The wavelength (in nanometer) associated with a proton moving at `1.0xx10^(3)` m/s is

Options

A. `0.032` nm
B. `0.40` nm
C. `2.5` nm
D. `14.0` nm

Explanation

To find the wavelength associated with a proton moving at a velocity of \(1.0 \times 10^3\) m/s, we can use the de Broglie wavelength formula:

\[ \lambda = \frac{h}{mv} \]

Where:
- \(\lambda\) is the wavelength,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \(m\) is the mass of the proton (\(1.67 \times 10^{-27} \, \text{kg}\)),
- \(v\) is the velocity of the proton (\(1.0 \times 10^3 \, \text{m/s}\)).

### Step-by-step Solution:

1. **Identify the values:**
- Planck's constant \(h = 6.626 \times 10^{-34} \, \text{Js}\)
- Mass of the proton \(m = 1.67 \times 10^{-27} \, \text{kg}\)
- Velocity \(v = 1.0 \times 10^3 \, \text{m/s}\)

2. **Substitute the values into the formula:**
\[ \lambda = \frac{6.626 \times 10^{-34}}{(1.67 \times 10^{-27})(1.0 \times 10^3)} \]

3. **Calculate the denominator:**
\[ m \cdot v = 1.67 \times 10^{-27} \, \text{kg} \times 1.0 \times 10^3 \, \text{m/s} = 1.67 \times 10^{-24} \, \text{kg m/s} \]

4. **Calculate the wavelength:**
\[ \lambda = \frac{6.626 \times 10^{-34}}{1.67 \times 10^{-24}} \approx 3.96 \times 10^{-10} \, \text{m} \]

5. **Convert the wavelength from meters to nanometers:**
- Since \(1 \, \text{nm} = 10^{-9} \, \text{m}\), we can convert:
\[ \lambda = 3.96 \times 10^{-10} \, \text{m} = 0.396 \, \text{nm} \]

6. **Round the value:**
- Rounding \(0.396\) gives approximately \(0.4 \, \text{nm}\).

### Final Answer:
The wavelength associated with the proton moving at \(1.0 \times 10^3\) m/s is approximately \(0.4 \, \text{nm}\).