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SUBJECTIVE UNSOLVED LEVEL - II (BRUSH UP YOUR CONCEPTS) 1. Find the binding energy of an electron in the ground state of hydrogen-like ions in whose spectrum the third line of the Balmer series is equal to 108.5 nm. 2. Calculate the de Broglie wavelengths of an electron, proton, and uranium atom, all having the same kinetic energy 100 eV. 3. What amount of energy should be added to an electron to reduce its de Broglie wavelength from 100 to 50 pm? 4. Find the binding energy of the nucleus of lithium isotope and hence find the binding energy per nucleon in it. Given atom amu; atom = 1.007825 amu; amu. 5. A neutron with kinetic energy MeV activates a nuclear reaction whose threshold MeV. Find the kinetic energy of the alpha particles outgoing at right angles to the incoming neutron’s direction. [Take 1 amu MeV] 6. A town has a population of 1 million. The average electric power needed per person is 300 W. A reactor is to be designed to supply power to this town. The efficiency with which thermal power is converted into electric power is aimed at 25%. (a) Assuming 200 MeV of thermal energy to come from each fission event on an average, find the number of events that should take place every day. (b) Assuming the fission to take place largely through , at what rate will the amount of decrease? Express your answer in kg/day. (c) Assume that uranium enriched to 3% in will be used. How much uranium is needed per month (30 days)? 7. The count rate of nuclear radiation coming from a radioactive sample containing varies with time as follows. Time t (minute): 0 25 50 75 100 Count rate : 30 16 8.0 3.8 2.0 (a) Plot against . (b) Roughly draw a straight line which passes through the points, and use its slope to find the decay constant . (c) Calculate the half-life . 8. Carbon with mass number 11 decays to boron . (a) Is it a -decay or a -decay? (b) The half-life of the decay scheme is 20.3 minutes. How much time will elapse before a mixture of 90% carbon-11 and 10% boron-11 (by the number of atoms) converts itself into a mixture of 10% carbon-11 and 90% boron-11? 9. When charcoal is prepared from a living tree, it shows a disintegration rate of 15.3 disintegrations of per gram per minute. A sample from an ancient piece of charcoal shows activity to be 12.3 disintegrations per gram per minute. How old is this sample? Half-life of is 5730y. 10. Electrons passing through the ionosphere are found to rotate at revolution per second. Estimate the strength of the earth’s magnetic field in the ionosphere ( for the electron ). 11. When 50 kV d.c. is applied across an X-ray tube, heat is produced in the target at the rate of 495 W. Assuming that 1% of the energy of the incident electrons gets converted to X-rays, find (a) the number of electrons striking the target per second (b) the velocity of the incident electron (c) the shortest wavelength of X-ray produced. Assume that the electrons are non-rel

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- 1. S SU UB BJ JE EC CT TI IV VE E U UN NS SO OL LV VE ED D L LE EV VE EL L - - I II I (BRUSH UP YOUR CONCEPTS) 1. Find the binding energy of an electron in the ground state of hydrogen-like ions in whose spectrum the third line of the Balmer series is equal to 108.5 nm. 2. Calculate the de Broglie wavelengths of an electron, proton, and uranium atom, all having the same kinetic energy 100 eV. 3. What amount of energy should be added to an electron to reduce its de Broglie wavelength from 100 to 50 pm? 4. Find the binding energy of the nucleus of lithium isotope 7 3 Li and hence find the binding energy per nucleon in it. Given 7 3 Li atom 7.016005 amu; 1 1 H atom = 1.007825 amu; 1 0 1.008665 n amu. 5. A neutron with kinetic energy 10 K MeV activates a nuclear reaction 12 9 n C Be whose threshold 6.17 th E MeV. Find the kinetic energy of the alpha particles outgoing at right angles to the incoming neutron’s direction. [Take 1 amu 931.1 MeV] 6. A town has a population of 1 million. The average electric power needed per person is 300 W. A reactor is to be designed to supply power to this town. The efficiency with which thermal power is converted into electric power is aimed at 25%. (a) Assuming 200 MeV of thermal energy to come from each fission event on an average, find the number of events that should take place every day. (b) Assuming the fission to take place largely through U 235 , at what rate will the amount of 235 U decrease? Express your answer in kg/day. (c) Assume that uranium enriched to 3% in 235 U will be used. How much uranium is needed per month (30 days)? 7. The count rate of nuclear radiation coming from a radioactive sample containing 128 I varies with time as follows. Time t (minute): 0 25 50 75 100 Count rate 9 1 (10 ) R s : 30 16 8.0 3.8 2.0 (a) Plot 0 ln( / ) R R against t . (b) Roughly draw a straight line which passes through the points, and use its slope to find the decay constant . (c) Calculate the half-life 1/ 2 t . 8. Carbon ( 6) Z with mass number 11 decays to boron ( 5) Z . (a) Is it a -decay or a -decay? (b) The half-life of the decay scheme is 20.3 minutes. How much time will elapse before a mixture of 90% carbon-11 and 10% boron-11 (by the number of atoms) converts itself into a mixture of 10% carbon-11 and 90% boron-11? 9. When charcoal is prepared from a living tree, it shows a disintegration rate of 15.3 disintegrations of 14 C per gram per minute. A sample from an ancient piece of charcoal shows 14 C activity to be 12.3 disintegrations per gram per minute. How old is this sample? Half-life of 14 C is 5730y. 10. Electrons passing through the ionosphere are found to rotate at 1.4 10 revolution per second. Estimate the strength of the earth’s magnetic field in the ionosphere ( / e m for the electron 11 -1 1.8 10 C kg ). 11. When 50 kV d.c. is applied across an X-ray tube, heat is produced in the target at the rate of 495 W. Assuming that 1% of the energy of the incident electrons gets converted to X-rays, find
- 2. (a) the number of electrons striking the target per second (b) the velocity of the incident electron (c) the shortest wavelength of X-ray produced. Assume that the electrons are non-relativistic. 12. If 10000 V is applied across an X-ray tube, compare the de Broglie wavelength of the incident electrons (on target) to the shortest wavelength of X-rays produced ( 11 -1 e/m = 1.8×10 C kg for electron). 13. An electron, in a hydrogen-like atom, is in an excited state. It has a total energy of -3.4 eV. Calculate (i) the kinetic energy and (ii) the de-Broglie wavelength of the electron. 14. At a given instant there are 25% undecayed radio-active nuclei in a sample. After 10 seconds, the number of undecayed nuclei reduces to 12.5%. Calculate (i) mean-life of the nuclei, and (ii) the time in which the number of undecayed nuclei will further reduce to 6.25% of the reduced number. 15. In a photoelectric effect set-up a point source of light of power 3 3.2 10 W emits monoenergetic photons of energy 5.0 eV. The source is located at a distance of 0.8 , from the centre of a stationary metallic sphere of work function 3.0 eV and of radius 3 8.0 10 m. The efficiency of photoelectron emission is one for every 6 10 incident photons. Assume that the sphere is isolated and initially neutral and that photoelectrons are instantly swept away after emission. (a) Calculate the number of photoelectrons emitted per second. (b) Find the ratio of the wavelength of incident light to the de-Brogile wavelength of the fastest photoelectrons emitted. (c) It is observed that the photoelectron emission stops at a certain time t after the light source is switched on why ? (d) Evaluate the timet . S SU UB BJ JE EC CT TI IV VE E U UN NS SO OL LV VE ED D L LE EV VE EL L - - I II II I (CHECK YOUR SKILLS) 1. A radionuclide 1 A with decay constant 1 transforms into a radionuclide 2 A with decay constant 2 . Assuming that at the initial moment the preparation contained only the radionuclide 1 A , find : (a) the equation describing the accumulation of the radionuclide 2 A with time. (b) the time interval after which the activity of radionuclide 2 A reaches its maximum value. 2. In an experiment on two radioactive isotopes of an element (which do not decay into each other), their mass ratio at a given instant was found to be 3. The rapidly decaying isotope has larger mass and an activity of 1.0 curie initially. The half lives of the two isotopes are known to be 12 hours and 16 hours. What would be the activity of each isotope and their mass ratio after two days ? 3. A sample of uranium is a mixture of three isotopes 234 235 238 92 92 92 , and U U U present in the ratio of 0.006%, 0.71% and 99.284% respectively. The half-lives of these isotopes are 5 2.5 10 years, 8 7.1 10 year and 9 4.5 10 years respectively. Calculate the contribution to activity (in %) of each isotope in this sample. 4. A nucleus X, initially at rest, undergoes alpha-decay according to the equation : 228 92 A Z X Y (a) Find the values of A and Z in the above process. (b) The alpha particle produced in the above process is found to move in a circular track of 3 Tesla. Find the energy (in MeV) released during the process and the binding energy of the parent nucleus X. 5. Hydrogen atom in its ground state is excited by means of monochromatic radiation of wavelength 975 Å. How many different lines are possible in the resulting spectrum ? Calculate the longest wavelength amongst them. You may assume the ionization for hydrogen atom is 13.6 eV.
- 3. 6. Assume that the de Broglie wave associated with an electron can form a standing wave between the atoms arranged in a one dimensional array with nodes at each of the atomic sites. It is found that one such standing wave is formed if the distance d is 2.0 Å. If this distance d is increased to 2.5 Å but not for any intermediate value of d . Find the energy of the electrons in electron volts and the least value of d for which the standing wave of the type described above can form. 7. The ionization energy of a hydrogen like Bohr atom is measured to be approximately 5 rydbergs. Using this information , calculate the following exactly. (i) What is the wavelength of radiation emitted when the electron jumps from first excited state to the ground state? (ii) What is the radius of first orbit for this atom ? Given the Bohr radius of hydrogen atom = 11 5 10 m, 1 rydberg = 18 2.2 10 J, h = 6.63 × 10-34 J-s, c = 3 × 108 m/s. 8. 1 A current flows in an X-ray tube which is operated at 10000 V. The target area is 4 2 10 m . Find the pressure on the target, assuming that the electrons strike the target normally and the photons leave the target normally. Consider the ideal situation where each incident electron gives rise to a photon of the same energy. 9. Calculate for a hydrogen atom and a He ion: (a) the radius of the first Bohr orbit and the velocity of an electron moving along it; (b) the kinetic energy and the binding energy of an electron in the ground state; (c) the ionization potential, the first excitation potential and the wavelength of the resonance line ( 2 1) n n . 10. Calculate the separation between the particles of a system in the ground state, the corresponding binding energy, and the wavelength of the first line of the Lyman series, if such a system is (a) a mesonic hydrogen atom whose nucleus is a proton (in a mesonic atom an electron is replaced by a meson whose charge is the same and mass is 207 times that of an electron); (b) a positronium consisting of an electron and a positron revolving around their common centre of masses. A position is the antiparticle of an electron i.e. it has positive charge (+e) and the same mass (m). Hint : Remember to use the concept of reduced mass or calculate from fundamentals. 11. A radioactive source, in the form of a metallic sphere of radius 2 10 m emits -particles at the rate of 10 5 10 particles per second. The source is electrically insulated. How long will it take for its potential to be raised by 2 volt, assuming that 40% of the emitted -particles escape the source. 12. A fusion reaction of the type given below 2 2 3 1 1 1 1 1 D D T p E is most promising for the production of power. Here D and T stand for deuterium and tritium, respectively. Calculate the mass of deuterium required per day for a power output of 9 10 W. Assume that efficiency of the process to be 50%. Given mass of 2 1 2.01458 D a.m.u mass of 3 1 3.01605 T a.m.u. mass of 1 1 1.00728 p a.m.u. and 1 a.m.u. = 930 MeV. 13. A nuclear explosion is designed to deliver 1 MW of heat energy; how many fission events must be required in a second to attain this power level? If this explosion is designed with a nuclear fuel consisting of uranium-235 to run a reactor at this power level for one year then calculate the amount of fuel needed. You can assume that the amount of energy released per fission events is 200 MeV. 14. Ultraviolet light of wavelength 830 Å and 700 Å when allowed to fall on hydrogen atom in their ground state is found to liberate electrons with kinetic energy 1.3 eV and 4.0 eV respectively. Find the value of Planck’s constant. 15. Light of wavelength 180 nm ejects photoelectrons from a plate of a metal whose work function is 2 eV. If a uniform magnetic field of 5 5 10 tesla is applied parallel to plate, what would be the radius of the path followed by electrons ejected normally from the plate with maximum energy.