A yellow line of sodium corresponds to a frequency `5.09 xx 10^(14)s^(-1).` Calculate the wavelength and the wave number of this line.

Answer

To solve the problem of calculating the wavelength and wave number of a yellow line of sodium corresponding to a frequency of \(5.09 \times 10^{14} \, s^{-1}\), we will follow these steps:

### Step 1: Write down the known values
- Frequency (\( \nu \)) = \( 5.09 \times 10^{14} \, s^{-1} \)
- Speed of light (\( c \)) = \( 3.00 \times 10^{8} \, m/s \)

### Step 2: Calculate the wavelength (\( \lambda \))
The formula to calculate the wavelength is given by:
\[ \lambda = \frac{c}{\nu} \]
Substituting the known values:
\[ \lambda = \frac{3.00 \times 10^{8} \, m/s}{5.09 \times 10^{14} \, s^{-1}} \]

### Step 3: Perform the calculation
Calculating the above expression:
\[ \lambda \approx \frac{3.00 \times 10^{8}}{5.09 \times 10^{14}} \approx 5.89 \times 10^{-7} \, m \]

### Step 4: Convert wavelength to Angstroms
1 Angstrom (\( \text{Å} \)) = \( 1 \times 10^{-10} \, m \), so:
\[ \lambda \approx 5.89 \times 10^{-7} \, m \times \frac{1 \times 10^{10} \, \text{Å}}{1 \, m} \approx 5890 \, \text{Å} \]

### Step 5: Calculate the wave number (\( \bar{\nu} \))
The wave number is defined as:
\[ \bar{\nu} = \frac{1}{\lambda} \]
Substituting the wavelength in meters:
\[ \bar{\nu} = \frac{1}{5.89 \times 10^{-7} \, m} \]

### Step 6: Perform the calculation for wave number
Calculating the above expression:
\[ \bar{\nu} \approx 1.698 \times 10^{6} \, m^{-1} \]

### Step 7: Convert wave number to cm\(^{-1}\)
1 m = 100 cm, thus:
\[ \bar{\nu} \approx 1.698 \times 10^{6} \, m^{-1} \times \frac{1 \, \text{cm}}{0.01 \, m} \approx 169800 \, cm^{-1} \]

### Final Results
- Wavelength (\( \lambda \)) = \( 5890 \, \text{Å} \)
- Wave number (\( \bar{\nu} \)) = \( 169800 \, cm^{-1} \)