Question

Sodium atoms emit a spectral line with a wavelength in the yellow, 589.6 nm. What is the difference in energy between the two energy levels involved in the emission of this spectral line?

• A. 2.6 eV
• B. 2.9 eV
• C. 2.1 eV
• D. None

Hint:

The energy of a photon can be calculated using \( \Delta E = \frac{hc}{\lambda} \), where \( \lambda \) is the wavelength of the emitted light.

Answer 1

The Correct Option is C

To find the energy difference between the two energy levels in sodium responsible for emitting a spectral line at 589.6 nm, we use the relationship between energy, frequency, and wavelength given by the formula:

Energy (E) = h × c / λ

Where:

• h = Planck's constant = 6.626 × 10^-34 J·s
• c = Speed of light in a vacuum = 3.00 × 10⁸ m/s
• λ = Wavelength = 589.6 nm = 589.6 × 10^-9 m

Substitute these values into the formula to find the energy:

E = (6.626 × 10^-34 J·s) × (3.00 × 10⁸ m/s) / (589.6 × 10^-9 m)

E ≈ 3.37 × 10^-19 J

To convert this energy from joules to electronvolts (eV), use the conversion factor:

• 1 eV = 1.602 × 10^-19 J

Thus,

E ≈ 3.37 × 10^-19 J / 1.602 × 10^-19 J/eV

E ≈ 2.1 eV

Therefore, the difference in energy between the two energy levels is 2.1 eV, which matches the correct answer.

Answer 2

The energy difference \( \Delta E \) between two energy levels is related to the wavelength \( \lambda \) of the emitted light by the equation: \[ \Delta E = \frac{hc}{\lambda} \] Where:
- \( h = 6.626 \times 10^{-34} \, \text{J s} \) is Planck's constant,
- \( c = 3 \times 10^8 \, \text{m/s} \) is the speed of light,
- \( \lambda = 589.6 \, \text{nm} = 589.6 \times 10^{-9} \, \text{m} \). Substituting the values: \[ \Delta E = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{589.6 \times 10^{-9}} \approx 2.1 \, \text{eV} \]

Answer 3

The sodium yellow line comes from an electron dropping between two specific energy levels in the atom, and the wavelength of the emitted photon fixes the energy gap between those levels. We are given four possible values for this gap and need to check each one against the wavelength quoted, 589.6 nm.

1. 2.6 eV: A photon carrying 2.6 eV of energy has a wavelength of about \( \frac{1240}{2.6} \approx 477 \) nm, which falls in the blue-green part of the visible spectrum, not yellow. This value is too large for the 589.6 nm line.
2. 2.9 eV: A photon of 2.9 eV corresponds to a wavelength of about \( \frac{1240}{2.9} \approx 427 \) nm, which is violet light. This is even further from yellow, so 2.9 eV cannot be the gap for this line.
3. 2.1 eV: A photon of 2.1 eV corresponds to a wavelength of about \( \frac{1240}{2.1} \approx 590 \) nm, which sits right at the yellow part of the spectrum and matches the given 589.6 nm almost exactly.
4. None: Since one of the listed values, 2.1 eV, does line up with the given wavelength, this option cannot be right.

Checking each option against its corresponding wavelength shows that only 2.1 eV reproduces a wavelength close to 589.6 nm, the yellow sodium line.

So the correct answer is 2.1 eV.