) `0.031 m//s`

Answer

To solve the problem of finding the minimum uncertainty in the velocity of a hydrogen atom given the uncertainty in its position, we will use Heisenberg's Uncertainty Principle. The principle states that:

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

Where:
- \(\Delta p\) is the uncertainty in momentum,
- \(\Delta x\) is the uncertainty in position,
- \(h\) is Planck's constant, approximately \(6.626 \times 10^{-34} \, \text{Js}\).

1. **Identify the given values**:
- Mass of the hydrogen atom, \(m = 1.7 \times 10^{-27} \, \text{kg}\)
- Uncertainty in position, \(\Delta x = 2.0 \times 10^{-6} \, \text{m}\)

2. **Calculate the uncertainty in momentum (\(\Delta p\))**:
From the uncertainty principle, we can rearrange it to find \(\Delta p\):
\[ \Delta p = \frac{h}{4\pi \Delta x} \]

3. **Substitute the values**:
- Substitute \(h = 6.626 \times 10^{-34} \, \text{Js}\) and \(\Delta x = 2.0 \times 10^{-6} \, \text{m}\):
\[ \Delta p = \frac{6.626 \times 10^{-34}}{4 \pi (2.0 \times 10^{-6})} \]

4. **Calculate the denominator**:
- First, calculate \(4\pi\):
\[ 4\pi \approx 12.566 \]
- Now calculate \(4\pi \Delta x\):
\[ 4\pi \Delta x \approx 12.566 \times 2.0 \times 10^{-6} \approx 2.5132 \times 10^{-5} \]

5. **Calculate \(\Delta p\)**:
\[ \Delta p = \frac{6.626 \times 10^{-34}}{2.5132 \times 10^{-5}} \approx 2.64 \times 10^{-29} \, \text{kg m/s} \]

6. **Relate \(\Delta p\) to \(\Delta v\)**:
The uncertainty in momentum is also related to the uncertainty in velocity:
\[ \Delta p = m \Delta v \]
Rearranging gives:
\[ \Delta v = \frac{\Delta p}{m} \]

7. **Substitute the values**:
\[ \Delta v = \frac{2.64 \times 10^{-29}}{1.7 \times 10^{-27}} \approx 0.155 \times 10^{-2} \approx 0.0155 \, \text{m/s} \]

8. **Round to appropriate significant figures**:
The minimum uncertainty in the atom's velocity is approximately \(0.016 \, \text{m/s}\).

Thus, the answer is **(d) 0.016 m/s**.