Case study (4 Marks)

Question 14 Marks

The orbital wave function or $\psi$ for an electronin an atom has no physical meaning. It issimply a mathematical function of thecoordinates of the electron. However, fordifferent orbitals the plots of correspondingwave functions as a function of r (the distancefrom the nucleus) are different. According to the German physicist, MaxBorn, the square of the wave function(i.e.,$\psi^2$) at a point gives the probability densityof the electron at that point. Boundary surface diagrams of constantprobability density for different orbitals give afairly good representation of the shapes of theorbitals. In this representation, a boundarysurface or contour surface is drawn in spacefor an orbital on which the value of probabilitydensity $\mid\psi\mid2$ is constant. In principle manysuch boundary surfaces may be possible.However, for a given orbital, only thatboundary surface diagram of constantprobability density* is taken to be goodrepresentation of the shape of the orbital whichencloses a region or volume in which theprobability of finding the electron is very high,say, 90%.
In hydrogen atom, electron has the same energy when it is in the2s orbital as when it is present in 2p orbital.The orbitals having the same energy are calleddegenerate. The 1s orbital in a hydrogenatom, as said earlier, corresponds to the moststable condition and is called the ground stateand an electron residing in this orbital is moststrongly held by the nucleus.
An electron inthe 2s, 2p or higher orbitals in a hydrogen atomis in excited state.The filling of electrons into the orbitals ofdifferent atoms takes place according to theaufbau principle which is based on the Pauli’sexclusion principle, the Hund’s rule ofmaximum multiplicity and the relativeenergies of the orbitals. Theaufbausprinciple states : In the ground state of theatoms, the orbitals are filled in order oftheir increasing energies. In other words,electrons first occupy the lowest energy orbitalavailable to them and enter into higher energyorbitals only after the lower energy orbitals arefilled.The number of electrons to be filled in variousorbitals is restricted by the exclusion principle,given by the Austrian scientist Wolfgang Pauli(1926). According to this principle : No twoelectrons in an atom can have the sameset of four quantum numbers. Pauliexclusion principle can also be stated as : “Onlytwo electrons may exist in the same orbitaland these electrons must have oppositespin.” This means that the two electrons canhave the same value of three quantum numbersn, l and $m_l$, but must have the opposite spinquantum number.Hund’s Rule of Maximum Multiplicity rule deals with the filling of electrons into the orbitals belonging to the same subshell. It states : pairing ofelectrons in the orbitals belonging to thesame subshell (p, d or f) does not take placeuntil each orbital belonging to thatsubshell has got one electron each i.e., itis singly occupied.
The distribution of electrons into orbitals of anatom is called its electronic configuration.If one keeps in mind the basic rules whichgovern the filling of different atomic orbitals,the electronic configurations of different atomscan be written very easily.The electronic configuration of differentatoms can be represented in two ways. Forexample :

$s^a p^bd^c$…… notation
Orbital diagram

…at a point gives the probability density of the electron at that point.

$\psi\times2$
$-\psi^2$
$\psi$
$\psi^2$

Only …. electrons may exist in the same orbital and these electrons must have opposite spin.

One
Two
Three
Four

…deals with the filling of electrons into the orbitals belonging to the same subshell.

Hund’s Rule of Maximum Multiplicity rule
Pauli’s exclusion principle
Aufbau principle
Werner Heisenberg

Electrons first occupy the …. energy orbital available to them and enter into … energy orbitals.

Lowest, Higher
Higher, Lowest
Middle, Higher
Higher, Middle

Answer

(d) $\psi^2$

(b) Two

(a) Hund’s Rule of Maximum Multiplicity rule

(a)Lowest, Higher

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Question 24 Marks

Read the passage given below and answer the following questions from (i) to (v).
The presence of positive charge on thenucleus is due to the protons in the nucleus.As established earlier, the charge on the proton is equal but opposite to that of electron.Atomic number $(Z)=$ number of protons inthe nucleus of an atom = number of electrons in a nuetral atom. protons and neutrons present in thenucleus are collectively known as nucleons. The total number of nucleons is termed asmass number $(A)$ of the atom.
mass number $(A)=$ number of protons $(Z)+$ number of neutrons $( n )$.
Isobars are the atoms with same massnumber but different atomic number forexample, ${ }_6^4 C$ and ${ }_7^{14} N$. On the other hand, atomswith identical atomic number but differentatomic mass number are known as Isotopes. For example, considering of hydrogen atom again, $99.985 \%$ of hydrogen atoms contain only one proton.This isotope is called protium $\left(1^1 H \right)$. Rest of thepercentage of hydrogen atom contains two otherisotopes, the one containing 1 proton and 1 neutron is called deuterium ( ${ }^2{ }_1 D , 0.015 \%$ )and the other one possessing 1 proton and 2 neutrons is called tritium ( ${ }^3 T$ )..the studies of interactions of radiations with matter haveprovided immense information regarding thestructure of atoms and molecules. Neils Bohrutilised these results to improve upon themodel proposed by Rutherford. Twodevelopments played a major role in theformulation of Bohr's model of atom. Thesewere:
1. Dual character of the electromagneticradiation which means that radiations possess both wave like and particle likeproperties, and
2. Experimental results regarding atomicspectra.

James Maxwell (1870) was the first to givea comprehensive explanation about theinteraction between the charged bodies andthe behaviour of electrical and magnetic fieldson macroscopic level. He suggested that whenelectrically charged particle moves underaccelaration, alternating electrical and magnetic fields are produced and transmitted.These fields are transmitted in the forms ofwaves called electromagnetic waves orelectromagnetic radiation.radiations are characterised by theproperties, namely, frequency $(v)$ and wavelength $(\lambda)$.The SI unit for frequency $(v)$ is hertz $\left( Hz , s ^{-1}\right)$, after Heinrich Hertz. It is defined asthe number of waves that pass a given pointin one second. Wavelength should have the units of lengthand as you know that the SI units of length ismeter ( m ). Since electromagnetic radiationconsists of different kinds of waves of muchsmaller wavelengths, smaller units are used.In vaccum all types of electromagneticradiations, regardless of wavelength, travel atthe same speed, i.e., $3.0 \times 10^8 m s ^{-1}$ ( $2.997925 \times 10^8 ms^{-1}$, to be precise). This is called speedof light and is given the symbol ' c '. Thefrequency $( V )$, wavelength $(\lambda)$ and velocity of light(c) are related by the following equation.
$c=v \lambda$
The other commonly used quantityspecially in spectroscopy, is the wavenumber.It is defined as the number of wavelengthsper unit length. Its units are reciprocal ofwavelength unit, i.e., $m^{–1}$. However commonlyused unit is $cm^{–1}$

The presence of positive charge on the nucleus is due to the …. in the nucleus.

Protons
Neutrons
Electron
Nucleons

Atomic Number is denoted by:

Atomic Mass number is denoted by:

… are the atoms with same mass number but different atomic number.

Isotopes
Allotropes
Isobars
None of above

Atoms with identical atomic number but different atomic mass number are known as ..

Isotopes
Allotropes
Isobars
None of above

Answer

(a) Protons

(b) Z

(b) Z

(a) Isotopes

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Question 34 Marks

Read the passage given below and answer the following questions from (i) to (v).
The first concreteexplanation for the phenomenon of the blackbody radiation was given byMax Planck in 1900.An ideal body, which emits and absorbs radiations of allfrequencies uniformly, is called a black bodyand the radiation emitted by such a body is called black body radiation. Max Planck arrived at a satisfactory relationshipbymaking an assumption that absorption andemmission of radiation arises from oscillatori.e., atoms in the wall of black body.He suggested that atoms andmolecules could emit or absorb energy onlyin discrete quantities and not in a continuousmanner. He gave the name quantum to thesmallest quantity of energy that can be emitted or absorbed in the form of electromagnetic radiation. The energy (E) of aquantum of radiation is proportionalto its frequency (ν) and is expressed byequation .
$E = hυ.$
The proportionality constant, ‘h’ is knownas Planck’s constant and has the value6.$626\times 10^{–34}$ Js.In 1887, H. Hertz performed a very interestingexperiment in which electrons (or electriccurrent) were ejected when certain metals (forexample potassium, rubidium, caesium etc.)were exposed to a beam of light. The phenomenon is calledPhotoelectric effect. The results observed inthis experiment were:

The electrons are ejected from the metalsurface as soon as the beam of light strikesthe surface, i.e., there is no time lagbetween the striking of light beam and theejection of electrons from the metal surface.
The number of electrons ejected is proportional to the intensity or brightness of light.
For each metal, there is a characteristicminimum frequency,ν0(also known asthreshold frequency) below which photoelectric effect is not observed. At afrequency $ν >ν_0$, the ejected electrons comeout with certain kinetic energy. The kineticenergies of these electrons increase withthe increase of frequency of the light used.

The particle nature of light posed a dilemmafor scientists. Theonly way to resolve the dilemma was to acceptthe idea that light possesses both particle andwave-like properties, i.e., light has dualbehaviour. Depending on the experiment, wefind that light behaves either as a wave or as astream of particles. Whenever radiationinteracts with matter, it displays particle likeproperties in contrast to the wavelike properties (interference and diffraction), whichit exhibits when it propagates. This conceptwas totally alien to the way the scientiststhought about matter and radiation and it tookthem a long time to become convincedof itsvalidity.
The study of emission or absorption spectra is referred to as spectroscopy.The emission spectra of atoms inthe gas phase, on the other hand, do not showa continuous spread of wavelength from redto violet, rather they emit light only at specificwavelengths with dark spaces between them.Such spectra are called line spectra or atomicspectra.The Swedishspectroscopist, Johannes Rydberg, noted that
all series of lines in the hydrogen spectrumcould be described by the following expression:
$\bar{\text{v}}=109,677\big(\frac{1}{\text{n}^2_1}-\frac{1}{\text{n}^2_2}\big)\text{cm}^{-1}$
The value $109,677 cm^{–1}$ is called theRydberg constant for hydrogen. The first fiveseries of lines that correspond to $n_1= 1, 2, 3,4, 5$ are known as Lyman, Balmer, Paschen,Bracket and Pfund series, respectively.Neils Bohr (1913) was the first to explainquantitatively the general features of thestructure of hydrogen atom and its spectrum.He used Planck’s concept of quantisation ofenergy. Though the theory is not the modernquantum mechanics, it can still be used to rationalize many points in the atomic structureand spectra. Bohr’s model for hydrogen atomis based on the following postulates:

The electron in the hydrogen atom canmove around the nucleus in a circular pathof fixed radius and energy. These paths arecalled orbits, stationary states or allowedenergy states. These orbits are arrangedconcentrically around the nucleus.
The energy of an electron in the orbit doesnot change with time. However, theelectron will move from a lower stationarystate to a higher stationary state whenrequired amount of energy is absorbedby the electron or energy is emitted when electron moves from higher stationarystate to lower stationary state. The energychange does not takeplace in a continuous manner.
The frequency of radiation absorbed oremitted when transition occurs between two stationary states that differ in energyby $\triangle\text{E},$ is given by:

$\text{v}=\frac{\triangle\text{E}}{\text{h}}=\frac{\text{E}_2-\text{E}_1}{\text{h}}$

Where E1 and E2 are the energies of the lower and higher allowed energy statesrespectively. This expression is commonly known as Bohr’s frequency rule.

The angular momentum of an electron isquantised. In a given stationary state itcan be expressed as in equation

$\text{m}_{\text{e}}\text{vr}=\text{n}.\frac{\text{h}}{2\pi}\text{n}=1,2,3.....$

The first concrete explanation for the phenomenon of the black body radiation was given by ….in 1900.

Max Planck
De Broglie
Albert Einstein,
Niels Bohr

Which of the following equation is Planck’s equation?

$E= mc^2$
$E = hυ$
$E= hc^2$
$E= vc^2.$

What is nature of light?

Wave
Particle
Wave and Particle
None of above

The value …. is called theRydberg constant for hydrogen.

$109,674cm^{–1}$
$109,675cm^{–1}$
$109,676cm^{–1}$
$109,677cm^{–1}$

…was the first to explain quantitatively the general features of the structure of hydrogen atom and its spectrum.

Max Planck
De Broglie
Albert Einstein,
Niels Bohr

Answer

(a) Max Planck

(b) E = hυ

(d) $109,677cm^{–1}$

(d) Niels Bohr

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Question 44 Marks

The French physicist, de Broglie, in 1924proposed that matter, like radiation, shouldalso exhibit dual behaviour i.e., both particleand wavelike properties. This means that justas the photon has momentum as well aswavelength, electrons should also havemomentum as well as wavelength, de Broglie,from this analogy, gave the following relationbetween wavelength $(\lambda)$ and momentum (p) ofa material particle
$\lambda=\frac{\text{h}}{\text{mv}}=\frac{\text{h}}{\text{p}}$
where m is the mass of the particle, v itsvelocity and p its momentum.
Werner Heisenberg a German physicist in1927, stated uncertainty principle which is theconsequence of dual behaviour of matter andradiation. It states that it is impossible todetermine simultaneously, the exact position and exact momentum (or velocity)of an electron.Mathematically, it can be given as inequation
$\triangle\text{x}\times\triangle\text{p}_{\text{x}}\geq\frac{\text{h}}{4\pi}$
or $\triangle\text{x}\times\triangle(\text{mv}_{\text{x}})\geq\frac{\text{h}}{4\pi}$
or $\triangle\text{x}\times\triangle(\text{v}_{\text{x}})\geq\frac{\text{h}}{4\pi\text{m}}$
where $\triangle\text{x}$ is the uncertainty in position and $\triangle\text{px}(\text{or}\triangle\text{vx})$ is the uncertainty in momentum (orvelocity) of the particle.
One of the important implications of theHeisenberg Uncertainty Principle is that itrules out existence of definite paths ortrajectories of electrons and other similarparticles. The effect of Heisenberg Uncertainty Principle is significant only for motion of microscopic objects and is negligible for that of macroscopic objects. It, therefore, means that theprecise statements of the position andmomentum of electrons have to bereplaced by the statements of probability,that the electron has at a given positionand momentum. This is what happens inthe quantum mechanical model of atom. In Bohr model, anelectron is regarded as a charged particlemovingin well defined circular orbits aboutthe nucleus. The wave character of the electronis not considered in Bohr model. Further, anorbit is a clearly defined path and this pathcan completely be defined only if both theposition and the velocity of the electron areknown exactly at the same time. This is notpossible according to the Heisenberguncertainty principle. Bohr model of thehydrogen atom, therefore, not only ignoresdual behaviour of matter but also contradictsHeisenberg uncertainty principle. The structure of the atom was needed which could account for wave-particle duality of matter and be consistent with Heisenberg uncertainty Principle. This came with the advent of Quantum mechanics. This is mainly becauseof the fact thatclassical mechanics ignores theconcept of dual behaviour of matter especiallyfor sub-atomic particles and the uncertaintyprinciple. The branch of science that takes intoaccount this dual behaviour of matter is calledquantum mechanics.Quantum mechanics is a theoreticalscience that deals with the study of the motionsof the microscopic objects that have bothobservable wave like and particle likeproperties.When Schrödinger equation is solved forhydrogen atom, the solution gives the possibleenergy levels the electron can occupy and thecorresponding wave function(s) $\psi$ of theelectron associated with each energy level. A large number of orbitals are possible in anatom. Qualitatively these orbitals can bedistinguished by their size, shape andorientation. An orbital of smaller size meansthere is more chance of finding the electron nearthe nucleus. Similarly shape and orientationmean that there is more probability of findingthe electron along certain directions thanalong others. Atomic orbitals are preciselydistinguished by what are known as quantumnumbers. Each orbital is designated by threequantum numbers labelled as n, l and $m_1$.
The principal quantum number ‘n’ isa positive integer with value of n = 1,2,3…….The principal quantum number determines thesize and to large extent the energy of theorbital. Azimuthal quantum number. ‘l’ is alsoknown as orbital angular momentum orsubsidiary quantum number. It defines thethree-dimensional shape of the orbital.. For agiven value of n, l can have n values rangingfrom 0 to n – 1, that is, for a given value of n,the possible value of l are : l = 0, 1, 2, ……….(n–1)
Magnetic orbital quantum number. ‘mlgives information about the spatialorientation of the orbital with respect tostandard set of co-ordinate axis. For anysub-shell (defined by ‘l’ value) 2l+1 valuesof ml are possible and these values are givenbuy :ml = – l, – (l –1), – (l–2)… 0,1… (l –2), (l–1)..
In 1925, George Uhlenbeck and SamuelGoudsmit proposed the presence of the fourthquantum number known as the electronspin quantum number (ms). electron has, besides charge and mass,intrinsic spin angular quantum number. Spinangular momentum of the electron — a vectorquantity, can have two orientations relative tothe chosen axis. These two orientations aredistinguished by the spin quantum numbersms which can take the values of $+½ or –½$.These are called the two spin states of theelectron and are normally represented by twoarrows, ↑ (spin up) and ↓ (spin down).the four quantum numbersprovide the following information :

n defines the shell, determines the size ofthe orbital and also to a large extent theenergy of the orbital.
There are n subshells in the n the shell. Lidentifies the subshell and determines the shape of the orbital (see section 2.6.2).There are (2l+1) orbitals of each type in asubshell, that is, one s orbital (l = 0), threep orbitals (l = 1) and five d orbitals (l = 2)per subshell. To some extent l alsodetermines the energy of the orbital in amulti-electron atom.
ml designates the orientation of the orbital.For a given value of l, mlhas (2l+1) values,the same as the number of orbitals persubshell. It means that the number oforbitals is equal to the number of ways inwhich they are oriented.
ms refers to orientation of the spin of the electron.

Uncertainty principle was given by:

Werner Heisenberg
George Uhlenbeck
Samuel Goudsmit
De Broglie

Quantum mechanics is a theoretical science that deals with the study of the motions of the ….. objects.

Macroscopic
Microscopic
Laparoscopic
All the above

The principal quantum number …

…is also known as orbital angular momentum or subsidiary quantum number.

Principal quantum number
Electron spin quantum number
Magnetic orbital quantum number.
Azimuthal quantum number

George Uhlenbeck and Samuel Goudsmit proposed the presence of the fourth quantum number known as the …

Principal quantum number
Electron spin quantum number
Magnetic orbital quantum number.
Azimuthal quantum number

Answer

(a) Werner Heisenberg

(b) Microscopic

(d) Azimuthal quantum number

(b) Electron spin quantum number

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Question 54 Marks

Read the passage given below and answer the following questions from (i) to (vi).
The atomic theory of matter was first proposed on afirm scientific basis by JohnDalton, a British schoolteacher in 1808. His theory, called Dalton’s atomictheory, regarded the atom as the ultimate particle ofmatter Dalton’s atomic theory was able to explainthe law of conservation of mass, law of constantcomposition and law of multiple proportion verysuccessfully. However, it failed to explain the results ofmany experiments.In mid 1850s many scientists mainlyFaraday began to study electrical dischargein partially evacuated tubes, known ascathode ray discharge tubes.Electrical discharge carried out in the modifiedcathode ray tube led to the discovery of canalrays carrying positively charged particles. Thecharacteristics of these positively chargedparticles are listed below.

Unlike cathode rays, mass of positivelycharged particles depends upon thenature of gas present in the cathode raytube. These are simply the positivelycharged gaseous ions.
The charge to mass ratio of the particlesdepends on the gas from which theseoriginate.
Some of the positively charged particlescarry a multiple of the fundamental unitof electrical charge.
The behaviour of these particles in themagnetic or electrical field is opposite tothat observed for electron or cathoderays.

The smallest and lightest positive ion wasobtained from hydrogen and was called
proton. This positively charged particle wascharacterised in 1919. Later, a need was feltfor the presence of electrically neutral particleas one of the constituent of atom. Theseparticles were discovered by Chadwick (1932)by bombarding a thin sheet of beryllium byα-particles. When electrically neutral particleshaving a mass slightly greater than that ofprotons were emitted. He named theseparticles as neutrons.J. J. Thomson, in 1898, proposed that an atom possesses a spherical shape (radiusapproximately 10–10 m) in which the positivecharge is uniformly distributed. The electronsare embedded into it in such a manner as togive the most stable electrostatic arrangementMany different names are given tothis model, for example, plum pudding, raisinpudding or watermelon. This model can be visualised as a pudding or watermelon ofpositive charge with plums or seeds (electrons)embedded into it. An important feature of thismodel is that the mass of the atom is assumed to be uniformly distributed over theatom.Rutherford and his students (Hans Geiger andErnest Marsden) bombarded very thin gold foilwith α–particles. Rutherford’s famous α–particle scattering experiment.The observations of Scattering experiment are as follows-:

most of the α–particles passed throughthe gold foil undeflected.
a small fraction of the α–particles wasdeflected by small angles.
a very few α–particles (∼1 in 20,000)bounced back, that is, were deflected bynearly 180°.

On the basis of observations andconclusions from this experiment, Rutherford proposed the nuclearmodel of atom. According to this model:

The positive charge and most of the massof the atom was densely concentrated inextremely small region. This very smallportion of the atom was called nucleusby Rutherford.
The nucleus is surrounded by electronsthat move around the nucleus with a veryhigh speed in circular paths called orbits.Thus, Rutherford’s model of atomresembles the solar system in which thenucleus plays the role of sun and theelectrons that of revolving planets.
Electrons and the nucleus are held together by electrostatic forces of attraction.

The atomic theory of matter was first proposed on afirm scientific basis by:

John Dalton
Ernest Rutherford
J.Thomson
Henry Moseley

The cathode rays start from … and move towards the….

Anode, Cathode
Centre, Anode
Cathod, Anode
Cathod, Centre

Negativelycharged particles in atoms, called…

Protons
Electrons
Neutron
Positron

The smallest and lightest positive ion wasobtained from …. and was called proton.

Oxygen
Nitrogen
Carbon
Hydrogen

Electrically neutral particles having a mass slightly greater than that of protons, these particles termed as:

Protons
Electrons
Neutron
Positron

J.J. Thomson’s atomic model is also named as:

Plum pudding
Raisin pudding
Watermelon
All the above

Answer

(a) John Dalton

(b) Electrons

(d) Hydrogen

(d) All the above

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Chemistry Case study (4 Marks)

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