A certain spectral line has wavelength of `4000 Å`. Calculate the frequency and the energy in electron volts of the photons associated with it.

Answer

To solve the problem of calculating the frequency and energy of a photon associated with a spectral line of wavelength \(4000 \, \text{Å}\), we will follow these steps:

### Step 1: Convert Wavelength to Meters
The given wavelength is in angstroms. We need to convert it to meters for our calculations.

\[ \lambda = 4000 \, \text{Å} = 4000 \times 10^{-10} \, \text{m} = 4.0 \times 10^{-7} \, \text{m} \]

### Step 2: Calculate Frequency
We use the relationship between the speed of light (\(c\)), wavelength (\(\lambda\)), and frequency (\(\nu\)):

\[ c = \lambda \nu \implies \nu = \frac{c}{\lambda} \]

The speed of light \(c\) is approximately \(3 \times 10^8 \, \text{m/s}\). Now we can substitute the values:

\[ \nu = \frac{3 \times 10^8 \, \text{m/s}}{4.0 \times 10^{-7} \, \text{m}} = 7.5 \times 10^{14} \, \text{Hz} \]

### Step 3: Calculate Energy
Next, we will calculate the energy (\(E\)) of the photon using Planck's equation:

\[ E = h \nu \]

Where \(h\) is Planck's constant, approximately \(6.626 \times 10^{-34} \, \text{J s}\). Substituting the frequency we found:

\[ E = 6.626 \times 10^{-34} \, \text{J s} \times 7.5 \times 10^{14} \, \text{Hz} \]

Calculating this gives:

\[ E = 4.97 \times 10^{-19} \, \text{J} \]

### Step 4: Convert Energy to Electron Volts
To convert energy from joules to electron volts, we use the conversion factor \(1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}\):

\[ E \, (\text{in eV}) = \frac{E \, (\text{in J})}{1.6 \times 10^{-19} \, \text{J/eV}} = \frac{4.97 \times 10^{-19} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}} \approx 3.1 \, \text{eV} \]

### Final Results
- Frequency: \(7.5 \times 10^{14} \, \text{Hz}\)
- Energy: \(3.1 \, \text{eV}\)

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