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1.2.1: Atomic Spectroscopy and the deBroglie Wavelength (Problems)

  • Page ID
    235734
  • PROBLEM \(\PageIndex{1}\)

    Assume the mass of an electron is 9.11 x 10-31 kg.

    a. Calculate the wavelength of an electron traveling with a speed of 2.65 x 106 m/s.

    b. What would the wavelength of the helium atom be if it were traveling at the same speed as the electron in part a (1 amu = 1.6606 x 10-24 g)?

    Answer a

    2.74 x 10-10 m

    Answer b

    3.77 x 10-14 m

    PROBLEM \(\PageIndex{2}\)

    What is the de Broglie wavelength of a 46 g baseball traveling at 94 mph?

    Answer

    3.43 x 10-34 m

    Click here for a video of the solution

    PROBLEM \(\PageIndex{3}\)

    Which of the following equations describe particle-like behavior? Which describe wavelike behavior?

    1. \(c = λν\)
    2. \(E=\dfrac{mν^2}{2}\)
    3. \(r=\dfrac{n^2a_0}{Z}\)
    4. \(E = hν\)
    5. \(λ=\dfrac{h}{mν}\)
    Answer

    e. describes particle-like behavior

    a&d describe wavelike behavior

    PROBLEM \(\PageIndex{4}\)

    RGB color television and computer displays use cathode ray tubes that produce colors by mixing red, green, and blue light. If we look at the screen with a magnifying glass, we can see individual dots turn on and off as the colors change. Using a spectrum of visible light, determine the approximate wavelength of each of these colors. What is the frequency and energy of a photon of each of these colors

    Answer

    Red: 660 nm; 4.54 × 1014 Hz; 3.01 × 10−19 J. Green: 520 nm; 5.77 × 1014 Hz; 3.82 × 10−19 J. Blue: 440 nm; 6.81 × 1014 Hz; 4.51 × 10−19 J.

    PROBLEM \(\PageIndex{5}\)

    A bright violet line occurs at 435.8 nm in the emission spectrum of mercury vapor. What amount of energy, in joules, must be released by an electron in a mercury atom to produce a photon of this light?

    Answer

    \(4.56\times 10^{-19}J\)

    PROBLEM \(\PageIndex{6}\)

    When rubidium ions are heated to a high temperature, two lines are observed in its line spectrum at wavelengths (a) 7.9 × 10−7 m and (b) 4.2 × 10−7 m. What are the frequencies of the two lines? What color do we see when we heat a rubidium compound?

    Answer

    The frequency of (a) would be 3.79 × 1014 s-1.

    The frequency of (b) would be 7.13 × 1014 s-1.

    Because (a) would be in the near-IR, the compound would appear purple/blue based on the wavelength of (b).

    Click here to see a video of the solution

    PROBLEM \(\PageIndex{7}\)

    The emission spectrum of cesium contains two lines whose frequencies are (a) 3.45 × 1014 Hz and (b) 6.53 × 1014 Hz. What are the wavelengths and energies per photon of the two lines? What color are the lines?

    Answer a

    λ = 8.69 × 10−7 m; E = 2.29 × 10−19 J; red

    Answer b

    λ = 4.59 × 10−7 m; E = 4.33 × 10−19 J; blue

     

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