The position of an electron in an atom is measured with an accuracy of 0.02 nm, then, what is the uncertainty in electron velocity ? (Give the answer in km/s)

To find the uncertainty in the electron's velocity, we can use Heisenberg's Uncertainty Principle. The principle states that the uncertainty in position (\(\Delta x\)) and the uncertainty in momentum (\(\Delta p\)) are related by the inequality:

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

where \( h \) is Planck's constant (\( h = 6.626 \times 10^{-34} \) Js). 

The momentum \( p \) is given by \( p = mv \), where \( m \) is the mass of the electron and \( v \) is its velocity. The uncertainty in momentum (\(\Delta p\)) can be expressed as:

\[
\Delta p = m \Delta v
\]

where \( \Delta v \) is the uncertainty in velocity. Thus, the uncertainty in velocity is given by:

\[
\Delta v = \frac{\Delta p}{m}
\]

Now, rearranging the uncertainty principle for velocity:

\[
\Delta v = \frac{h}{4\pi m \Delta x}
\]

We have:
- \( \Delta x = 0.02 \) nm \( = 0.02 \times 10^{-9} \) m
- \( m = 9.109 \times 10^{-31} \) kg (mass of an electron)

Substitute the values:

\[
\Delta v = \frac{6.626 \times 10^{-34}}{4\pi (9.109 \times 10^{-31})(0.02 \times 10^{-9})}
\]

Let's calculate \( \Delta v \).The uncertainty in the electron's velocity is approximately \( 2894.28 \) km/s.

Do you have any further questions or need additional details? Here are some related questions:

1. What is Heisenberg's Uncertainty Principle?
2. How is the mass of an electron determined?
3. Can the uncertainty in position be reduced to zero?
4. What are the implications of the Uncertainty Principle in quantum mechanics?
5. How does the Uncertainty Principle affect atomic and subatomic particles?
6. What is the significance of Planck's constant?
7. How is the uncertainty in momentum related to the uncertainty in velocity?
8. How does the Uncertainty Principle relate to measurement precision in physics?

**Tip:** The Heisenberg Uncertainty Principle is a fundamental concept in quantum mechanics, highlighting the limitations of measuring certain pairs of physical properties simultaneously.

Learn how to calculate the uncertainty in electron velocity using Heisenberg's Uncertainty Principle. This problem involves measuring electron position with a precision of 0.02 nm and calculating the resulting uncertainty in velocity. Suitable for college-level physics students.

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