Consider the four scenarios involving visible light.

Question

visible light has a frequency of 5.725×1014 s−1.5.725×1014 s−1. Determine its wavelength, energy per photon, and color.

wavelength ? nm

In scenario C, visible light is in the middle of the yellow region of the visible spectrum. Estimate its wavelength, frequency, and energy per photon.

wavelength?

In scenario D, visible light has a photon energy of 4.161×10−19 J.4.161×10−19 J. Determine its wavelength, frequency, and color.

frequency?

Answer 1

Three relationships do all of this, and they are really only two ideas: c = λν, and E = hν. Combine them and you get E = hc/λ, which saves a step whenever you are going between energy and wavelength directly.

Constants: c = 2.998 × 10⁸ m/s, h = 6.626 × 10^-34 J·s, and the combination hc = 1.9865 × 10^-25 J·m is worth writing on your formula sheet.

First scenario, ν = 5.725 × 10¹⁴ s^-1

λ = c/ν = (2.998 × 10⁸) / (5.725 × 10¹⁴) = 5.237 × 10^-7 m = 524 nm

E = hν = (6.626 × 10^-34)(5.725 × 10¹⁴) = 3.793 × 10^-19 J

524 nm falls in the 495-570 nm band, so the colour is green.

Scenario C, middle of the yellow region

Yellow runs roughly 570-590 nm, so take λ ≈ 580 nm.

ν = c/λ = (2.998 × 10⁸) / (580 × 10^-9) = 5.169 × 10¹⁴ s^-1

E = hc/λ = (1.9865 × 10^-25) / (580 × 10^-9) = 3.425 × 10^-19 J

The word estimate is doing real work here. Yellow is a band, not a number, so your answer only needs to be close; if you used 575 or 585 nm you are equally right.

Scenario D, E = 4.161 × 10^-19 J

ν = E/h = (4.161 × 10^-19) / (6.626 × 10^-34) = 6.280 × 10¹⁴ s^-1

λ = hc/E = (1.9865 × 10^-25) / (4.161 × 10^-19) = 4.774 × 10^-7 m = 477 nm

477 nm sits in the 450-495 nm band, so the colour is blue.

The colour boundaries, since the problem keeps asking for them: violet 400-450 nm, blue 450-495, green 495-570, yellow 570-590, orange 590-620, red 620-750.

Two habits worth taking from this set. First, every answer above landed between 400 and 700 nm, which it had to, because the problem said visible light. If a calculation had given you 250 nm or 1500 nm, that is a unit error, almost always nanometres left unconverted to metres. Second, watch the inverse relationship: scenario D had the largest photon energy and came out with the shortest wavelength. Energy and wavelength always move in opposite directions, so you can predict the ordering before you compute anything.

One note on the question itself: it mentions four scenarios, but only three appear in the posted text, so I have answered those three. If the missing one gives you a wavelength directly, use ν = c/λ and E = hc/λ exactly as in scenario C.