Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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What is the minimum uncertainty in a helium atoms velocity if the position is known within 1.1 A
![**Problem Statement:**
"What is the minimum uncertainty in a helium atom's velocity (\(\Delta v_{\text{min}}\)) if the position is known within 1.1 Å?"
**User Input:**
\[
\Delta v_{\text{min}} = 5.27 \times 10^5 \, \text{m/s}
\]
**Feedback:**
- Incorrect
**Explanation:**
This problem is addressing the concept of uncertainty as described by Heisenberg's Uncertainty Principle. The aim is to calculate the minimum uncertainty in the velocity of a helium atom when its position is known to an accuracy of 1.1 Ångströms. The user provided an answer, 5.27 x 10^5 m/s, which is marked as incorrect.
In order to find the correct answer, one would typically use the Heisenberg Uncertainty Principle formula:
\[
\Delta x \cdot \Delta p \geq \frac{\hbar}{2}
\]
Where:
- \(\Delta x\) is the uncertainty in position (1.1 Å)
- \(\Delta p\) is the uncertainty in momentum
- \(\hbar\) is the reduced Planck's constant \(\left(\frac{h}{2\pi}\right)\)
Since momentum (\(p\)) is the product of mass (\(m\)) and velocity (\(v\)), the formula becomes:
\[
\Delta x \cdot m\Delta v \geq \frac{\hbar}{2}
\]
Rearrange to solve for \(\Delta v\):
\[
\Delta v \geq \frac{\hbar}{2m\Delta x}
\]
To calculate \(\Delta v_{\text{min}}\), you'd need to know the mass of the helium atom and use the value for \(\hbar\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1cfcdea6-a22e-4898-9e64-7f9b12d8c774%2F3ff284f2-e413-419c-acca-ec19755be12b%2Fgukok19_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
"What is the minimum uncertainty in a helium atom's velocity (\(\Delta v_{\text{min}}\)) if the position is known within 1.1 Å?"
**User Input:**
\[
\Delta v_{\text{min}} = 5.27 \times 10^5 \, \text{m/s}
\]
**Feedback:**
- Incorrect
**Explanation:**
This problem is addressing the concept of uncertainty as described by Heisenberg's Uncertainty Principle. The aim is to calculate the minimum uncertainty in the velocity of a helium atom when its position is known to an accuracy of 1.1 Ångströms. The user provided an answer, 5.27 x 10^5 m/s, which is marked as incorrect.
In order to find the correct answer, one would typically use the Heisenberg Uncertainty Principle formula:
\[
\Delta x \cdot \Delta p \geq \frac{\hbar}{2}
\]
Where:
- \(\Delta x\) is the uncertainty in position (1.1 Å)
- \(\Delta p\) is the uncertainty in momentum
- \(\hbar\) is the reduced Planck's constant \(\left(\frac{h}{2\pi}\right)\)
Since momentum (\(p\)) is the product of mass (\(m\)) and velocity (\(v\)), the formula becomes:
\[
\Delta x \cdot m\Delta v \geq \frac{\hbar}{2}
\]
Rearrange to solve for \(\Delta v\):
\[
\Delta v \geq \frac{\hbar}{2m\Delta x}
\]
To calculate \(\Delta v_{\text{min}}\), you'd need to know the mass of the helium atom and use the value for \(\hbar\).
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