In an atom, an electron is moving with a speed of 600 m/s with an uncertainty of 0.005%. Uncertainty with which the position of the electron can be located is Xm. The value of X is... In an atom, an electron is moving with a speed of 600 m/s with an uncertainty of 0.005%. Uncertainty with which the position of the electron can be located is Xm. The value of X is.
Understand the Problem
The question is asking for the uncertainty in the position of an electron in an atom, given its speed and the uncertainty in that speed. This can be solved using the principles of quantum mechanics, particularly Heisenberg's uncertainty principle.
Answer
The uncertainty in the position of the electron is approximately $1.93 \times 10^{-3} \, \text{m}$.
Answer for screen readers
The uncertainty in the position of the electron, $X$, is approximately $1.93 \times 10^{-3} , \text{m}$.
Steps to Solve
- Identify Given Values
Identify the values provided in the problem:
- Speed of the electron, $v = 600 , \text{m/s}$
- Uncertainty in speed, $\Delta v = 0.005% , \text{of} , 600 , \text{m/s}$
- Calculate Uncertainty in Speed
Calculate the uncertainty in speed: [ \Delta v = 0.005% \times 600 = \frac{0.005}{100} \times 600 = 0.03 , \text{m/s} ]
- Apply Heisenberg’s Uncertainty Principle
Use Heisenberg's uncertainty principle, which states: [ \Delta x \Delta p \geq \frac{h}{4\pi} ] Where $\Delta p$ is the uncertainty in momentum.
- Calculate the Uncertainty in Momentum
Calculate the uncertainty in momentum, where $p = mv$: [ \Delta p = m \Delta v = (9.1 \times 10^{-31} , \text{kg}) \times (0.03 , \text{m/s}) = 2.73 \times 10^{-32} , \text{kg m/s} ]
- Calculate Uncertainty in Position
Rearranging the uncertainty principle to find $\Delta x$: [ \Delta x \geq \frac{h}{4 \pi \Delta p} ] Substituting in the values: [ \Delta x \geq \frac{6.6 \times 10^{-34} , \text{kg m}^2 \text{s}^{-1}}{4 \pi (2.73 \times 10^{-32} , \text{kg m/s})} ]
- Calculate Numerical Value for Uncertainty in Position
Perform the calculation: [ \Delta x \geq \frac{6.6 \times 10^{-34}}{4 \pi \times 2.73 \times 10^{-32}} \approx 1.93 \times 10^{-3} , \text{m} ]
The uncertainty in the position of the electron, $X$, is approximately $1.93 \times 10^{-3} , \text{m}$.
More Information
This result demonstrates the application of Heisenberg's uncertainty principle, showing how the uncertainty in speed leads to a significant uncertainty in position for particles at the quantum level.
Tips
- Miscalculating Uncertainty in Speed: Ensure that the percentage is converted correctly before using it in calculations.
- Forgetting Units: Keep track of units throughout the calculations to avoid confusion.
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