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 involves calculating the uncertainty in the position of an electron based on its speed and the given uncertainty in that speed. This is related to the Heisenberg Uncertainty Principle, where the uncertainty in position is inversely related to the uncertainty in momentum.
Answer
$X \approx 0.00593 \, \text{m}$
Answer for screen readers
The value of ( X ) is approximately $0.00593 , \text{m}$.
Steps to Solve
- Calculate the Uncertainty in Velocity ($\Delta v$)
First, we need to calculate the uncertainty in velocity based on the given percentage uncertainty of 0.005%.
The formula for uncertainty in velocity is:
[ \Delta v = v \times \text{percentage uncertainty} ]
Given that the speed of the electron ( v = 600 , \text{m/s} ):
[ \Delta v = 600 , \text{m/s} \times \frac{0.005}{100} ]
Calculating this gives:
[ \Delta v = 600 \times 0.00005 = 0.03 , \text{m/s} ]
- Calculate the Uncertainty in Momentum ($\Delta p$)
Next, we use the uncertainty in velocity to find the uncertainty in momentum, which is given by:
[ \Delta p = m \cdot \Delta v ]
Where ( m = 9.1 \times 10^{-31} , \text{kg} ). Substituting the values gives:
[ \Delta p = 9.1 \times 10^{-31} , \text{kg} \times 0.03 , \text{m/s} ]
Calculating this gives:
[ \Delta p = 2.73 \times 10^{-32} , \text{kg m/s} ]
- Apply the Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle states that:
[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} ]
Where ( h = 6.6 \times 10^{-34} , \text{kg m}^2/\text{s} ).
We need to rearrange this to find the uncertainty in position ( \Delta x ):
[ \Delta x \geq \frac{h}{4\pi \Delta p} ]
Substituting the known values:
[ \Delta x \geq \frac{6.6 \times 10^{-34}}{4\pi \cdot 2.73 \times 10^{-32}} ]
- Calculate the Minimum Uncertainty in Position ($\Delta x$)
Calculating the value gives:
[ \Delta x \geq \frac{6.6 \times 10^{-34}}{34.2958 \times 10^{-32}} \approx 5.93 \times 10^{-3} , \text{m} ]
This value can be rounded to obtain ( X ).
- Final Result
Final value ( X ) is approximately ( 0.00593 , \text{m} ) or ( 5.93 , \text{mm} ).
The value of ( X ) is approximately $0.00593 , \text{m}$.
More Information
The uncertainty in position of the electron is derived from the limitations set by the Heisenberg Uncertainty Principle, which illustrates the intrinsic limits of measuring subatomic particle properties accurately.
Tips
- Confusing the percentage uncertainty calculation with absolute uncertainty. Remember to convert percentages to decimals correctly.
- Forgetting to convert units if necessary. Ensure all measurements are in compatible units.
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