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 use the following steps:

### Step 1: Understand the relationship between energy and wavelength
The energy (E) of a photon is related to its wavelength (λ) by the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \( E \) is the energy of the photon,
- \( h \) 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.

### Step 2: Convert energy from electron volts to joules
1 eV is equal to \( 1.602 \times 10^{-19} \) joules. Therefore, to convert 2 eV to joules:
\[ E = 2 \, \text{eV} \times 1.602 \times 10^{-19} \, \text{J/eV} = 3.204 \times 10^{-19} \, \text{J} \]

### Step 3: Rearrange the formula to solve for wavelength
We need to rearrange the equation to solve for \( \lambda \):
\[ \lambda = \frac{hc}{E} \]

### Step 4: Substitute the values into the equation
Now, substitute the values of \( h \), \( c \), and \( E \) into the equation:
\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{J s}) \times (3 \times 10^{8} \, \text{m/s})}{3.204 \times 10^{-19} \, \text{J}} \]

### Step 5: Calculate the wavelength
Now, perform the calculation:
1. Calculate the numerator:
\[ 6.626 \times 10^{-34} \times 3 \times 10^{8} = 1.9878 \times 10^{-25} \, \text{J m} \]
2. Now divide by the energy in joules:
\[ \lambda = \frac{1.9878 \times 10^{-25}}{3.204 \times 10^{-19}} \]
\[ \lambda \approx 6.20 \times 10^{-7} \, \text{m} \]

### Step 6: Convert the wavelength to nanometers
To convert meters to nanometers (1 m = \( 10^{9} \) nm):
\[ \lambda \approx 6.20 \times 10^{-7} \, \text{m} \times 10^{9} \, \text{nm/m} \approx 620 \, \text{nm} \]

### Final Answer
The wavelength of a photon having an energy of 2 eV is approximately **620 nm**.

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