Ch 12 (atoms)

In these slides, I covered the following topics with PYQ's of CH-12 (Atom) of class 12th Physics:
-Alpha-particle scattering experiment
-Rutherford's model of the atom
-Bohr model,
-Energy levels,
-Hydrogen spectrum

1. CH-12 ATOMS

2. SYLLABUS Alpha-particle scattering experiment Rutherford's model of atom Bohr model, Energy levels, H ydrogen spectrum

3. INTRODUCTION Indian Philosopher Maharishi Kanad if we go on dividing matter, a stage will come when particles obtained cannot be divided further. Maharishi Kanad called these indivisible particles Parmanu . Indian Philosopher Pakudha Katyayama Parmanu normally exist in a combined form which gives us various forms of matter . Greek Philosopher Democritus and Leucippus if we go on dividing matter, a stage will come when particles obtained cannot be divided further. Democritus called these indivisible particles atoms .

4. Experimental Evidences By the 19 th century , enough evidence had accumulated in favour of atomic hypothesis of matter . In 1897 , the experiments on electric discharge through gases carried out by J . J. Thomson revealed that atoms of different elements contain negatively charged constituents (electrons) that are identical for all atoms . In 1885 , J. J. Balmer obtained a simple empirical formula which gave the wavelengths of a group of lines emitted by atomic hydrogen. In 1906 , E. Rutherford (student of J. J. Thomson) proposed a classic experiment of scattering of α -particles (emitted by some radioactive elements)by atoms to investigate the atomic structure . This experiment was later performed by Hans Geiger and Ernst Marsden in 1911 .

5. Experimental Evidences Neils Bohr’s model of the atom overcome the objections raised against Rutherford’s model of the atom . In 1932 , J. Chadwick discovered another subatomic particle which had no charge and a mass nearly equal to that of a proton. It was eventually named as neutron .

6. Thomson’s model of an atom In 1898 J.J. Thomson proposed the model of an atom which is similar to that of a Christmas pudding. Thomson proposed that: An atom consists of a positively charged sphere and the electrons are embedded in it like currants (dry fruits) in a spherical Christmas pudding. The negative and positive charges are equal in magnitude. So, the atom as a whole is electrically neutral. Thomson’s model explained that atoms are electrically neutral . Failure: It could not explain the origin of spectral series in the case of hydrogen & other atoms. It failed to explain the origin of the large angle scattering of α - particles in Rutherford’s experiment.

7. Rutherford’s model of an atom & α - particles scattering experiment Ernest Rutherford was interested in knowing how the electrons are arranged within an atom . At the suggestion of Ernst Rutherford , in 1911, H. Geiger and E. Marsden performed experiment called α -particles scattering experiment. In this experiment, fast moving alpha ( α )-particles were made to fall on a thin gold foil . 5.5 Mev Alpha-particles emitted by a radioactive source & were collimated into a narrow beam by their passage through lead bricks. The beam was allowed to fall on a thin foil of gold of thickness 2.1 × 10 –7 m .  

8. Alpha particle Scattering Experiment The scattered alpha-particles were observed through a rotatable detector consisting of zinc sulphide screen and a microscope. The scattered alpha-particles on striking the screen produced brief light flashes. These flashes may be viewed through a microscope . It was expected that α -particles would be deflected by the sub-atomic particles in the gold atoms. Since the α -particles were much heavier than the protons, he did not expect to see large deflections.

9. Alpha particle Scattering Experiment Observations: Most of the fast moving α -particles passed straight through the gold foil. Only about 0.14% of the incident α -particles scatter by more than 1 ° about 1 in 8000 deflect by more than 90° Surprisingly one out of every 12000 particles appeared to rebound.

10. Alpha particle Scattering Experiment Results: As most of the α -particles passed straight through the gold foil which indicates that most of the space within the atom must be empty. Large angle of scattering indicates that atom has a small positively charged core called ‘nucleus’ at centre of atom which deflects α -particles at different angles. Very few α - particle suffers deflection of 180 . This shows that size of nucleus is very small. The nucleus is surrounded by cloud of electron whose total –ve charge is equal to total +ve charge on the nucleus so that atom as a whole is electrically neutral .

11. Reason of Scattering The scattering of α -particles is due to the Columbic repulsion between +vely α -particles charge & the nucleus. α -particles which passes through the nucleus at large distance, experiences less repulsion & passes undeflected. α -particles which passes close to the nucleus, experiences more repulsive & thus scatter at large angles. When α -particles travels head on towards the nucleus, the strong repulsive force slows down the α -particles which finally stopped & repelled back along the same original path.

12. Distance of Closest Approach: Estimation of Nuclear Size Suppose an α -particles of mass m & initial velocity v moves directly towards the centre of the nucleus of an atom. As α -particles approaches the positive nucleus, it experiences Columbic repulsion & its kinetic energy gets converted into electric potential energy. At certain distance r from the nucleus, α -particles stops for a moment and begins to move back along the same path. This distance is called distance of closest approach. At this distance r , the entire initial kinetic energy of the α -particles gets converted into electrostatic potential energy. Acc. to conservation of energy Charge on α- particles, q 1 = +2e Charge on scattering nucleus, q 2 = + Ze Charge on nucleus of Gold, q 2 = +79e KE of α- particles = 5.5Mev Distance of closest approach, r = 41.3 fm  

13. Impact Parameter The impact parameter is defined as the perpendicular distance of the velocity vector of the α- particles from the nucleus, when it is far away from the atom. an α- particle close to the nucleus (small impact parameter ) suffers large scattering. In case of head-on collision, the impact parameter is minimum and the α- particle rebounds back ( ). For a large impact parameter, the α- particle goes nearly undeviated and has a small deflection ( ).  

14. total energy of the electron An electron revolving in an orbit of atom, has both kinetic energy and electrostatic potential energy . To keep the electron in its orbit, the electrostatic force of attraction (F e ), provides the requisite centripetal force (F c ) Divide both side by 2   Total energy of electron in the n th orbit is The total energy of the electron is negative. This implies that the electron is bound to the nucleus. If E were positive, an electron will not follow a closed orbit around the nucleus.  

15. Rutherford’s model of atom Based on the above observations and conclusions, Rutherford proposed the atomic structure of elements. According to the Rutherford atomic model: The positively charged particles and most of the mass of an atom was concentrated in an extremely small volume. He called this region of the atom as a nucleus. Rutherford model proposed that the negatively charged electrons surround the nucleus of an atom. He also claimed that the electrons surrounding the nucleus revolve around it with very high speed in circular paths. He named these circular paths as orbits. Electrons being negatively charged and nucleus being a densely concentrated mass of positively charged particles are held together by a strong electrostatic force of attraction.

16. Limitations of Rutherford’s model of atom Although the Rutherford atomic model was based on experimental observations it failed to explain certain things. Rutherford model was not in accordance with Maxwell’s theory Rutherford proposed that the electrons revolve around the nucleus in fixed paths called orbits. According to Maxwell, accelerated charged particles emit electromagnetic radiations according to electromagnetic theory and hence an electron revolving around the nucleus should emit electromagnetic radiation. Since an accelerated charge emits energy, the radius of the circular path of a revolving electron should go on decreasing and ultimately it should fall into the nucleus. So, it could not explain the structure of the atom. As matter is stable, we cannot expect the atoms to collapse. He did not say anything about the arrangement of electrons in an atom which made his theory incomplete . Rutherford's model is not able to explain the spectrum of even most simplest H-spectrum.

17. Questions (PYQ’s) Why is the classical (Rutherford) model for an atom of electron orbiting around the nucleus not able to explain the atomic structure ? Using Rutherford's model of the atom, derive the expression for the total energy of the electron in hydrogen atom. What is the significance of total negative energy possessed by the electron ? In an experiment on α -particle scattering by a thin foil of gold, draw a plot showing the number of particles scattered versus the scattering angle . Why is it that a very small fraction of the particles are scattered at >90° ? Write two important conclusions that can be drawn regarding the structure of the atom from the study of this experiment . Draw a schematic arrangement of the Geiger - Marsden experiment for studying α- particle scattering by a thin foil of gold. Describe briefly , by drawing trajectories of the scattered α -particle, how this study can be used to estimate the size of the nucleus.  

18. questions 5. State the basic assumption of the Rutherford model of the atom. Explain, in brief, why this model cannot account for the stability of an atom. 6. In Rutherford scattering experiment, draw the trajectory traced by a-particles in the coulomb field of target nucleus and explain how this led to estimate the size of the nucleus?

19. BOHR’S MODEL OF ATOM

20. Bohr’s Model of Atom Or Bohr’s postulates Drawbacks of Rutherford’s atomic model lead to the Bohr’s Model where he came up with 3 postulates 1 st Postulate (Stationary orbit) An electron in an atom could revolve in certain stable orbits without the emission of radiant energy. E ach atom has certain definite stable states in which it can exist, and each possible state has definite total energy. These are called the stationary states of the atom . 2 nd Postulate (Quantum condition) The electron revolves around the nucleus only in those orbits for which the angular momentum is integral multiple of Thus the angular momentum (L) of the orbiting electron is quantized .   a statement that is accepted without proof

21. Bohr’s postulates 3 rd Postulate (Frequency condition) An atom can emit or absorb radiation in the form of discrete energy photons. A photon is emitted when an electron make a transition from stationary orbit (non-radiating orbits) of higher energy to stationary orbit of lower energy. Photon have energy equal E i and E f are the energies of the initial and final states and E i > E f .  

22. DE BROGLIE’S EXPLANATION OF BOHR’S SECOND POSTULATE OF QUANTISATION de Broglie’s hypothesis that material particles, such as electrons, also have a wave nature Hence a circular orbit can be taken as stationary states only when it contains integral no. of de-Broglie wavelength, i.e. From de-Broglie wavelength equation L n = angular momentum of electron in nth orbit v n = velocity of electron in nth orbit r n = radius of nth orbit n= permitted orbits on which electron revolve (called principle quantum number)  

23. Bohr’s theory of Hydrogen Atom & Hydrogen-like Atoms A hydrogen-like atom consists of a positively-charged nucleus with a positive charge + Ze and an electron revolving around the nucleus in a stable orbit . Bohr's Radius : The electrostatic force of attraction between the nucleus & electron To keep the electron in its orbit , the electrostatic force of attraction, provides the requisite centripetal force ( F c )   ( i ) (ii)

24. Bohr’s theory of Hydrogen Atom & Hydrogen-like Atoms According to Bohr’s Quantization condition Comparing eqa n (ii) & eqa n ( iii ) Put in eqa n ( iii)   (iii)

25. Bohr’s theory of Hydrogen Atom & Hydrogen-like Atoms Radii of permitted orbit is proportional to n 2 & increases in the ratio of 1:2:4:9:16…. n is called principle quantum number Radius of innermost orbit of the hydrogen atom, called the n = 1 , Z = 1Bohr radius Orbital speed of electron For 1 st orbit  

26. Energy of the electron It includes electrons K.E. & electrostatic P.E. in n th orbit To keep the electron in its orbit, the electrostatic force of attraction (F e ), provides the requisite centripetal force (F c ) Divide both side by 2   Total energy of electron in the n th orbit is  

27. Total Energy of the electron in terms of K.E. & P.E. Total energy of electron in terms of K.E. Total energy of electron in terms of P.E. (U)  

28. Formulas de-Broglie wavelength of the electron orbiting in the n th state Quantum condition Frequency condition Velocity of electron in n th orbit of atom Radius of n th orbit of atom   Radius of n th orbit of Hydrogen atom Velocity of electron in n th orbit of Hydrogen atom Total energy of electron in the n th orbit is Total energy of electron in terms of K.E. Total energy of electron in terms of P.E. (U)  

29. Questions (PYQ’s) What is the ratio of radii of the orbits corresponding to first excited state and ground state in a hydrogen atom ? (9) State Bohr's quantisation condition for defining stationary orbits . State Bohr postulate of hydrogen atom that gives the relationship for the frequency of emitted photon in a transition . Using Bohr's postulates of the atomic model, derive the expression for the radius of n th electron orbit. Hence obtain the expression for Bohr's radius. Show that the radius of the orbit in hydrogen atom varies as n 2 , where n is the principal quantum number of the atom . Using Bohr's postulates for hydrogen atom, show that the total energy (E) of the electron in the stationary states can be expressed as the sum of kinetic energy (K) and potential energy (U ), where K = -2U. Hence deduce the expression for the total energy in the n th energy level of hydrogen atom.

30. Questions (PYQ’s) In the ground state of hydrogen atom, its Bohr radius is given as 5.3 x 10 -11 m. The atom is excited such that the radius becomes 21.2 x 10 -11 m. Find the value of the principal quantum number and the total energy of the atom in this excited state . (15) (a) The radius of the innermost electron orbit of a hydrogen atom is 5.3 x 10 -11 m. Calculate its radius in n = 3 orbit. (b) The total energy of an electron in the first excited state of the hydrogen atom is -3.4 eV. Find out its ( i ) kinetic energy and (ii) potential energy in this state . (16) Write two important limitations of Rutherford model which could not explain the observed features of atomic spectra. How were these explained in Bohr's model of hydrogen atom? (24) If electron in the atom is replaced by a particle (muon) having the same charge but mass about 200 times as that of the electron to form a muonic atom, how would ( i ) the radius and ( i i) the ground state energy of this be affected?

31. Questions (PYQ’s) Calculate the de-Broglie wavelength of the electron orbiting in the n = 2 state of hydrogen atom . Use de-Broglie's hypothesis to write the relation for the n th radius of Bohr orbit in terms of Bohr's quantization condition of orbital angular momentum.

32. Spectral Series of Hydrogen Atom From Bohr 3 rd Postulate We know from Bohr’s theory, the energy of electron in the n th orbit of hydrogen atom is given by For hydrogen atom, Z=1   Divide both side by c Where is called wave number R is called Rydberg’s constant whose value is  

33. Spectral Series of Hydrogen Atom Lyman Series: When electron jumps from higher energy orbitals (n 2 = 2,3,4……) to a lower energy orbital (n 1 =1), we get a spectral series called Lyman series. These spectral series belong to ultraviolet region . Balmer Series: When electron jumps from higher energy orbitals (n 2 = 3,4,5……) to a lower energy orbital (n 1 =2), we get a spectral series called Balmer series. These spectral series belong to visible region .  

34. Spectral Series of Hydrogen Atom Paschen Series: When electron jumps from higher energy orbitals (n 2 = 4,5,6,……) to a lower energy orbital (n 1 =3), we get a spectral series called Paschen series. These spectral series belong to infrared region . Brackett Series: When electron jumps from higher energy orbitals (n 2 = 5,6,7,……) to a lower energy orbital (n 1 =4), we get a spectral series called Brackett series. These spectral series belong to infrared region .  

35. Spectral Series of Hydrogen Atom Pfund Series: When electron jumps from higher energy orbitals (n 2 = 6,7,8,……) to a lower energy orbital (n 1 =5) , we get a spectral series called Pfund series. These spectral series belong to infrared region .  

36. Maximum possible number of Spectral lines Maximum possible number of Spectral lines when the hydrogen atom is in its excited state is given by Eg .:- For 2 nd excited state maximum possible number of Spectral lines For 4 th excited state maximum possible number of Spectral lines  

37. Energy level Diagram for Hydrogen It is a diagram in which energies of different stationary states of an atom are represented by parallel horizontal lines. Energy levels of hydrogen For hydrogen atom, Z=1 Substituting n =1,2,3,4,……, we get the energies of different orbits   Clearly, an electron can have only certain values of energy while revolving in different orbits. This is called energy quantization.  

38. Energy level Diagram for Hydrogen The energy state corresponding to n =1, has the lowest energy equal to -13.6 eV. This state or orbit is called the ground or normal state of the atom.

39. Normally the electron in hydrogen atom remains in the ground state. When hydrogen atom receives energy from outside, the electron may make transition to some higher energy states which are called excited states. Excitation Energy It is defined as the energy required by its electron to jump from the ground state to any one of the excited states. 1 st excitation energy of hydrogen 2 nd excitation energy of hydrogen   Excitation & Ionization (Energy & Potential)

40. Ionization energy: It is defined as the energy required to knock the electron completely out of the atom. After the removal of electron atom is left with positive charge & it is said to ionized. Ionization energy of hydrogen = Excitation Potential: The potential difference through which an electron should be accelerated to acquire the value of excitation energy is called excitation potential. 1 st excitation potential of hydrogen = 2 nd excitation potential of hydrogen =  

41. Ionization Potential ( V ion ): The potential difference through which an electron should be accelerated to acquire the value of ionization energy is called Ionization potential. Ionization potential of hydrogen  

42. It was primarily for hydrogen l ike single electron atoms & fails in the case of with two or more electrons . In the spectrum of hydrogen, certain spectral lines are not single lines but a group of closed lines with slightly different frequencies. Bohr’s theory could not explain these fine feature of hydrogen spectrum. Bohr’s theory does not tell anything about the relative intensities of various spectral lines. It does not explain the further splitting of spectral lines in a magnetic field (Zeeman effect) or in an electric field (Stark effect). It does not explain why only circular orbits should be chosen. Limitations of Bohr’s Theory

43. Questions (PYQ’s) Using Bohr's postulates, obtain the expression for the total energy of the electron in the stationary states of the hydrogen atom. Hence draw the energy level diagram showing how the line spectra corresponding to Balmer series occur due to transition between energy levels . (17) Define ionization energy. What is its value for a hydrogen atom ? Using the postulates of Bohr's model of hydrogen atom, obtain an expression for the frequency of radiation emitted when atom make a transition from the higher energy state with quantum number n i to the lower energy state with quantum number n f ( n f < n i ). What is the maximum possible number of spectral lines observed when the hydrogen atom is in its second excited state? Justify your answer. Calculate the ratio of the maximum and minimum wavelengths of the radiations emitted in this process . (22)

44. Questions (PYQ’s) Calculate the shortest wavelength in the Balmer series of hydrogen atom . In which region of hydrogen spectrum does this wavelength lie ? (7) The second member of Lyman series in hydrogen spectrum has wavelength 5400Å . Find the wavelength of the first member . (8) Define ionization energy. How would the ionization energy change when electron in hydrogen atom is replaced by a particle of mass 200 times that of the electron but having the same charge ? (31) An electron jumps from fourth to first orbit in an atom . How many maximum number of spectral lines can be emitted by the atom? To which series these lines correspond ? (32)

45. Questions (PYQ’s) In hydrogen atom, an electron undergoes transition from 2 nd excited state to the first excited state and then to the ground state. Identify the spectral series to which these transitions belong . ( ii) Find out the ratio of the wavelengths of the emitted radiations in the two cases . (34) Using Rydberg formula, calculate the longest wavelength belonging to Lyman and Balmer series of hydrogen spectrum. In which region these transitions lie ? (35) A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. Upto which energy level the hydrogen atoms would be excited ? (36) The value of ground state energy of hydrogen atom is -13.6 eV . ( i ) Find the energy required to move an electron from the ground state to the first excited state of the atom . ( ii) Determine (a) the kinetic energy and (b ) orbital radius in the first excited state of the atom .

46. Questions (PYQ’s) The energy levels of a hypothetical atom are shown below. Which of the shown transitions will result in the emission of a photon of wavelength 275 nm? Which of these transitions correspond to emission of radiation of ( i ) maximum & (ii ) minimum wavelength ? (19)