An atom has mass of 0.2 kg and uncertainty in its velocity is 6.218×`10^(−6)`m/s then uncertainty in position is (h=6.626×`10^(−34)`Js)

Options

A. 4.2×`10^(−29)`m
B. 2.86×`10^(−23)`m
C. 4.86×`10^(−23)`m
D. None

Explanation

To find the uncertainty in position (Δx) of an atom given its mass and the uncertainty in its velocity, we can use the Heisenberg Uncertainty Principle, which states:

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

Where:
- Δx = uncertainty in position
- m = mass of the atom
- Δv = uncertainty in velocity
- h = Planck's constant (6.626 × 10^(-34) Js)

**Step 1: Identify the given values**
- Mass of the atom (m) = 0.2 kg
- Uncertainty in velocity (Δv) = 6.218 × 10^(-6) m/s
- Planck's constant (h) = 6.626 × 10^(-34) Js

**Step 2: Substitute the values into the uncertainty principle equation**
We rearrange the equation to solve for Δx:

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

**Step 3: Calculate the denominator**
First, calculate \(4\pi m \Delta v\):

1. Calculate \(4\pi\):
\[ 4\pi \approx 4 \times 3.14 = 12.56 \]

2. Now substitute the values:
\[ 4\pi m \Delta v = 12.56 \times 0.2 \times 6.218 \times 10^{-6} \]

3. Calculate \(0.2 \times 6.218 \times 10^{-6}\):
\[ 0.2 \times 6.218 \times 10^{-6} = 1.2436 \times 10^{-6} \]

4. Now multiply by \(12.56\):
\[ 12.56 \times 1.2436 \times 10^{-6} \approx 1.564 \times 10^{-5} \]

**Step 4: Calculate Δx**
Now substitute back into the equation for Δx:

\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{1.564 \times 10^{-5}} \]

Calculating this gives:

\[ \Delta x \geq 4.24 \times 10^{-29} \text{ m} \]

**Final Answer:**
The uncertainty in position (Δx) is approximately \(4.24 \times 10^{-29} \text{ m}\).

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