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NCERT Exemplar Class 12 Physics Solutions Chapter 12 Atoms

NCERT Exemplar Class 12 Physics Solutions Chapter 12 Atoms

Edited By Safeer PP | Updated on Sep 14, 2022 01:19 PM IST | #CBSE Class 12th

NCERT Exemplar Class 12 Physics solutions chapter 12 deals with the fundamental building blocks of any material that exists in this Universe. The term "atom" comes from the Greek word "atomos," which means indivisible because it was considered that atoms were the smallest things and could not be divided further. NCERT Exemplar Class 12 Physics chapter 12 solutions discuss the atomic particles inside the atom, namely protons, electrons, and neutrons. The nucleus is the central sphere of the atom, which contains the protons (positively charged) and the neutrons (no charge). The atom's outermost region is called an electron shell, which includes the electrons (negatively charged).

As we learn about atoms and their structure, we would find that we start from minute particles and achieve something substantial. Class 12 Physics NCERT Exemplar solutions chapter 12 given on this page would help you understand the concepts from scratch and NCERT is helpful for the 12 boards or competitive exams. NCERT Exemplar Class 12 Physics solutions chapter 12 PDF download is also available for students, for further learning. The topics and subtopics covered are as follows:

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NCERT Exemplar Class 12 Physics Solutions Chapter 12 MCQI


Taking the Bohr radius as a0 = 53pm, the radius of Li++ ion in its ground state, on the basis of Bohr’s model, will be about
A. 53 pm
B. 27 pm
C. 18 pm
D. 13 pm


The answer is the option (c).
According to Bohr’s radius of the orbit, the necessary centripetal force is provided by the electrostatic force of attraction for hydrogen and H2 like atoms. So, the correct option is (c)18 pm as -
i.e.i.e\; \; \; \frac{1}{4\pi \varepsilon _{0}}\frac{\left ( Ze \right )e}{r^{2}}=\frac{mv^{2}}{r} ....(i)\\ Also\; \; \; mvr=\frac{nh}{2\pi} .........(ii)
From equation (i) and (ii) radius of nth orbit
r_{n}=\frac{n^{2}h^{2}}{4 \pi^{2}kZme^{2}}=\frac{n^{2}h^{2}\varepsilon _{0}}{\pi m Ze^{2}}=0.53\frac{n^{2}}{Z}A^{\circ}\; \; \; \; \; \; \left ( k=\frac{1}{4\pi \varepsilon _{0}} \right )\\ \\ \Rightarrow r_{n}\alpha \frac{n^{2}}{Z}\; or\; r_{n}\alpha \frac{1}{Z}\\ r_{n}=a_{0}\frac{n^{2}}{Z}, \; where\; a_{0}=the \; Bohr\; radius=53pm
The atomic number (Z) of lithium is 3
As r_{a}=a_{0}\frac{n^{2}}{Z}
Therefore the radius of Li++ ion in-ground state on the basis of Bohr's model, will be about \frac{1}{3} times to that of Bohr radius.
Therefore the radius of lithium ion is near r= \frac{53}{3}\approx 18 pm


The binding energy of a H-atom, considering an electron moving around a fixed nuclei (proton), is
B=-\frac{me^{2}}{8n^{2}\varepsilon _{0}^{2}h^{2}} . (m = electron mass). If one decides to work in a frame of reference where the electron is at rest, the proton would be moving around it. By similar arguments, the binding energy would be

B=-\frac{me^{2}}{8n^{2}\varepsilon _{0}^{2}h^{2}} (M = proton mass)
This last expression is not correct because
A. n would not be integral.
B. Bohr-quantization applies only to electron
C. the frame in which the electron is at rest is not inertial.
D. the motion of the proton would not be in circular orbits, even approximately.


The answer is the option (c) the frame in which the electron is at rest is not inertial.


The simple Bohr model cannot be directly applied to calculate the energy levels of an atom with many electrons. This is because
A. of the electrons not being subject to a central force.
B. of the electrons colliding with each other
C. of screening effects
D. the force between the nucleus and an electron will no longer be given by Coulomb’s law.


The answer is the option (a).
The above-mentioned Bohr’s model is not relevant for calculating energy levels of atoms containing multiple electrons because of the assumption that the centripetal force is provided by the nucleus’ electrostatic force of attraction and it is also used as an assumption while deriving formulas for radius/energy levels, etc. Therefore, the model is only applicable to single-electron atoms. If applied in multiple electron atoms, another factor, repulsion due to the electrons need to be accounted for. This proves that the simple Bohr’s model cannot be used to calculate the energy level of atoms with numerous electrons.


For the ground state, the electron in the H-atom has an angular momentum =? according to the simple Bohr model. Angular momentum is a vector, and hence there will be infinitely many orbits with the vector pointing in all possible directions. In actuality, this is not true,

A. because Bohr model gives incorrect values of angular momentum.
B. because only one of these would have a minimum energy.
C. angular momentum must be in the direction of spin of electron.
D. because electrons go around only in horizontal orbits.


Option (a) is the correct answer as -
Neil Bohr proposed that the magnitude of electron’s angular momentum is quantized
i.e\; L=mv_{n}r_{n}=n\left ( \frac{h}{2\pi} \right )where\; n=1,2,3.........
each value of n corresponds to a permitted value of the orbit radius.
rn=Radius of nth, vn=corresponding speed
Angular momentum given by the Bohr’s model is a vector quantity, and the model only give the magnitude of the angular momentum. Therefore, angular momentum is not described to a full extent by the Bohr’s model. So, the values given of the angular momentum of revolving electron by the Bohr’s model are not correct.


O2 molecule consists of two oxygen atoms. In the molecule, nuclear force between the nuclei of the two atoms
A. is not important because nuclear forces are short-ranged.
B. is as important as electrostatic force for binding the two atoms.
C. cancels the repulsive electrostatic force between the nuclei.
D. is not important because oxygen nucleus have equal number of neutrons and protons.


The answer is the option (a).
Nuclear forces are responsible for keeping the nucleons bound in the nucleus.
These forces come under short-range forces, and the maximum distance of their existence is from 10~15 m. These are one of the strongest forces present in nature. They bring stability of the nucleus, and they are attractive in nature. They are non-central and does not have any charge dependency.
Due to their strong nature, nuclear forces dominate the force between protons in the nucleus, also known as repulsive Coulomb force. And as the distance increases more than a few femtometres, the nuclear forces reduce to zero between two nucleons.
And in case of O2 molecule made up of two oxygen atom molecules, the nuclear forces are not accounted s they work within the nucleus so (a) is the correct option.


Two H atoms in the ground state collide in-elastically. The maximum amount by which their combined kinetic energy is reduced is
A. 10.20 eV
B. 20.40 eV
C. 13.6 eV
D. 27.2 eV


The answer is the option (a).
Total energy: Total energy (E) is the sum of both potential energy and kinetic energy, I.e. E = K + U
(a) 10.20 eV
The lowest state of the atom, called the ground state is that of the lowest energy the energy of this state (n=1), E1 is -13.6 eV
Let us take two hydrogen atoms in the ground state. When two atoms will collide inelastically, total energy associated with the H atoms –
=2 X (13.6 eV)=27.2 eV
The maximum amount of energy by which their combined kinetic energy is reduced when any one of them goes into a first excited state (n = 2) after the inelastic collision.
Thus, the total energy associated with the two H atoms after the collision is -
=\left ( \frac{13.6}{2^{2}} \right )+(13.6)=17.0 eV
Hence the maximum loss of their combined kinetic energy
=27.2-17.0=10.2 eV
Therefore option (a) 10.20 eV is the right option.


A set of atoms in an excited state decays.
A. in general to any of the states with lower energy.
B. into a lower state only when excited by an external electric field.
C. all together simultaneously into a lower state.
D. to emit photons only when they collide.


A set of atoms in an excited state decays in general to any of the states with lower energy.

NCERT Exemplar Class 12 Physics Solutions Chapter 12 MCQII


An ionized H-molecule consists of an electron and two protons. The protons are separated by a small distance of the order of angstrom. In the ground state,
A. the electron would not move in circular orbits.
B. the energy would be (2)4 times that of an H-atom.
C. the electrons, the orbit would go around the protons.
D. the molecule will soon decay in a proton and an H-atom.


The correct answers are the options (a, c). The electron present in the hydrogen atom revolves in a circular path around a fixed nucleus. This easily explained by the Bohr’s model. But the Bohr’s model cannot explain the protonated structure of hydrogen atom when t contains two protons around the nucleus separated by a small distance in the same order of angstrom. So, in this ground state scenario, the electrons orbit will go around the protons instead of moving in their usual circular path around the nucleus.


Consider aiming a beam of free electrons towards free protons. When they scatter, an electron and a proton cannot combine to produce a H-atom,
A. because of energy conservation.
B. without simultaneously releasing energy in the form of radiation.
C. because of momentum conservation.
D. because of angular momentum conservation.


The correct answers are the options (a, b). Due to the law of conservation of energy, a moving electron and a proton cannot collectively form a hydrogen atom as they will simultaneously release energy in the form of radiation.


The Bohr model for the spectra of a H-atom
A. will not be applicable to hydrogen in the molecular from.
B. will not be applicable as it is for a He-atom.
C. is valid only at room temperature.
D. predicts continuous as well as discrete spectral lines.


The correct answers are the options (a, b). The model of the atom proposed by Neil Bohr was applicable for the hydrogen atom and some other light atoms containing only one electron around the nucleus which is stationary having a positive charge, Ze (called a hydrogen-like atom, e.g., H, He+, Li+2, Na+1 etc). But one of the limitations of the model is that it is not applicable to hydrogen in its molecular form and also in case of He atom.


The Balmer series for the H-atom can be observed
A. if we measure the frequencies of light emitted when an excited atom falls to the ground state.
B. if we measure the frequencies of light emitted due to transitions between excited states and the first excited state.
C. in any transition in a H-atom.
D. as a sequence of frequencies with the higher frequencies getting closely packed.


The correct answers are the options (b, d) is the correct option as,
To observe the Balmer series for H-atom, simply by measuring the frequencies of light that are emitted because of consequences such as the transition between higher excited states and the first excited state and as a sequence of frequencies with the higher frequencies getting closely packed.


Let E_{n}=\frac{-1}{8\varepsilon_{0}^{2}}\frac{me^{2}}{n^{2}h^{2}} be the energy of the nth level of H-atom. If all the H-atoms are in the ground state and radiation of frequency (E2-E1)/h falls on it,
A. it will not be absorbed at all
B. some of atoms will move to the first excited state.
C. all atoms will be excited to the n = 2 state.
D. no atoms will make a transition to the n = 3 state.


The correct answers are the options (b, d). Let us assume E2 and E2 as the energy corresponding to n = 2 and n = 1 respectively. According to the Bohr’s model of atom few of the atoms will reach to the first excited state if the radiation of energy on a sample in which all the hydrogen atoms at ground state is \DeltaE = (E2 – E1) = hf incident. But as the energy is insufficient for the transition to take place from n = 1 to n =3, therefore none of the atoms present will reach up to the n = 3 state.


The simple Bohr model is not applicable to He^{4} atom because
A. He^{4} is an inert gas.
B. He^{4} has neutrons in the nucleus.
C. He^{4} has one more electron.
D. electrons are not subject to central forces.


The correct option of the mentioned problem is (c), (d) as,
The famous model of atomic structure proposed by Neil Bohr is only applicable to light atoms j=having only one electron revolving around the nucleus, and it is mostly used for H-atom. The electron must revolve around a nucleus which stationary and positively charged. It is also applicable for - H, He+, Li+2, Na+1 etc.
Because of having only one electron and electrons are not subjected to centripetal forces; thus, it is also applicable for He4.

NCERT Exemplar Class 12 Physics Solutions Chapter 12 Very Short Answer


The mass of an H-atom is less than the sum of the masses of a proton and electron. Why is this?


Mass defect: It is found in that the mass of the nucleus is always less than the masses of its constituent nucleons in a free state. This difference in masses is called the mass defect. Hence mass defect
\Deltam=Sum of masses of nucleons-mass of nucleus
={Zmp+(A-Z)mn} M={Zmp+Zme+(A-Z)mn}-M'
Where mp=Mass of Proton, mn=Mass of each neutron, me= Mass of each electron
M=Mass of nucleus, Z= Atomic number, A=Mass member,M'=Mass of atom as a whole
Binding Energy: The neutron and protons in a stable nucleus are held together by nuclear forces and energy is needed to pull them indefinitely apart (or the same energy is released during the formation of the nucleus). This energy is called the binding energy of the nucleus.
The binding of the nucleus can also be described as the energy equivalent tot eh mass defect of the nucleus.


Imagine removing one electron from He4 and He3. Their energy levels, as worked out on the basis of the Bohr model, will be very close. Explain why.


One of the biggest limitations of the Bohr’s model of the atom is that it is only applicable for either light atoms having one electron revolving around the nucleus or for H-atoms. The nucleus of these atoms must be positively charged and stationery, such atoms are termed as Ze (called hydrogen-like atom). But if one electron is removed from both He4 and He3, they will fulfil the condition, and the model can be used to find their energy levels.


When an electron falls from higher energy to a lower energy level, the difference in the energies appears in the form of electromagnetic radiation. Why cannot it be emitted as other forms of energy?


Electrons are negatively charged particles, and when they undergo a transition from higher energy level to lower energy level, they get accelerated. Thus being accelerated particles, they emit energy in the form of electromagnetic radiations.


Would the Bohr formula for the H-atom remain unchanged if proton had a charge (+4/3) e and electron a charge (−3/4) e, where e = 1.6 × 10–19C. Give reasons for your answer.


Neil Bohr proposed that if an electron is moving around a stationary nucleus, the require centripetal force is provided by the electrostatic force of attraction.
i.e \frac{1}{4 \pi \epsilon _{0}}\frac{q_{1}q_{2}}{r^{2}}=\frac{mv^{2}}{r} ...............(i)
Therefore the magnitude of electrostatic force F\propto q_{1}\times q_{2}. The Bohr’s formula for the H-atom remains unchanged if the charge on the proton is (+4/3)e whereas the charge carried by the electron is (-3/4)e. This is possible as the Bohr’s formula requires the product of the charges which remain constant for given values of charges.


Consider two different hydrogen atoms. The electron in each atom is in an excited state. Is it possible for the electrons to have different energies but the same orbital angular momentum according to the Bohr model


According to Bohr model electrons having different energies belong to different levels having different values of n. So, their angular momenta will be different, as
L=\frac{nh}{2\pi}\;or\;L\infty n

NCERT Exemplar Class 12 Physics Solutions Chapter 12 Short Answer


Positronium is just like an H-atom with the proton replaced by the positively charged anti-particle of the electron (called the positron which is as massive as the electron). What would be the ground state energy of positronium?


The total energy of the electron in the stationary states of the hydrogen atom is given by
E_{n}=-\left \{ \frac{\mu}{2h^{2}}\left (\frac{e^[2]}{4\pi\epsilon _{0}} \right )^{2} \right \}\frac{1}{n^{2}}
\mu that occurs in the Bohr formula is the reduced mass of electron and proton For hydrogen atom :
\mu=\frac{m_{e}m_{p}}{m_{e}+m_{p}}=\frac{m_{e}m_{p}}{m_{p}}=m_{e}(As m_{p}>>m_{e})
me mp are the mass of electron and proton which are the same for positronium:
(As mass of positron is equal to the mass of electron)
E_{n}=\frac{1}{2}\left \{ \frac{m_{e}}{2h^{2}\left ( \frac{e^{2}}{4 \pi \varepsilon _{0}} \right )^{2}} \right \}\frac{1}{n^{2}}-\frac{13.6eV}{2}\frac{1}{n^{2}}=\frac{-6.8eV}{n^{2}}
The energy is half of the hydrogen level.
the lowest energy level of positronium (n=1)is -6.8 electron volts(eV).
The next highest energy level (n=2) is 1.7 eV the negative sign implies a bound state


Assume that there is no repulsive force between the electrons in an atom, but the force between positive and negative charges is given by Coulomb’s law as usual. Under such circumstances, calculate the ground state energy of a He-atom.


For a he-nucleus z=2 and for ground n=1
Thus ground state energy of a He atom
Thus the ground state will have two electrons each of energy R and the total ground state energy would be -(4 X 13.6)eV=54.4eV


Using Bohr model, calculate the electric current created by the electron when the H-atom is in the ground state.


The equivalent electric current due to rotation of charge is given by
i=\frac{Q}{T}=Q \times f, where\; f\; is\; the\; frequency
In hydrogen atom an electron in ground state revolves on a circular orbit whose radius is equal to the Bohr radius. Let the velocity of the electron is v
Number of revolution per unit time
f=\frac{2\pi a_{0}}{v}
The electric current is given by
i=e\left ( \frac{2 \pi a_{0}}{v} \right )=\frac{2 \pi a_{0}}{v}e


Show that the first few frequencies of light that is emitted when electrons fall to the nth level from levels higher than n, are approximate harmonics (i.e. in the ratio 1: 2: 3...) when n >>1


Frequency of emitted radiation:
\Delta E=hf\Rightarrow f=\frac{\Delta E}{h}=\frac{E_{2}-E_{1}}{h}=RcZ^{2}\left ( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right )
Wavenumber is the number of waves in unit length
\bar{v}=\frac{1}{\lambda}=\frac{f}{c}\Rightarrow \frac{1}{\lambda}=RZ^{2}\left ( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right )=\frac{13.6Z^{2}}{hc}\left ( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right )
The number of spectral lines: If an electron jumps from higher energy orbit to lower energy orbit, it emits radiation with various spectral line.
If electron falls from n2 to n1, then the number of spectral lines emitted is given by
N_{E}=\frac{\left ( n_{2}-n_{1}+1 \right )\left ( n_{2}-n_{1} \right )}{2}
If electron falls from nth orbit to ground state (i.e n2=n and n1=1), then number of spectral lines emitted N_{E}=\frac{\left ( n(n-1) \right )}{2}
The frequency of any line in a series in the spectrum of hydrogen-like atoms corresponding to the transition of an electron from (n + p) level to the nth level can be expressed as a difference of two terms.
f_{min}=cRZ^{2}\left [ \frac{1}{\left ( n+p \right )^{2}}-\frac{1}{n^{2}} \right ]
where m=n+p,(p=1,2,3,.....) and R is Rydberg constant for p<<n
f_{min}=cRZ^{2}\left [ \frac{1}{n^{2}}\left ( 1+\frac{p}{n} \right )^{-2}-\frac{1}{n^{2}} \right ]\\ f_{min}=cRZ^{2}\left [ \frac{1}{n^{2}}-\frac{2p}{n^{3}}-\frac{1}{n^{2}}\right]\\ \left [ by \; Binomial\; theorem \left ( 1+x \right )^{2}=1+nx\; \; If |x|<1 \right ]\\ f_{min}=cRZ^{2}\frac{2p}{n^{3}}=\left ( \frac{2cRZ^{2}}{n^{3}} \right )p
Hence, the first few frequencies of light that are emitted when the electrons fall to the nth level from levels higher than n, are approximately harmonic (i.e., in ratio 1:2:3 …) when n>> 1.


What is the minimum energy that must be given to an H atom in the ground state so that it can emit an H line in Balmer series. If the angular momentum of the system is conserved, what would be the angular momentum of such H photon?


“H-alpha” that corresponds to transition 3 -> 2. And “H-beta” corresponds to 4 -> 2 and the “H-gamma” corresponds to transition 5 -> 2.
Where H is the element hydrogen, four of the Balmer lines are in the technically visible part of the spectrum, with wavelengths larger than 400nm and shorter than 700 nm.
From the above discussion, it is clear that H-gamma in Balmer series corresponds to transition n = 5 to n = 2. Initially, the electron in the ground state must be placed in state n = 5.
Energy required for the transition from n=1 to n=5 is given by
=\left ( \frac{-13.6}{5^{2}} \right )-\left ( \frac{-13.6}{1^{2}} \right )=13.06 eV
As angular momentum should be conserved hence angular momentum corresponding to Hy photon =change in angular momentum of electron
=5\left ( \frac{5}{2\pi} \right )-2\left ( \frac{h}{2\pi} \right )=\frac{3h}{2\pi}\\ =\frac{3 \times 6.6 \times 10^{-34}}{2 \times 3.14}=3.17 \times 10^{-34}Js

NCERT Exemplar Class 12 Physics Solutions Chapter 12 Long Answer


The first four spectral lines in the Lyman series of a H-atom are \lambda = 1218 Å, 1028Å, 974.3 Å and 951.4Å. If instead of hydrogen, we consider deuterium, calculate the shift in the wavelength of these lines.


Let us take \mu H as the reduced masses of electron of hydrogen and \mu D as the reduced masses of electron of deuterium.
we know that
\frac{1}{\lambda}=R\left [ \frac{1}{n_{f}^{2}} -\frac{1}{n_{i}^{2}}\right ]
As ni and nf are fixed for by mass series for hydrogen and deuterium
\lambda \propto \frac{1}{R}or\; \frac{\lambda_{D}}{\lambda_{H}}=\frac{R_{H}}{R_{D}}........................(i)\\ R_{R}=\frac{m_{e}e^{4}}{8\varepsilon _{0}ch^{3}}=\frac{\mu_{H}e^{4}}{8\varepsilon _{0}ch^{3}}\\ R_{D}=\frac{m_{e}e^{4}}{8\varepsilon _{0}ch^{3}}=\frac{\mu_{D}e^{4}}{8\varepsilon _{0}ch^{3}}\\ \therefore \frac{R_{H}}{R_{D}}=\frac{\mu_{H}}{\mu_{D}}.................(ii)\\
From equation (i) and (ii)
Reduced mass for hydrogen,
\mu_{H}=\frac{m_{e}}{1+m_{e}1M}\simeq m_{e}\left ( 1-\frac{m_{e}}{M} \right )
Reduced mass for deuterium,
\mu_{D}=\frac{2M.m_{e}}{2M\left (1+\frac{m_{e}}{2M} \right )}\simeq m_{e}\left ( 1-\frac{m_{e}}{2M} \right )
where M is mass of the proton
\frac{\mu_{H}}{\mu_{D}}=\frac{m_{e}\left (1- \frac{m_{e}}{2M} \right )}{m_{e}\left (1- \frac{m_{e}}{2M} \right )}=\left ( 1-\frac{m_{e}}{M} \right )\left ( 1-\frac{m_{e}}{2M} \right )^{-1}\\ =\left ( 1-\frac{m_{e}}{M} \right )\left ( 1+\frac{m_{e}}{2M} \right )\\ \Rightarrow \frac{\mu_{H}}{\mu_{D}}=\left ( 1-\frac{m_{e}}{2M} \right )\\ or \;\;\frac{\mu_{H}}{\mu_{D}}=\left ( 1-\frac{1}{2 \times 1840} \right )=0.99973\lambda_{H}...........(iv)\\ \left ( \because M=1840 m_{e} \right )
From (iii) and (iv)
\frac{\lambda_{D}}{\lambda_{H}}=0.99973, \;\; \lambda_{D}=0.99973 \lambda_{H}\\\\ using \; \lambda_{H}=1218 \AA ,1028\AA ,974.3\AA \; and\;951.4\AA \; we\;get \\\\ \lambda_{D}=1217.7\AA ,1027\AA ,974.04\AA ,951.1\AA \\\\ Shift\;in\;wavelength\left ( \lambda_{H}-\lambda_{D} \right )\approx \approx 0.3\AA


Deuterium was discovered in 1932 by Harold Urey by measuring the small change in wavelength for a particular transition in 1H and 2H. This is because the wavelength of transition depends to a certain extent on the nuclear mass. If the nuclear motion is taken into account, then the electrons and nucleus revolve around their common centre of mass. Such a system is equivalent to a single particle with a reduced mass , revolving around the nucleus at a distance equal to the electron-nucleus separation. Here = meM/ (me + M) where M is the nuclear mass and me is the electronic mass. Estimate the percentage difference in wavelength for the 1st line of the Lyman series in 1H and 2H. (Mass of 1H nucleus is 1.6725 × 10–27 kg, the mass of 2H nucleus is 3.3374 × 10–27 kg, the mass of electron = 9.109 × 10-31 kg.)


According to Bohr’s theory for a hydrogen-like atom of atomic number Z, the total energy of the electron in the nth state is
E_{n}=-\left \{ \frac{\mu}{2h^{2}}\left (\frac{e^{2}}{4\pi \epsilon _{0}} \right )^{2} \right \}\frac{Z^{2}}{n^{2}}=\frac{\mu Z^{2}e^{4}}{8 \epsilon_{0}^{2}h^{2} }\left ( \frac{1}{n^{2}} \right )
Where signs are usual, and the u that occurs in the Bohr’s formula is the reduced mass of electron and proton.
For hydrogen atom:
\mu_{H}=\frac{m_{e}M_{H}}{\left ( m_{e}+M_{H} \right )}
For Deutrium
:\mu_{D}=\frac{m_{e}M_{D}}{\left ( m_{e}+M_{D} \right )}
Let \mu_{H} be the reduced mass of hydrogen and \mu_{D} that of Deuetrium. Then the frequency of the 1st Lyman line in hydrogen is
hf_{H}=\frac{\mu_{H}e^{4}}{8\epsilon _{0}^{2}h^{2}}\left ( 1-\frac{1}{4} \right )=\frac{\mu_{H}e^{4}}{8\epsilon _{0}^{2}h^{2}}\times \frac{3}{4}
Thus, the wavelength of the transition is
\lambda_{H}=\frac{3}{4}\frac{\mu_{H}e^{4}}{8\epsilon _{0}^{2}h^{2}c}
The wavelength of the transition for the same line in detutrium is
\lambda_{D}=\frac{3}{4}\frac{\mu_{D}e^{4}}{8\epsilon _{0}^{2}h^{2}c}
\therefore Difference in wavelength \Delta \lambda=\lambda_{D}-\lambda_{H}
Hence the percentage difference is
100 \times \frac{\Delta \lambda}{\lambda_{H}}=\frac{\lambda_{D}-\lambda_{H}}{\lambda_{H}}\times100=\frac{\mu_{D-\mu_{H}}}{\mu_{H}}\times 100\\\\ =\frac{\frac{m_{e}M_{D}}{\left ( m_{e}+M_{D} \right )}-\frac{m_{e}M_{H}}{\left ( m_{e}+M_{H} \right )}}{\frac{m_{e}M_{H}}{\left ( m_{e}+M_{H} \right )}}\times 100\\\\ =\left [ \left ( \frac{m_{e}+M_{H}}{m_{e}+M_{D}} \right )\frac{M_{D}}{M_{H}}-1 \right ]\times 100
Since me<<MH<<MD
\frac{\Delta \lambda}{\lambda_{H}}\times=\left [ \frac{M_{H}}{M_{D}}\times \frac{M_{D}}{M_{H}}\left ( \frac{1+\frac{m_{e}}{M_{H}}}{1+\frac{m_{e}}{M_{D}}} \right )-1 \right ]\times 100\\\\ =\left [ \left (1+\frac{m_{e}}{M_{H}} \right )\left (1+\frac{m_{e}}{M_{D}} \right )^{-1}-1 \right ]\times 100\\\\ =\left [ \left (1+\frac{m_{e}}{M_{H}} \right )\left (1+\frac{m_{e}}{M_{D}} \right )-1 \right ]\times 100\\\\ (By \; binomial\;theorem\; (1+x)^{n}=1+nx \; is \; |x|<1)\\\\ \frac{\Delta \lambda }{\lambda _{H}}\times 100=\left [ 1+\frac{m_{e}}{M_{H}}-\frac{m_{e}}{M_{D}}-\frac{(m_{e})^{2}}{M_{H}M_{D}}-1 \right ]\times 100\\\\ Neglecting \;\frac{(m_{e})^{2}}{M_{H}M_{D}}\;as \; it \; is\; very\; small


If a proton had a radius R and the charge was uniformly distributed, calculate using Bohr theory, the ground state energy of a H-atom when (i) R = 0.1Å, and (ii) R = 10 Å.


The electron present in H-atom revolves around the point size proton of a defined radius rB (Bohr’s radius) in a circular path when the atom is in the ground state.
As\;mvr_{B}=h\; and\; \frac{Mv^{2}}{r_{B}}=\frac{-1 \times e \times e}{4 \pi \varepsilon_{0}r_{B}^{2} }\\ \frac{m}{r_{B}}\left ( \frac{-h^{2}}{m^{2}r_{B}^{2}} \right )=\frac{e^{2}}{4 \pi \epsilon_{0}r_{B}^{2}}\\ r_{B}=\frac{4 \pi \epsilon_{0}}{e^{2}}\frac{h^{2}}{m}=0.53\AA\\ K.E=\frac{1}{2}mv^{2}=\left (\frac{m}{2} \right )\left (\frac{h}{mr_{B}} \right )^{2}\\ =\frac{h^{2}}{2mr_{B}^{2}}=13.6eV
total enrgy of the electron i.e
(i)when r=0.1Å:R<rB (as rB =0.51 Å )and the ground state energy is the same as obtained earlier for point size proton i.e -13.6 eV
(ii)when r=10Å:R>>rB the electrons moves inside the proton (assumed to be a sphere of radius R ) with new Bohr's radius r'B
r'_{B}=\frac{4 \pi \epsilon_{0}}{m(e)(e')}
[Replacing e2 by (e)(e') where e' is the charge on the sphere of radius r'B ]

e'=\left [\frac{e}{(4 \pi/3)R^{3}} \right ]\left [ \left ( \frac{4\pi}{3} \right ) {r_{B}^{3}}'\right ]=\frac{{r_{B}^{3}}'}{R^{3}}\\\\ {r_{B}^{3}}'=\frac{4 \pi \epsilon_{0}h^{2}}{m_{e}\left ( e{r_{B}^{3}}'R^{3} \right )}=\left (\frac{4 \pi \epsilon_{0}h^{2}}{m_{e}^{2}} \right )\left ( \frac{R^{3}}{{r_{B}^{3}}'} \right )\\\\ or\;\;{r_{B}^{3}}'= \left (\frac{4 \pi \epsilon_{0}h^{2} }{me^{2}} \right )R^{3}\\\\ =(0.51\AA)(10\AA)=510\AA
r'B =4.8Å which is less than R(=10Å)
KE of the electron
K'=\frac{1}{2}mv^{2}=\left ( \frac{m}{2} \right )\left ( \frac{h^{2}}{m^{2}{r_{B}^{2}}'} \right )=\frac{h}{m{r_{B}^{2}}'}\\ =\left (\frac{h^{2}}{m{r_{B}^{2}}'} \right )\left (\frac{ r_{B}}{{r_{B}}'} \right )=13.6eV\left ( \frac{0.51\AA}{4.8\AA} \right )=0.16eV
The potential at a point inside the change proton,
V=\frac{k_{e}}{R}\left (3-\frac{{r_{B}^{2}}'}{R^{2}} \right )=k_{e}e\left ( \frac{3R^{2}-{r_{B}^{2}}'}{R^{3}} \right )
Potential energy of electron and proton u=-eV
=-e(k_{e}e)\left [ \frac{3R^{2}-{r_{B}^{2}}'}{R^{3}} \right ]\\ =-\left ( \frac{e^{2}}{4 \pi \varepsilon _{0}r_{B}} \right )\left [ \frac{r_{B} \left ( 3R^{2}-{r_{B}^{2}}' \right ) }{R^{3}} \right ]\\ =-\left ( 24.2eV \right )\left [ \frac{(0.51\AA)(300\AA-23.03\AA)}{1000\AA} \right ]\\\\ =3.83eV
Total energy of the electron


In the Auger process, an atom makes a transition to a lower state without emitting a photon. The excess energy is transferred to an outer electron which may be ejected by the atom. (This is called an Auger electron). Assuming the nucleus to be massive, calculate the kinetic energy of an n = 4 Auger electron emitted by Chromium by absorbing the energy from a n = 2 to n = 1 transition.


Auger Effect: While responding to the downward transition by another electron in the atom, the characteristic energies of the atoms are ejected, and this process is called Auger effect. The bombardment with the high energy electrons results in the formation of a vacancy in Auger spectroscopy. But the effect is possible when other interaction leads to vacancy formation. It is an electron arrange method after the nucleus captures the electron.
During removal of electron shell from the atom, a simultaneous response by a higher-level electron will result in a downward transition and fill the vacancy. Multiple times there is a release of photons along with the vacancy filling, the energy of these photons is equal to the energy gap between the upper and lower level. This is also known as x-ray fluorescence due to tot eh presence of all these heavy atoms in the x-ray region. But the same emission process for lighter atoms or outer electrons forms a line spectrum.
But in most cases, a higher-level electron fills the vacancy and emits a photon.
Because the nucleus is large and thus the momentum of the atom is neglected, and the complete transitional energy is transferred to the Auger electron. The energy states represented by the Bohr’s model is similar to the states of Cr due to the presence of a single valance electron in its atom.
The energy of the nth state E_{n}=-Rch\frac{Z^{2}}{n^{2}}=-13.6\frac{Z^{2}}{n^{2}}eV where R is the Rydberg constant and Z=24

The energy released in a tramsition from 2 to 1 is
\Delta E=13.6Z^{2}\left ( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right )=13.6Z^{2}\left ( \frac{1}{1^{2}}-\frac{1}{2^{2}} \right )
=13.6Z^{2}\left ( 1-\frac{1}{4} \right )=13.6Z^{2}\times \frac{3}{4}
The energy required of the Auger electron is
KE=13.6Z^{2}\left ( \frac{3}{4}-\frac{1}{16} \right )=5385.6eV


The inverse square law in electrostatics is \left | F \right |=\frac{e^{2}}{\left (4 \pi \epsilon _{0} \right )r^{2}} for the force between an electron and a proton. The \frac{1}{r} dependence of |F| can be understood in quantum theory as being due to the fact that the ‘particle’ of light (photon) is massless. If photons had a mass mp, force would be modified to \left | F \right |=\frac{e^{2}}{\left (4 \pi \epsilon _{0} \right )r^{2}}\left [ \frac{1}{r^{2}} +\frac{\lambda}{r}\right ].\exp \left ( -\lambda r \right ) where \lambda=m_{p}c/h and h=\frac{h}{2\pi} .

Estimate the change in the ground state energy of a H-atom if mp were 10–6 times the mass of an electron.


We are given
\lambda=\frac{m_{p}c}{h}=\frac{m_{p}c^{2}}{hc}=\frac{(10^{2}m_{e})c^{2}}{hc}\\ =\frac{10^{-6}[0.51][1.6 \times 10^{-13}J]3\times 10^{8}ms^{-1}}{(1.05 \times 10^{-34}Js)(3 \times 10^{8}ms^{-1})}\\ =0.26\times 10^{7}m^{-1},\left [ \because m_{e}C^{2}=0.51MeV \right ] \\r_{B}(\ Bohrs\ radius=0.51\AA=0.51 \times 10^{-10}m)\; \\or \lambda r_{B}=(0.26 \times 10^{7}m^{-1})(0.51 \times 10^{-10}m) =0.14 \times 10^{-13}<<1
Further\; as\; \left | F \right |=\left ( \frac{e^{2}}{4\pi \epsilon _{0}} \right )\left [ \frac{1}{r^{2}+\frac{\lambda}{r}} \right ]e^{-\lambda}r............(i)\\ and \; \left |F \right |=\frac{dU}{dr}\\ U_{r}=\int \left | F \right |dr=\left ( \frac{e^{2}}{4\pi \epsilon _{0}} \right )\int \left ( \frac{\lambda e^{-\lambda r}}{r}+\frac{ e^{-\lambda r}}{r^{2}} \right )dr \\ If \; z=\frac{ e^{-\lambda r}}{r}=\frac{1}{r}(e^{-\lambda r})\\ \frac{dz}{dr}=\left [\frac{1}{r}(e^{-\lambda r})(-\lambda)+(e^{-\lambda r})\left ( \frac{1}{r^{2}} \right ) \right ]\\ or\; dz=-\left [\left ( \frac{\lambda e^{-\lambda r}}{r}+\frac{ e^{-\lambda r}}{r^{2}} \right ) \right ]dr
Thus\;\int \left ( \frac{\lambda e^{-\lambda r}}{r}+\frac{ e^{-\lambda r}}{r^{2}} \right )dr\Rightarrow -\int dz=-z=-\frac{e^{\lambda r}}{r}\\ =-\left ( \frac{e^{2}}{4\pi \epsilon _{0}} \right )\left ( \frac{e^{\lambda r}}{r} \right ).......(ii)
we know that
mvr=h\Rightarrow v=\frac{h}{mr}; \; and \\ \frac{mv^{2}}{r}=F=\left ( \frac{e^{2}}{4 \pi \epsilon_{0} } \right )\left ( \frac{1}{r^{2}}+\frac{\lambda}{r} \right )\\\left [ putting\; e^{-\lambda r}\approx 1 \;in \; eqn.(i) \right ]\\ Thus \; \left ( \frac{m}{r} \right )\left (\frac{h^{2}}{4\pi \epsilon_{0}} \right )=\left ( \frac{e^{2}}{4 \pi \epsilon_{0} } \right )\left ( \frac{1}{r^{2}}+\frac{\lambda}{r} \right )\\ or\;\frac{h^{3}}{mr^{3}}=\left ( \frac{e^{2}}{4 \pi \epsilon_{0} } \right )\left ( \frac{r+\lambda r^{2}}{r^{3}} \right )\\ or\;\frac{h^{3}}{mr^{3}}=\left ( \frac{e^{2}}{4 \pi \epsilon_{0} } \right )\left ( r+ \lambda r^{2} \right )...............(iii)\\ when \; \lambda=0, r={r_{B}}'\; and \\ \frac{h^{3}}{mr^{3}}=\left ( \frac{e^{2}}{4 \pi \epsilon_{0} } \right )r_{B}......................(iv)
from eq.(iii) and (iv)
r_{B}+r+\lambda r^{2}\\ let\; r=r+{B}+\delta \; so\;that\;from(iii)\\ r_{B}=(r_{B}+\delta)+\lambda(r_{B}^{2}+\delta^{2}+2\delta r_{B})\\ or 0=\lambda r_{B}^{2}+\delta(1+2 \lambda r_{B})\;(neglectinf \delta^{2})\\
or\; \delta =\frac{\lambda r_{B}^{2}}{(1+2 \lambda r_{B})}=(-\lambda r_{B}^{2})(1+\lambda r_{B}^{2})^{-1}\\ =(-\lambda r_{B}^{2})(1+\lambda r_{B}^{2})=\lambda r_{B}^{2} \;\;\left ( \because \lambda r_{B}^{2}<<1 \right )\\ from \;eq.(ii)\\ \mu_{i}=- \left ( \frac{e^{2}}{4 \pi \epsilon_{0} } \right )\frac{e^{-\lambda (r_{B}+\delta)}}{(r_{B}+\delta)}\\ =- \left ( \frac{e^{2}}{4 \pi \epsilon_{0} } \frac{1}{r_{B}}\right )\left ( 1-\frac{\delta}{r_{B}} \right )\left ( 1- \lambda r_{B} \right )=-\frac{e^{2}}{4 \pi \epsilon_{0}r_{B} }\\ =-24.2eV
\left [\because e^{-\lambda (r_{B}+\delta)}\approx 1-\lambda(r_{B}-\delta) =1-\lambda r_{B} - \lambda \delta \approx 1- \lambda e_{B}\right ]\\ and \frac{1}{(r_{B}+\delta)}=\frac{1}{r_{B}(1+\delta/r_{B})}=\frac{1}{r_{B}}\left ( 1+\frac{\delta}{r_{B}} \right )^{-1}\\ =\frac{1}{r_{B}}\left ( 1-\frac{\delta}{r_{B}} \right )
Further KE of the electron
K=\frac{1}{2}mv^{2}=\frac{1}{2}m\left ( \frac{h^{2}}{m^{2}r^{2}} \right )\\ =\frac{h^{2}}{2mr^{2}}=\frac{h^{2}}{2m(r_{B}+\delta)^{2}}=\frac{h^{2}}{2mr_{B}^{2}+(1+\delta/r_{B})^{2}}\\ =\left ( \frac{h^{2}}{2mr_{B}^{2}} \right )\left ( 1+\frac{\delta}{r_{B}} \right )^{-2}=\left ( \frac{h^{2}}{2mr^{2}B} \right )\left ( 1-\frac{2\delta}{r_{B}} \right )\\ =(13.6)(1+2\lambda r_{B})eV \left ( as\;\frac{h^{2}}{2mr_{B}}=13.6 eV \; and \; \delta=-\lambda r_{B}^{2} \right )
The total energy of H-atom in the ground state=finalbenergy - initial energy
=(-13.6+27.2 \lambda r_{B})eV-(-13.6 eV)\\ =(27.2 \lambda r_{B})eV


The Bohr model for the H-atom relies on the Coulomb’s law of electrostatics. Coulomb’s law has not directly been verified for very short distances of the order of angstroms. Supposing Coulomb’s law between two opposite charge +q1, –q2 is modified to

\left | F \right |=\frac{q_{1}q_{2}}{\left (4\pi \epsilon_{0} \right )}\frac{1}{r^{2}},r\geq R_{0}\\ =\frac{q_{1}q_{2}}{\left (4\pi \epsilon_{0} \right )}\frac{1}{R_{0}^{2}} \left ( \frac{R_{0}}{r} \right ),r\geq R_{0}\\
Calculate in such a case, the ground state energy of an H-atom, if \epsilon = 0.1, R_{0} = 1\AA


Let the case consider the case when r\leq R_{0}=1 \AA
Let \epsilon =2+\delta
F=\frac{q_{1}q_{2}}{4\pi \epsilon_{0} }.\frac{R_{0}^{\delta}}{r^{2+\delta}}= \frac{xR_{0}^{\delta}}{r^{2+\delta}}
where F=\frac{q_{1}q_{2}}{4\pi \epsilon_{0} }=x=\left ( 1.6 \times 10^{-19} \right )^{2}\times 9 \times 10^{9}=2.04 \times 10^{-29}Nm^{2}
The electrostatic force of attraction between the positively charged nucleus and negatively charged electrons provides the necessary centripetal force.
\frac{mv^{2}}{r}=\frac{xR_{0}^{\delta}}{r^{2+\delta}}=\; or\; v^{2}=\frac{xR_{0}^{\delta}}{mr^{1+\delta}}...........(i)\\ mvr=nh \Rightarrow r=\frac{nh}{mv}=\frac{nh}{m}\left [ \frac{m}{xR_{0}^{\delta}} \right ]^{1/2}r^{(1+\delta)/2}\\ (a\\lying \; Bohr \; second \;postulates)\\ Solving\;this\;fir\;r,we\;ger\; r_{n}=\left ( \frac{n^{2}h^{2}}{xR_{0}^{\delta}} \right )^{\frac{1}{1-\delta}}\\
Where rn is the redius of nth orbit of electron for n=1 and substituting the values of constant we get
r_{1}=\left [ \frac{h^{2}}{mxR_{0}^{2}} \right ]^{\frac{1}{1-\delta}}\\ \Rightarrow r_{1}=\left [ \frac{1.05^{2} \times 10^{-68}}{9.1 \times 10^{-31} \times 2.3 times 10^{-28}times 10^{+19}} \right ]^{\frac{1}{29}}\\ =8times 10^{-11}=0.08 nm(<0.1 nm)
This is the radius of orbit of electron in ground state of hydrogen atom again using bohr second postulate the speed of electron
v_{n}=\frac{nh}{mr_{n}}=nh\left ( \frac{mxR_{0}^{\delta}}{n^{2}h^{2}} \right )^{\frac{1}{1-\delta}}
For n=1 the speed of electron in ground state v_{1}=\frac{h}{mr_{1}}=1.44 \times 10^{6}m/s
The kinetic energy of electron in ground state
KE=\frac{1}{2}mv_{1}^{2} \times -^{-19}J= eV
Potential energy of electron in ground state till R0
U=\int_{0}^{R_{0}}F dr =\int_{0}^{R_{0}}\frac{x}{r^{2}}dr=\frac{x}{R_{0}}
Potential energy from R0 to r, U=\int_{0}^{R_{0}}F dr =\int_{0}^{R_{0}}\frac{xR_{0}^{\delta}}{r^{2+\delta}}dr \\ U=+xR_{0}^{\delta}\int _{R_{0}}^{r} \frac{dr}{r^{2+\delta}}=+\frac{xR_{0}^{\delta}}{-1-\delta}\left [ \frac{1}{r^{1+\delta}} \right ]_{R_{0}}^{r}\\ U=\frac{xR_{0}^{\delta}}{1+\delta}\left [ \frac{1}{r^{1+\delta}}-\frac{1}{R_{0}^{1+\delta}} \right ]=\frac{x}{1+\delta}\left [ \frac{R_{0}^{\delta}}{r^{1+\delta}}-\frac{1}{R_{0}} \right ]\\ U=-x\left [ \frac{R_{0}^{\delta}}{r^{1+\delta}}-\frac{1}{R_{0}} +\frac{1+\delta}{R_{0}}\right ]\\ u=-x\left [ \frac{R_{0}^{-19}}{r^{-0.9}}-\frac{1.9}{R_{0}} \right ]\\ =\frac{2.3}{0.9}\times^{-18}\left [ (0.8)^{0.9}-1.9 \right ]J=-17.3eV\\ Hence\;total \; energy\; of\; electron\; in\; ground\; state=\left ( -17.3+5.9 \right )=-11.4eV

Main Subtopics Covered in NCERT Exemplar Class 12 Physics Solutions Chapter 12 Atoms

  • Introduction
  • Alpha-Particle Scattering And Rutherford's Nuclear Model Of Atom
  • Alpha-particle trajectory
  • Electron orbits
  • Atomic Spectra
  • Spectral series
  • Bohr Model Of The Hydrogen Atom
  • Energy levels
  • The Line Spectra Of The Hydrogen Atom
  • De Broglie's Explanation Of Bohr's Second Postulate Of Quantisation
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NCERT Exemplar Class 12 Physics Solutions Chapter 12 Atoms-What will the students learn?

Class 12 Physics NCERT Exemplar solutions chapter 12 explores the different atoms' different models over time and the experiments that led to discovering atomic particles. NCERT Exemplar solutions for Class 12 Physics chapter 12 will also throw light on the other characteristics and properties of elements depending on the atomic particles' location and number. We study atoms because the Universe revolves around the properties of elements, not necessarily the properties of an electron or proton.

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Yes, these solutions can help in understanding the topic better and thus get a good grip on topics for entrance exams.

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  • Passing NIOS in October 2024 will make you eligible for NIT admissions in 2025 . NIT admissions are based on your performance in entrance exams like JEE Main, which typically happen in January and April. These exams consider the previous year's Class 12th board results (or equivalent exams like NIOS).

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg


An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)


Option 2)

\; K\;

Option 3)


Option 4)


In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)


Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)


Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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