Answer
For both wavelengths, the energy in \( \text{kJ/mol} \) is approximately \( 202 \, \text{kJ/mol} \).
Work Step by Step
The frequency of a photon can be calculated using the equation \( \nu = \frac{c}{\lambda} \), where \( \nu \) is the frequency, \( c \) is the speed of light (\( 3.00 \times 10^8 \, \text{m/s} \)), and \( \lambda \) is the wavelength.
For the 589.0 nm wavelength:
\( \nu = \frac{3.00 \times 10^8 \, \text{m/s}}{589.0 \times 10^{-9} \, \text{m}} \)
\( \nu \approx 5.09 \times 10^{14} \, \text{Hz} \)
For the 589.6 nm wavelength:
\( \nu = \frac{3.00 \times 10^8 \, \text{m/s}}{589.6 \times 10^{-9} \, \text{m}} \)
\( \nu \approx 5.08 \times 10^{14} \, \text{Hz} \)
The energy of a photon can be calculated using the equation \( E = h\nu \), where \( E \) is the energy, \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)), and \( \nu \) is the frequency.
For the 589.0 nm wavelength:
\( E = (6.626 \times 10^{-34} \, \text{J s})(5.09 \times 10^{14} \, \text{Hz}) \)
\( E \approx 3.37 \times 10^{-19} \, \text{J} \)
For the 589.6 nm wavelength:
\( E = (6.626 \times 10^{-34} \, \text{J s})(5.08 \times 10^{14} \, \text{Hz}) \)
\( E \approx 3.36 \times 10^{-19} \, \text{J} \)
To convert the energies to \( \text{kJ/mol} \), we can use Avogadro's number and convert from joules to kilojoules:
\( 1 \, \text{J} = \frac{1}{1000} \, \text{kJ} \)
\( 1 \, \text{mol} \) of photons = \( 6.022 \times 10^{23} \) photons
For both wavelengths, the energy in \( \text{kJ/mol} \) is approximately \( 202 \, \text{kJ/mol} \).
