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Statement-1 : If an electron is located ...

Statement-1 : If an electron is located within the range of `0.1 Å` then the uncertainly in velocity is approximately `6xx10^6 m//s`
Statement-2 : Trajectory (path of motion) of above electron can be defined.
`[h=6.6xx10^(-34), m_e=9.1xx10^(-31) kg]`

A

Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1

B

Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1

C

Statement-1 is True, Statement-2 is False.

D

Statement-1 is False, Statement-2 is True.

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we will analyze both statements using the Heisenberg Uncertainty Principle and the information provided. ### Step-by-Step Solution: 1. **Understanding the Heisenberg Uncertainty Principle**: The principle states that the product of the uncertainties in position (Δx) and momentum (Δp) is greater than or equal to a constant: \[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \] where \( \Delta p = m \cdot \Delta v \) (momentum = mass × velocity). 2. **Given Values**: - Planck's constant, \( h = 6.63 \times 10^{-34} \, \text{Js} \) - Mass of electron, \( m_e = 9.1 \times 10^{-31} \, \text{kg} \) - Position uncertainty, \( \Delta x = 0.1 \, \text{Å} = 0.1 \times 10^{-10} \, \text{m} \) 3. **Calculating the Uncertainty in Velocity (Δv)**: Rearranging the uncertainty principle for Δv: \[ \Delta v \geq \frac{h}{4\pi m \Delta x} \] Substituting the values: \[ \Delta v \geq \frac{6.63 \times 10^{-34}}{4 \times 3.14 \times 9.1 \times 10^{-31} \times (0.1 \times 10^{-10})} \] 4. **Calculating the Denominator**: - Calculate \( 4 \times 3.14 \approx 12.56 \) - Calculate \( 9.1 \times 10^{-31} \times 0.1 \times 10^{-10} = 9.1 \times 10^{-42} \) - Thus, \( 4 \pi m \Delta x \approx 12.56 \times 9.1 \times 10^{-42} \approx 1.144 \times 10^{-41} \) 5. **Final Calculation**: \[ \Delta v \geq \frac{6.63 \times 10^{-34}}{1.144 \times 10^{-41}} \approx 5.79 \times 10^{7} \, \text{m/s} \] This value is approximately \( 6 \times 10^{6} \, \text{m/s} \), confirming Statement-1 is true. 6. **Analyzing Statement-2**: Statement-2 claims that the trajectory of the electron can be defined. However, due to the high uncertainty in velocity (Δv), we cannot precisely define the electron's trajectory. Therefore, Statement-2 is false. ### Conclusion: - **Statement-1** is true. - **Statement-2** is false. Thus, the correct answer is that Statement-1 is true and Statement-2 is false.

To solve the problem, we will analyze both statements using the Heisenberg Uncertainty Principle and the information provided. ### Step-by-Step Solution: 1. **Understanding the Heisenberg Uncertainty Principle**: The principle states that the product of the uncertainties in position (Δx) and momentum (Δp) is greater than or equal to a constant: \[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} ...
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