Calculate the wavelength ( in nanometer ) associated with a proton moving at ` 1.0xx 10^3` m/s (Mass of proton ` =1.67 xx 10^(-27) kg ` and ` h=6.63 xx 10^(-34) is)` :

Options

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

Answer

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

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

where:
- \(\lambda\) is the wavelength,
- \(h\) is the Planck's constant,
- \(m\) is the mass of the proton,
- \(v\) is the velocity of the proton.

### Step 1: Identify the given values
- Mass of proton, \(m = 1.67 \times 10^{-27}\) kg
- Velocity of proton, \(v = 1.0 \times 10^3\) m/s
- Planck's constant, \(h = 6.63 \times 10^{-34}\) J·s

### Step 2: Substitute the values into the de Broglie wavelength formula
We substitute the known values into the formula:

\[ \lambda = \frac{6.63 \times 10^{-34}}{(1.67 \times 10^{-27}) \times (1.0 \times 10^3)} \]

### Step 3: Calculate the denominator
First, calculate the denominator:

\[ m \cdot v = (1.67 \times 10^{-27}) \times (1.0 \times 10^3) = 1.67 \times 10^{-24} \text{ kg·m/s} \]

### Step 4: Calculate the wavelength
Now substitute back into the equation:

\[ \lambda = \frac{6.63 \times 10^{-34}}{1.67 \times 10^{-24}} \]

Calculating this gives:

\[ \lambda = 3.97 \times 10^{-10} \text{ meters} \]

### Step 5: Convert the wavelength to nanometers
To convert meters to nanometers, we use the conversion factor \(1 \text{ meter} = 10^9 \text{ nanometers}\):

\[ \lambda = 3.97 \times 10^{-10} \text{ m} \times 10^9 \text{ nm/m} = 0.397 \text{ nm} \]

### Step 6: Round the answer
Rounding \(0.397\) nm gives approximately:

\[ \lambda \approx 0.40 \text{ nm} \]

### Final Answer
The wavelength associated with the proton moving at \(1.0 \times 10^3\) m/s is approximately \(0.40\) nanometers.

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