According to Heisenberg's uncertainly principle, the product of uncertainties in position and velocities for an electron of mass `9.1 xx 10^-31 kg` is.

Options

A. `2.8 xx 10^-3 m^2 s^-1`
B. `3.8 xx 10^-5 m^2 s^-1`
C. `5.8 xx 10^-5 m^2 s^-1`
D. `6.8 xx 10^-6 m^2 s^-1`

Explanation

To solve the problem using Heisenberg's Uncertainty Principle, we will follow these steps:

### Step 1: Understand Heisenberg's Uncertainty Principle
Heisenberg's Uncertainty Principle states that the product of the uncertainties in position (Δx) and momentum (Δp) of a particle cannot be smaller than a certain value. Mathematically, it is expressed as:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where \( h \) is Planck's constant.

### Step 2: Relate Momentum to Mass and Velocity
Momentum (p) is defined as the product of mass (m) and velocity (v):
\[ p = mv \]
Thus, the uncertainty in momentum (Δp) can be expressed as:
\[ \Delta p = m \cdot \Delta v \]
where Δv is the uncertainty in velocity.

### Step 3: Substitute Δp in the Uncertainty Principle
Substituting Δp in the uncertainty principle gives us:
\[ \Delta x \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi} \]
This can be rearranged to find the product of uncertainties:
\[ \Delta x \cdot \Delta v \geq \frac{h}{4\pi m} \]

### Step 4: Use Known Values
Planck's constant \( h \) is approximately \( 6.626 \times 10^{-34} \, \text{Js} \) and the mass of the electron \( m \) is given as \( 9.1 \times 10^{-31} \, \text{kg} \).

### Step 5: Calculate the Product of Uncertainties
Now, we can plug in the values into the equation:
\[ \Delta x \cdot \Delta v \geq \frac{6.626 \times 10^{-34}}{4\pi \times 9.1 \times 10^{-31}} \]

Calculating the denominator:
\[ 4\pi \times 9.1 \times 10^{-31} \approx 1.141 \times 10^{-30} \]

Now calculating the right-hand side:
\[ \Delta x \cdot \Delta v \geq \frac{6.626 \times 10^{-34}}{1.141 \times 10^{-30}} \approx 5.8 \times 10^{-4} \]

### Final Answer
Thus, according to Heisenberg's uncertainty principle, the product of uncertainties in position and velocity for an electron of mass \( 9.1 \times 10^{-31} \, \text{kg} \) is approximately:
\[ \Delta x \cdot \Delta v \geq 5.8 \times 10^{-5} \, \text{m}^2/\text{s} \]

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