The uncertainties in position and the velocity of a particle are `10^(-10)` m and 1×`10^(−24).sec^(−1)` respectively. The mass of the particle in kg is

Options

A. `h/
(4π)`×`10 ^(−34)`
B. h×4×`10^23`
C. h/4π×`10^(−34)`
D. None

Explanation

To solve the problem, we will use the Heisenberg Uncertainty Principle, which states:

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

Where:
- \(\Delta x\) is the uncertainty in position,
- \(m\) is the mass of the particle,
- \(\Delta v\) is the uncertainty in velocity,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)).

### Step 1: Identify the given values
- \(\Delta x = 10^{-10} \, \text{m}\)
- \(\Delta v = 1 \times 10^{-24} \, \text{s}^{-1}\)

### Step 2: Rearrange the uncertainty principle formula to solve for mass \(m\)
From the uncertainty principle, we can rearrange it to find the mass \(m\):

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

### Step 3: Substitute the known values into the formula
Substituting the values of \(h\), \(\Delta x\), and \(\Delta v\):

\[ m \geq \frac{6.626 \times 10^{-34}}{4\pi \times (10^{-10}) \times (1 \times 10^{-24})} \]

### Step 4: Calculate the denominator
Calculate \(4\pi \times (10^{-10}) \times (1 \times 10^{-24})\):

\[ 4\pi \approx 12.566 \]
\[ 4\pi \times (10^{-10}) \times (1 \times 10^{-24}) \approx 12.566 \times 10^{-34} \]

### Step 5: Calculate the mass \(m\)
Now, substituting this back into the equation for \(m\):

\[ m \geq \frac{6.626 \times 10^{-34}}{12.566 \times 10^{-34}} \approx \frac{6.626}{12.566} \approx 0.528 \, \text{kg} \]

### Step 6: Final result
Thus, the mass of the particle is approximately:

\[ m \approx 0.528 \, \text{kg} \]