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Physics A: Problem Set 25: Mass-Energy

recommended reading

Barron's Let's Review: 13.6 Energy and Mass
physics.info: n/a
Wikipedia: Mass-energy equivalence, Nuclear reaction, Pair production, Annihilation, Antimatter
HyperPhysics: Relativistic Energy
Khan Academy: Nuclear chemistry (the first four chapters)

nuclear reactions

  1. Certain heavy, unstable nuclei will eject a helium nucleus (also known as an alpha particle) in a process known as alpha decay. The following nuclei undergo alpha decay. Write these reactions out in symbolic form.
    1. 23892U
    2. 20984Po
    3. 22688Ra
    4. the daughter nucleus from part c.
    5. the daughter nucleus from part d.

    Alpha decay lowers the atomic number by 2 and the mass number by 4.

    1. 23892U → 23490Th + 42He
    2. 20984Po → 20582Pb + 42He
    3. 22688Ra → 22286Rn + 42He
    4. 22288Rn → 21884Po + 42He
    5. 21884Po → 21482Pb + 42He
  2. In certain unstable nuclei, a neutron will transmute into a proton, an electron, and an electron antineutrino. The proton remains in the nucleus and the electron (also known as a beta particle) and electron antineutrino (often just called a neutrino) are both ejected from the nucleus. The following nuclei undergo beta decay. Write these reactions out in symbolic form.
    1. 146C
    2. 20982Pb
    3. 31H

    Beta decay raises the atomic number by 1 but keeps the mass number unchanged.

    1. 146C → 147N + 0−1e + 00ν̅
    2. 20982Pb → 20983Bi + 0−1e + 00ν̅
    3. 31H → 32He + 0−1e + 00ν̅
  3. Technetium is an artificially produced element that is used in nuclear medicine as a tracer. For each step in the process from production to use to eventual decay into a stable nucleus, write out the corresponding reaction in symbolic form.
    1. Molybdenum 98 (9842Mo) is bombarded with neutrons so that one sticks to the nucleus. This is known as neutron capture.
    2. The daughter nucleus of the previous reaction undergoes beta decay.
    3. The daughter nucleus of the previous reaction is metastable and undergoes gamma decay. (The gamma ray from this decay is detected and used to create an image of an affected tissue.)
    4. The daughter nucleus of the previous reaction undergoes beta decay to a stable nucleus.

    Solutions.

    1. Neutron capture raises the mass number by 1, but leaves the atomic number (and thus the element) unchanged.

      10n + 9842Mo → 9942Mo

    2. Beta decay raises the atomic number by 1 (which changes the element), but leaves the mass number unchanged. In this case, molybdenum decays into technetium.

      9942Mo → 99m43Tc + 0−1e + 00ν̅

    3. Gamma decay keeps both the atomic number and the mass number the same. Indicate that the nucleus has fallen from an excited (or metastable) state back to its ground state by deleting the "m" from the end of the mass number.

      99m43Tc → 9943Tc + 00γ

    4. A second beta decay raises the atomic number by 1 again. Consult your favorite periodic table if you don't know which element has this increased number of protons.

      9943Tc → 9944Ru + 0−1e + 00ν̅

  4. Neutron induced fission is the class of nuclear reaction powering nuclear reactors and nuclear weapons. A particular isotope of uranium or plutonium is bombarded with a bullet neutron. The now slightly heavier nucleus is unstable and splits (fissions) into two roughly equal halves plus 2–3 free neutrons. For each of the incomplete reactions below, determine the number of free neutrons.
    1. 10n + 23592U → 8735Br + 14657La + …
    2. 10n + 23592U → 9236Kr + 14156Ba + …
    3. 10n + 23592U → 9037Rb + 14455Cs + …
    4. 10n + 23592U → 9038Sr + 14354Xe + …
    5. 10n + 23994Pu → 9436Kr + 14458Ce + …

    Check the mass numbers on both sides of the reaction. They should be the same, but they aren't (because the reaction was deliberately written as incomplete). The number you're short on the daughters' side (the right side) is the number of neutrons.

    1. 10n + 23592U → 8735Br + 14657La + 3 10n
    2. 10n + 23592U → 9236Kr + 14156Ba + 3 10n
    3. 10n + 23592U → 9037Rb + 14455Cs + 2 10n
    4. 10n + 23592U → 9038Sr + 14354Xe + 3 10n
    5. 10n + 23994Pu → 9436Kr + 14458Ce + 2 10n

for practice

  1. A photon has just enough energy to form an electron-positron pair.
    1. Write this reaction out in symbolic form.
    2. What is the energy of this photon?
    3. What is its frequency?
    4. What is its wavelength?
    5. What type of electromagnetic radiation is this?
    1. Here's the reaction in symbolic form.

      00γ → 0−1e + 0+1

    2. A positron has the same mass as an electron. Only the charge is different. Calculate the energy equivalent to the mass of two electrons using Einstein's famous equation.

      E = mc2
      E = 2(9.11 × 10−31 kg)(3.00 × 108 m/s)2
      E = 1.64 × 10−13 J
    3. Use Planck's equation to determine the frequency of the photon.

      f = 
      E
      h
      f =  1.64 × 10−13 J
      6.63 × 10−34 Js
      f =  2.47 × 1020 Hz  
       
    4. Planck's equation also comes in a version that uses wavelength instead of frequency. That's one way to solve this part of the problem.

      λ = 
      hc
      E
      λ =  (6.63 × 10−34 Js)(3.00 × 108 m/s)
      1.64 × 10−13 J
      λ =  1.21 × 10−12 m  
       
    5. Most reference sources call this a gamma ray photon.

  2. The fuel used in most high-yield thermonuclear weapons is solid lithium 6 deuteride. These weapons, commonly known as "hydrogen bombs" or "H-bombs", use the energy released when a nucleus of light lithium (63Li, m = 6.015121 u) and heavy hydrogen, also known as deuterium (21H, m = 2.0140 u), fuse to form two nuclei of ordinary helium (42He, m = 4.00260 u).
    1. Write this reaction out in symbolic form.
    2. What is the mass defect when one molecule of lithium 6 deuteride is transformed into two atoms of helium? State your answer in…
      1. atomic mass units
      2. megaelectronvolts
      3. joules
      4. kilograms
    1. Here's the reaction in symbolic form.

      63Li + 21H → 2 42He

    2. Most of the rest of this problem is an exercise in conversion — conversion between mass and energy in SI units and acceptable non-SI units.

      1. Mass before does not equal mass after. The parents weigh more than the daughters in this reaction. This means that the reaction releases energy.

        + 6.015121 u
        + 2.014000 u
        + 8.029121 u
        + 4.00260 u
        + 4.00260 u
        + 8.00520 u

        m = 0.023921 u

      2. Convert atomic mass units (u) to megaelectronvolts (MeV) using the conversion factor.

        0.023921 u
        × 931 MeV/u
        22.27 MeV
      3. Convert megaelectronvolts to joules using the conversion factor better known as the elementary charge. Remember that mega means 106.

        22.27 × 106 V
        × 1.60 × 10−19 C
        3.563 × 10−12 J
      4. Joules convert to kilograms using Einstein's (rearranged) equation — but there are other methods that give the same answer.

        m = 
        E
        c2
         
         
        m =  (3.563 × 10−12 J)
        (3.00 × 108 m/s)2  
        m =  3.959 × 10−29 kg  
         

for real

  1. A standard measure of explosive energy is the ton of TNT. By definition, one ton of TNT possesses 4.184 gigajoules of chemical energy.
    1. The "Little Boy" uranium fission bomb dropped on Hiroshima, Japan on 6 August 1945 had an explosive yield of 12.5 kilotons of TNT. How much mass was transformed into energy by this weapon? About how big is this?
    2. The nuclear weapon with the greatest yield was the 50 megaton, two-stage fusion "Tsar Bomba" (Цар Бомба) detonated over Novaya Zemlya, Russia on 30 October 1961. How much mass was transformed into energy by this weapon? About how big is this?

    Use the famous equation…

    E = mc2

    …and solve it for mass

    m =  E
    c2
    1. The kilo in kilotons means 103.

      m = 
      E
      c2
       
       
      m =  (12.5 × 103)(4.184 × 109 J)  
      (3.00 × 108 m/s)2  
      m =  0.00058 kg  
       

      When the Little Boy exploded, it destroyed about half a gram — the mass of a paperclip.

    2. The mega in megatons means 106.

      m = 
      E
      c2
       
       
      m =  (50 × 106)(4.184 × 109 J)  
      (3.00 × 108 m/s)2  
      m =  2.3 kg  
       

      The Tsar of bombs destroyed the mass of a small cat, or a big chicken.

  2. The luminosity of the sun is 3.827 × 1026 watts. How much mass is destroyed in the sun every second? About how big is this?

    Use the famous equation…

    E = mc2

    …and solve it for mass

    m =  E
    c2

    A watt is a joule per second — or a joule is a watt for a second.

    m = 
    E
    c2
     
     
    m =  (3.827 × 1026 J)  
    (3.00 × 108 m/s)2  
    m =  4.3 × 109 kg  
     

    This is about 4,000,000 tonnes — the mass of a small railway bridge — every second. The sun is a busy place.