The uncertainty in position and velocity of the particle are 0.1 nm and `5.27xx10^(-24) ms^(-1)` respectively then find the approximate integral mass of the particle (in g ) . `(h=6.625xx10^(-34) Js)`

Answer

To solve the problem, we will use the Heisenberg Uncertainty Principle, which states that the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) is equal to or greater than a constant value. The formula is given by:

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

Where:
- \( \Delta x \) is the uncertainty in position
- \( \Delta p \) is the uncertainty in momentum
- \( h \) is Planck's constant

### Step-by-Step Solution:

1. **Identify the given values**:
- Uncertainty in position, \( \Delta x = 0.1 \, \text{nm} = 0.1 \times 10^{-9} \, \text{m} = 1 \times 10^{-10} \, \text{m} \)
- Uncertainty in velocity, \( \Delta v = 5.27 \times 10^{-24} \, \text{m/s} \)
- Planck's constant, \( h = 6.625 \times 10^{-34} \, \text{Js} \)

2. **Calculate the uncertainty in momentum**:
The uncertainty in momentum \( \Delta p \) is given by:
\[ \Delta p = m \cdot \Delta v \]
where \( m \) is the mass of the particle we want to find.

3. **Substitute into the Heisenberg Uncertainty Principle**:
\[ \Delta x \cdot \Delta p = \Delta x \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi} \]
Rearranging gives:
\[ m \geq \frac{h}{4\pi \Delta x \Delta v} \]

4. **Substituting the values**:
Substitute \( \Delta x \), \( \Delta v \), and \( h \) into the equation:
\[ m \geq \frac{6.625 \times 10^{-34}}{4 \cdot 3.14 \cdot (1 \times 10^{-10}) \cdot (5.27 \times 10^{-24})} \]

5. **Calculate the denominator**:
\[ 4 \cdot 3.14 \cdot (1 \times 10^{-10}) \cdot (5.27 \times 10^{-24}) = 4 \cdot 3.14 \cdot 5.27 \times 10^{-34} \approx 6.67 \times 10^{-33} \]

6. **Calculate the mass**:
\[ m \geq \frac{6.625 \times 10^{-34}}{6.67 \times 10^{-33}} \approx 0.099 \, \text{kg} \approx 0.1 \, \text{kg} \]

7. **Convert to grams**:
Since \( 1 \, \text{kg} = 1000 \, \text{g} \):
\[ m \approx 0.1 \, \text{kg} = 100 \, \text{g} \]

### Final Answer:
The approximate integral mass of the particle is **100 g**.