Calculate the wavelength of photon having energy 5eV.

Options

A. `2.47 xx 10^(-6)cm`
B. `2.47 xx 10^(-5)cm`
C. `24.7 xx 10^(-5)cm`
D. `24.7 xx 10^(-6)cm`

Explanation

To calculate the wavelength of a photon with energy of 5 eV, we can follow these steps:

### Step 1: Convert Energy from Electron Volts to Joules
The energy of the photon is given in electron volts (eV). To use it in the formula, we need to convert it to joules (J). The conversion factor is:
1 eV = \(1.6 \times 10^{-19}\) J.

So, for 5 eV:
\[ E = 5 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 8.0 \times 10^{-19} \, \text{J} \]

### Step 2: Use the Energy-Wavelength Relation
The energy of a photon is related to its wavelength by the equation:
\[ E = \frac{hc}{\lambda} \]
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 (\(3.0 \times 10^{8} \, \text{m/s}\)),
- \(\lambda\) is the wavelength in meters.

### Step 3: Rearranging the Formula to Solve for Wavelength
We can rearrange the formula to solve for the wavelength \(\lambda\):
\[ \lambda = \frac{hc}{E} \]

### Step 4: Substitute the Values into the Equation
Now we substitute the values of \(h\), \(c\), and \(E\) into the equation:
\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{J s})(3.0 \times 10^{8} \, \text{m/s})}{8.0 \times 10^{-19} \, \text{J}} \]

### Step 5: Calculate the Wavelength
Calculating the numerator:
\[ 6.626 \times 10^{-34} \times 3.0 \times 10^{8} = 1.9878 \times 10^{-25} \, \text{J m} \]
Now, divide by the energy:
\[ \lambda = \frac{1.9878 \times 10^{-25}}{8.0 \times 10^{-19}} = 2.485 \times 10^{-7} \, \text{m} \]

### Step 6: Convert to Nanometers
To express the wavelength in nanometers (1 nm = \(10^{-9}\) m):
\[ \lambda = 2.485 \times 10^{-7} \, \text{m} = 248.5 \, \text{nm} \]

### Final Answer
The wavelength of the photon with energy 5 eV is approximately **248.5 nm**.

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