SCI 270: On Nuclear Technology Practice Problems  
Answers are given for your reference only. Please do your part of the learning, because no one else will be able to do that for you.

Your skills of problem solving are tested and useful in the future. Answers are useless. Acquiring skills and abilities are the goals of learning, marks indications of your performances on these tasks.

3. Atoms - tiny wonders

  1. When heated to very high temperature or passing a electric discharge through, a hydrogen gas glows, emitting a purplish light. When analyzed by a prism, the light consists of a few lines with wavelengths listed in the Table.

    Wavelength
    l nm
    Wavenumber
    1/l /106 m-1
    Frequency
    c/l /1014 Hz
    Photon energy
    h v /10-19 J
    n in Balmer
    series.
    656.31.524 4.571 3.029 3
    486.12.057 6.172 4.090 4
    434.02.304 6.912 4.580 5
    410.12.438 7.315 4.847 6
    396.92.520 7.556 5.007 7
    389.02.570 7.712 5.110 8

    Convert the wavelengths l to wavenumbers and frequencies using the given formulas and units. Also evaluate the n in the Balmer series (see page 62 of the lecture notes).

    I hope the repetitive calculation will let you learn the formulas and theory well.

    Wavenumber = 1/l
    Frequency = c/l
    Photon energy = h v
        = h c/l

    In order to avoid the repitition of common factors and units, I have used /106 m-1 to mean millions per meter.

    Unfortunately, some students considered the the factors after the "/" as part of the formula without knowing why and calculate the numbers.

  2. From the values above, draw a spectrum based on a linear scale using the frequencies, and evaluate the Rydberg constant R both in wavenumbers (m-1) and in Hz.

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    |4____________________5____________________6____________________7___________/e14 Hz
    

    The line spectrum in in e14 Hz are given above. If you have plotted the frequency as a function of 1/n2, we will not mark you wrong.

    The Rydberg constant R is
    __3.29212e15 Hz__

    There are many methods to calculate R. If you use a plot of frequency vs. 1/n2 to evaluate it, that is fine. The simplest way is to convert the known value in m-1 to Hz by multiplying it by the speed of light or use any two lines to evaluate it.

  3. The Bohr model and the quantum mechanical approach result in an expression for the Rydberg constant R:

            2 p2 m Z2 e4
    R = --------------- = 10967700 m-1
                c h3

    where m, Z, c, h and e are the mass of the electron, atomic number, speed of light, Planck's constant, and charge of an electron respectively.

    Apply this number to calculate the wave numbers of the 5 lines (nf = 2, 3, 4, 5, and 6; ni = 1) of the Lyman series.

    The formula used:
    wavenumber = -R(1/n2 - 1)
    Wave numbers of the five lines
    8.226e6 9.750e610.28e6 10.53e610.66e6 m-1

    You can also plot a spectrum in the UV region for the Lyman series. Having generated these numbers, you have learned how photons are emitted. The energy differences from various energy levels are given out as photons.

    In what regions are these lines within the electromagnetic radiation spectrum, visible, UV, IR, Microwave, or X-ray?

    These lines are in the region of
    __ Ultraviolet (UV) __

  4. The characteristic X-rays of some elements are listed here,

    Atomic
    number
    Photon energy
    h v /10-15 J
    ElementWavelength
    nm
    Frequency
    /1018 s
    Frequency(1/2)
    /109
    23 0.794 V 0.2503 1.199 1.095
    24 0.868 Cr 0.2289 1.310 1.145
    25 0.946 Mn 0.2102 1.427 1.195
    26 1.027 Fe 0.1936 1.550 1.245
    27 1.111 Co 0.1789 1.677 1.295
    28 1.199 Ni 0.1657 1.810 1.345
    29 1.290 Cu 0.1541 1.947 1.395
    30 1.385 Zn 0.1435 2.090 1.446
    42 2.664 Mo 0.0746 4.020 2.005
    47 3.554 Ag 0.0559 5.363 2.316
    79 11.03 Au 0.0180 16.650 4.080

    Complete the Table by taking the square root of the frequencies, and plot them against the atomic number Z on a graph to see if these data obey the Moseley's law.

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    |   You should get a linear plot
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    Evaluate the photon energies and plot them against Z2 to see if they fit a straight line.

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    |   You should get a linear plot
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  5. Apply the Moseley's law to estimate the wavelength of characteristic X-rays from hydrogen, (H, Z = 1), aluminium (Al, Z = 13), tin (Sn, Z = 50), and uranium (U, Z = 92).

    The wavelengths are given below
    H Al Sn U

    _____________

    _____________

    _____________

    _____________

    Element, ZSQRT(f) FrequencyWavelength Wavenumber /m-1
    H, 141924543 2.2968e15131 nm 7.63e6
    Al, 136.23e8 3.8815e170.773 nm1.29e9
    Sn, 502.396e9 5.742e180.0522 nm 1.92e10
    U, 924.409e91.944e191.54e-11 m
    0.0154 nm
    6.49e10

    ing on the two points you may choose, the value you estimated may be slightly different. If you used a least-squares line to get the slope, you are doing it properly. Since we do not require math skills here, we choose to use a simple approach in this hint.

    Note the wavenumbers for the Lyman series range from 8 to 11 million and the estimate for H here is 7.6 million. Depending on the slope one uses, the agreement can be very good.

    © cchieh@uwaterloo.ca
    You should get a linear plot