Question

Given, the mass of electron is $9.11 \times 10^{-31}\, kg$, Planck's constant is $6.626 \times 10^{-34}\, Js$, the uncertainty involved in the measurement of velocity within a distance of $0.1 \mathring{A}$ is

• A. $5.79 \times 10^6 ms^{-1}$
• B. $5.79 \times 10^7 ms^{-1}$
• C. $5.79 \times 10^8 ms^{-1}$
• D. $5.79 \times 10^5 ms^{-1}$

Answer 1

The Correct Option is A

To determine the uncertainty in the measurement of velocity of an electron, we apply Heisenberg's Uncertainty Principle. The principle states that it is impossible to measure both the position and momentum (or velocity) of a particle, like an electron, with absolute precision. The more precisely one is known, the less precise the measurement of the other is. The principle is mathematically represented as:

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

where:

• \(\Delta x\) is the uncertainty in position.
• \(\Delta p\) is the uncertainty in momentum.
• h is Planck's constant.

Since momentum p = mv, where m is the mass and v is the velocity, the uncertainty in momentum can also be expressed as:

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

Substituting this into the uncertainty principle equation gives:

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

Solving for \(\Delta v\) gives the expression for the uncertainty in velocity:

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

Given values:

• Mass of electron, m = 9.11 \times 10^{-31} \, \text{kg}
• Planck's constant, h = 6.626 \times 10^{-34} \, \text{Js}
• Uncertainty in position, \(\Delta x = 0.1 \, \mathring{A} = 0.1 \times 10^{-10} \, \text{m}\)

Substitute these values into the formula:

\[ \Delta v \geq \frac{6.626 \times 10^{-34} \, \text{Js}}{4 \pi \cdot 9.11 \times 10^{-31} \, \text{kg} \cdot 0.1 \times 10^{-10} \, \text{m}} \]

Calculate the denominator:

\[ 4 \pi \cdot 9.11 \times 10^{-31} \cdot 0.1 \times 10^{-10} = 1.14244 \times 10^{-40} \]

Thus, the uncertainty in velocity is:

\[ \Delta v \geq \frac{6.626 \times 10^{-34}}{1.14244 \times 10^{-40}} = 5.79 \times 10^{6} \, \text{m/s} \]

Therefore, the uncertainty involved in the measurement of velocity within a distance of 0.1 \mathring{A} is 5.79 \times 10^6 \, \text{ms}^{-1}.