Physics · Textbook solutions

Structure of Atoms and Nuclei

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 45 questions

15. Structure of Atoms and Nuclei — worked examples

13 q

Solved Examples

Worked · 13
  1. Solved Ex.15.1
    Calculate the radius of the 3rd orbit of the electron in hydrogen atom.
  2. Solved Ex.15.2
    In a Rutherford scattering experiment, assume that an incident alpha particle (radius 1.80 fm) is moving directly toward a target gold nucleus (radius 6.23 fm). If the alpha particle stops right at the surface of the gold nucleus, how much energy did it have to start with?
  3. Solved Ex.15.3
    Determine the energies of the first two excited states of the electron in hydrogen atom. What are the excitation energies of the electrons in these orbits?
  4. Solved Ex.15.4
    Calculate the wavelengths of the first three lines in Paschen series of hydrogen atom.
  5. Solved Ex.15.5
    Calculate the radius and density of 70Ge^{70}\text{Ge} nucleus, given its mass to be approximately 69.924 u.
  6. Solved Ex.15.6
    Calculate the binding energy of 37Li^{7}_{3}\text{Li}, the masses of hydrogen and lithium atoms being 1.007825 u and 7.016 u respectively.
  7. Solved Ex.15.7
    Calculate the energy released in the alpha decay of 238Pu^{238}\text{Pu} to 234U^{234}\text{U}, the masses involved being mPu=238.04955m_{Pu} = 238.04955 u, mU=234.04095m_U = 234.04095 u and mHe=4.002603m_{He} = 4.002603 u.
  8. Solved Ex.15.8
    Calculate the maximum kinetic energy of the beta particle (positron) emitted in the decay of 1122Na^{22}_{11}\text{Na}, given the mass of 1122Na=21.994437^{22}_{11}\text{Na} = 21.994437 u, 1022Ne=21.991385^{22}_{10}\text{Ne} = 21.991385 u and me=0.00055m_e = 0.00055 u.
  9. Solved Ex.15.9
    The half-life of a nuclear species NX^{N}\text{X} is 3.2 days. Calculate its (i) decay constant, (ii) average life and (iii) the activity of its sample of mass 1.5 mg.
  10. Solved Ex.15.10
    The activity of a radioactive sample decreased from 350 s−1\text{s}^{-1} to 175 s−1\text{s}^{-1} in one hour. Determine the half-life of the species.
  11. Solved Ex.15.11
    In an alpha decay, the daughter nucleus produced is itself unstable and undergoes further decay. If the number of parent and daughter nuclei at time tt are NpN_p and NdN_d respectively and their decay constants are λp\lambda_p and λd\lambda_d respectively. What condition needs to be satisfied in order for NdN_d to remain constant?
  12. Solved Ex.15.12
    Calculate the energy released in the reaction 92236U→53137I+3997Y+2n^{236}_{92}\text{U} \rightarrow {}^{137}_{53}\text{I} + {}^{97}_{39}\text{Y} + 2n. The masses of 92236U^{236}_{92}\text{U}, 53137I^{137}_{53}\text{I} and 3997Y^{97}_{39}\text{Y} are 236.04557, 136.91787 and 96.91827 respectively.
  13. Solved Ex.15.13
    Calculate the energy released in the fusion reaction taking place inside the Sun, 4p→α+2e++4p \rightarrow \alpha + 2e^{+} + neutrinos, neglecting the energy given to the neutrinos. Mass of alpha particle being 4.001506 u.

Exercises

32 q

Choose the correct option

Practice · 5
  1. Choose the correct option.
    Ex Q.1 (i)
    In which of the following systems will the radius of the first orbit of the electron be smallest?
    1. A.
      hydrogen
    2. B.
      singly ionized helium
    3. C.
      deuteron
    4. D.
      tritium
  2. Ex Q.1 (ii)
    The radius of the 4th4^{\text{th}} orbit of the electron will be smaller than its 8th8^{\text{th}} orbit by a factor of
    1. A.
      2
    2. B.
      4
    3. C.
      8
    4. D.
      16
  3. Ex Q.1 (iii)
    In the spectrum of hydrogen atom which transition will yield longest wavelength?
    1. A.
      n=2n = 2 to n=1n = 1
    2. B.
      n=5n = 5 to n=4n = 4
    3. C.
      n=7n = 7 to n=6n = 6
    4. D.
      n=8n = 8 to n=7n = 7
  4. Ex Q.1 (iv)
    Which of the following properties of a nucleus does not depend on its mass number?
    1. A.
      radius
    2. B.
      mass
    3. C.
      volume
    4. D.
      density
  5. Ex Q.1 (v)
    If the number of nuclei in a radioactive sample at a given time is NN, what will be the number at the end of two half-lives?
    1. A.
      N/2N/2
    2. B.
      N/4N/4
    3. C.
      3N/43N/4
    4. D.
      N/8N/8

Answer in brief

Practice · 5
  1. Answer in brief.
    Ex Q.2 (i)
    State the postulates of Bohr's atomic model.
  2. Ex Q.2 (ii)
    State the difficulties faced by Rutherford's atomic model.
  3. Ex Q.2 (iii)
    What are alpha, beta and gamma decays?
  4. Ex Q.2 (iv)
    Define excitation energy, binding energy and ionization energy of an electron in an atom.
  5. Ex Q.2 (v)
    Show that the frequency of the first line in Lyman series is equal to the difference between the limiting frequencies of Lyman and Balmer series.

Solve the following

Practice · 22
  1. Ex Q.3
    State the postulates of Bohr's atomic model and derive the expression for the energy of an electron in the atom.
  2. Ex Q.4
    Starting from the formula for energy of an electron in the nthn^{\text{th}} orbit of hydrogen atom, derive the formula for the wavelengths of Lyman and Balmer series spectral lines and determine the shortest wavelengths of lines in both these series.
  3. Ex Q.5
    Determine the maximum angular speed of an electron moving in a stable orbit around the nucleus of hydrogen atom.
  4. Ex Q.6
    Determine the series limit of Balmer, Paschen and Bracket series, given the limit for Lyman series is 911.6 Å.
  5. Ex Q.7
    Describe alpha, beta and gamma decays and write down the formulae for the energies generated in each of these decays.
  6. Ex Q.8
    Explain what are nuclear fission and fusion giving an example of each. Write down the formulae for energy generated in each of these processes.
  7. Ex Q.9
    Describe the principles of a nuclear reactor. What is the difference between a nuclear reactor and a nuclear bomb?
  8. Ex Q.10
    Calculate the binding energy of an alpha particle given its mass to be 4.00151 u.
  9. Ex Q.11
    An electron in hydrogen atom stays in its second orbit for 10−810^{-8} s. How many revolutions will it make around the nucleus in that time?
  10. Ex Q.12
    Determine the binding energy per nucleon of the americium isotope 95244Am^{244}_{95}\text{Am}, given the mass of 95244Am^{244}_{95}\text{Am} to be 244.06428 u.
  11. Ex Q.13
    Calculate the energy released in the nuclear reaction 37Li+p→2α^{7}_{3}\text{Li} + p \rightarrow 2\alpha given mass of 37Li^{7}_{3}\text{Li} atom and of helium atom to be 7.016 u and 4.0026 u respectively.
  12. Ex Q.14
    Complete the following equations describing nuclear decays. (a) 86226Ra→α+^{226}_{86}\text{Ra} \rightarrow \alpha + (b) 819O→e−+^{19}_{8}\text{O} \rightarrow e^{-} + (c) 90228Th→α+^{228}_{90}\text{Th} \rightarrow \alpha + (d) 712N→612C+^{12}_{7}\text{N} \rightarrow {}^{12}_{6}\text{C} +
  13. Ex Q.15
    Calculate the energy released in the following reactions, given the masses to be 88223Ra^{223}_{88}\text{Ra} : 223.0185 u, 82209Pb^{209}_{82}\text{Pb} : 208.9811, 614C^{14}_{6}\text{C} : 14.00324, 92236U^{236}_{92}\text{U} : 236.0456, 56140Ba^{140}_{56}\text{Ba} : 139.9106, 3694Kr^{94}_{36}\text{Kr} : 93.9341, 611C^{11}_{6}\text{C} : 11.01143, 511B^{11}_{5}\text{B} : 11.0093. Ignore neutrino energy. (a) 88223Ra→82209Pb+614C^{223}_{88}\text{Ra} \rightarrow {}^{209}_{82}\text{Pb} + {}^{14}_{6}\text{C} (b) 92236U→56140Ba+3694Kr+2n^{236}_{92}\text{U} \rightarrow {}^{140}_{56}\text{Ba} + {}^{94}_{36}\text{Kr} + 2n (c) 611C→511B+e++^{11}_{6}\text{C} \rightarrow {}^{11}_{5}\text{B} + e^{+} + neutrino
  14. Ex Q.16
    Sample of carbon obtained from any living organism has a decay rate of 15.3 decays per gram per minute. A sample of carbon obtained from very old charcoal shows a disintegration rate of 12.3 disintegrations per gram per minute. Determine the age of the old sample given the decay constant of carbon to be 3.839×10−123.839 \times 10^{-12} per second.
  15. Ex Q.17
    The half-life of 3890Sr^{90}_{38}\text{Sr} is 28 years. Determine the disintegration rate of its 5 mg sample.
  16. Ex Q.18
    What is the amount of 2760Co^{60}_{27}\text{Co} necessary to provide a radioactive source of strength 10.0 mCi, its half-life being 5.3 years?
  17. Ex Q.19
    Disintegration rate of a sample is 101010^{10} per hour at 20 hrs from the start. It reduces to 6.3×1096.3 \times 10^{9} per hour after 30 hours. Calculate its half life and the initial number of radioactive atoms in the sample.
  18. Ex Q.20
    The isotope 57Co^{57}\text{Co} decays by electron capture to 57Fe^{57}\text{Fe} with a half-life of 272 d. The 57Fe^{57}\text{Fe} nucleus is produced in an excited state, and it almost instantaneously emits gamma rays. (a) Find the mean lifetime and decay constant for 57Co^{57}\text{Co}. (b) If the activity of a radiation source 57Co^{57}\text{Co} is 2.0 μCi\mu\text{Ci} now, how many 57Co^{57}\text{Co} nuclei does the source contain? (c) What will be the activity after one year?
  19. Ex Q.21
    A source contains two species of phosphorous nuclei, 1532P^{32}_{15}\text{P} (T1/2=14.3 d)(T_{1/2} = 14.3\ \text{d}) and 1533P^{33}_{15}\text{P} (T1/2=25.3 d)(T_{1/2} = 25.3\ \text{d}). At time t=0t = 0, 90% of the decays are from 1532P^{32}_{15}\text{P}. How much time has to elapse for only 15% of the decays to be from 1532P^{32}_{15}\text{P}?
  20. Ex Q.22
    Before the year 1900 the activity per unit mass of atmospheric carbon due to the presence of 14C^{14}\text{C} averaged about 0.255 Bq per gram of carbon. (a) What fraction of carbon atoms were 14C^{14}\text{C}? (b) An archaeological specimen containing 500 mg of carbon, shows 174 decays in one hour. What is the age of the specimen, assuming that its activity per unit mass of carbon when the specimen died was equal to the average value of the air? Half-life of 14C^{14}\text{C} is 5730 years?
  21. Ex Q.23
    How much mass of 235U^{235}\text{U} is required to undergo fission each day to provide 3000 MW of thermal power? Average energy per fission is 202.79 MeV
  22. Ex Q.24
    In a periodic table the average atomic mass of magnesium is given as 24.312 u. The average value is based on their relative natural abundance on earth. The three isotopes and their masses are 1224Mg^{24}_{12}\text{Mg} (23.98504 u), 1225Mg^{25}_{12}\text{Mg} (24.98584 u) and 1226Mg^{26}_{12}\text{Mg} (25.98259 u). The natural abundance of 1224Mg^{24}_{12}\text{Mg} is 78.99% by mass. Calculate the abundances of other two isotopes.