What is the energy of a proton possessing wavelength `0.4A`?

Options

A. 0.51 eV
B. 1.51 eV
C. 10.51 eV
D. 100.51 eV

Explanation

To find the energy of a proton possessing a wavelength of \(0.4 \, \text{Å}\), we can follow these steps:

### Step 1: Convert Wavelength to Meters
The given wavelength is \(0.4 \, \text{Å}\). We need to convert this to meters:
\[ 0.4 \, \text{Å} = 0.4 \times 10^{-10} \, \text{m} \]

### Step 2: Use the De Broglie Wavelength Formula
The de Broglie wavelength \(\lambda\) is related to momentum \(p\) by the formula:
\[ \lambda = \frac{h}{p} \]
where \(h\) is Planck's constant, \(h = 6.63 \times 10^{-34} \, \text{J s}\).

### Step 3: Relate Momentum to Kinetic Energy
Momentum \(p\) can also be expressed in terms of mass \(m\) and velocity \(v\):
\[ p = mv \]
The kinetic energy \(K\) of the proton is given by:
\[ K = \frac{1}{2} mv^2 \]
From this, we can express \(v\) as:
\[ v = \sqrt{\frac{2K}{m}} \]

### Step 4: Substitute for Momentum in Terms of Kinetic Energy
Substituting \(v\) into the momentum equation gives:
\[ p = m \sqrt{\frac{2K}{m}} = \sqrt{2mK} \]
Now substituting this into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{\sqrt{2mK}} \]

### Step 5: Rearranging to Find Kinetic Energy
Rearranging the above equation to solve for kinetic energy \(K\):
\[ K = \frac{h^2}{2m\lambda^2} \]

### Step 6: Substitute Values
Now we can substitute the known values:
- \(h = 6.63 \times 10^{-34} \, \text{J s}\)
- \(m = 1.67 \times 10^{-27} \, \text{kg}\) (mass of a proton)
- \(\lambda = 0.4 \times 10^{-10} \, \text{m}\)

Calculating:
\[ K = \frac{(6.63 \times 10^{-34})^2}{2 \times (1.67 \times 10^{-27}) \times (0.4 \times 10^{-10})^2} \]

### Step 7: Calculate Kinetic Energy in Joules
After performing the calculations:
\[ K \approx 8.19 \times 10^{-19} \, \text{J} \]

### Step 8: Convert Joules to Electron Volts
To convert joules to electron volts, we use the conversion factor \(1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}\):
\[ K \approx \frac{8.19 \times 10^{-19}}{1.6 \times 10^{-19}} \approx 0.51 \, \text{eV} \]

### Final Answer
The energy of the proton possessing a wavelength of \(0.4 \, \text{Å}\) is approximately \(0.51 \, \text{eV}\).

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