Calculate the wavelength of a photon having an energy of 2 electron volt

Options

A. `6.204 xx 10^(-7)m`
B. `6.206 xx 10^(-6)m`
C. `6.204 xx 10^(-9)m`
D. `6.204xx10^(-8)m`

Explanation

To calculate the wavelength of a photon with an energy of 2 electron volts (eV), we can follow these steps:

### Step 1: Convert Energy from Electron Volts to Joules
1 eV is equivalent to \(1.6 \times 10^{-19}\) Joules. Therefore, to convert 2 eV to Joules:
\[ E = 2 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 3.2 \times 10^{-19} \, \text{J} \]

### Step 2: Use the Energy-Wavelength Relationship
The relationship between energy (E) and wavelength (\(\lambda\)) is given by the formula:
\[ E = \frac{hc}{\lambda} \]
Where:
- \(E\) is the energy in Joules,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{J s}\)),
- \(c\) is the speed of light (\(3 \times 10^{8} \, \text{m/s}\)),
- \(\lambda\) is the wavelength in meters.

To find \(\lambda\), we rearrange the formula:
\[ \lambda = \frac{hc}{E} \]

### Step 3: Substitute the Values into the Formula
Now, substituting the values of \(h\), \(c\), and \(E\) into the equation:
\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{J s}) \times (3 \times 10^{8} \, \text{m/s})}{3.2 \times 10^{-19} \, \text{J}} \]

### Step 4: Calculate the Wavelength
Calculating the numerator:
\[ hc = 6.626 \times 10^{-34} \times 3 \times 10^{8} = 1.9878 \times 10^{-25} \, \text{J m} \]
Now, substituting this into the wavelength equation:
\[ \lambda = \frac{1.9878 \times 10^{-25}}{3.2 \times 10^{-19}} \approx 6.204 \times 10^{-7} \, \text{m} \]

### Step 5: Convert to Nanometers (if needed)
To express the wavelength in nanometers (1 nm = \(10^{-9}\) m):
\[ \lambda \approx 6.204 \times 10^{-7} \, \text{m} = 620.4 \, \text{nm} \]

### Final Answer:
The wavelength of the photon with an energy of 2 eV is approximately \(620.4 \, \text{nm}\).

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