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JEE Mains | PYQ

Consider the following spectral lines for atomic hydrogen:
A. First line of Paschen series
B. Second line of Balmer series
C. Third line of Paschen series
D. Fourth line of Bracket series
The correct arrangement of the above lines in ascending order of energy is:

Radius of the first excited state of Helium ion is given as:
\( a_0 \to \) radius of first stationary state of hydrogen atom.

The shortest wavelength of hydrogen atom in Lyman series is \( \lambda \). The longest wavelength in Balmer series of \( \mathrm{He}^+ \) is:

The wave function \( (\psi) \) of \( 2s \) is given by:
\( \psi_{2s}=\dfrac{1}{2\sqrt{2\pi}}\left(\dfrac{1}{a_0}\right)^{1/2}\left(2-\dfrac{r}{a_0}\right)e^{-r/2a_0} \)
At \( r=r_0 \), radial node is formed. Thus, \( r_0 \) in terms of \( a_0 \)

The total number of anti bonding molecular orbitals, formed from \( 2s \) and \( 2p \) atomic orbitals in a diatomic molecule is __________.

The electron in the \( n^{\text{th}} \) orbit of \( \mathrm{Li^{2+}} \) is excited to \( (n + 1) \) orbit using the radiation of energy \( 1.47 \times 10^{-17} \, \text{J} \) (as shown in the diagram). The value of \( n \) is _______. (Given : \( R_H = 2.18 \times 10^{-18} \, \text{J} \))

If shortest wavelength of hydrogen atom in Lyman series is \\( x \\), then longest wavelength in Balmer series of \\( \\mathrm{He^+} \\) is:

Frequency of the de-Broglie wave of electron in Bohr’s first orbit of hydrogen atom is __________ \( \times 10^{13} \) \( \mathrm{Hz} \) (nearest integer).
[Given: \( R_H \) (Rydberg constant) \( = 2.18 \times 10^{-18} \ \mathrm{J} \), \( h \) (Plank’s constant) \( = 6.6 \times 10^{-34} \ \mathrm{J\cdot s} \).]

Match List I with List II.

Choose the correct answer from the options given below:

For hydrogen like species, which of the following graphs provides the most appropriate representation of \( E \) vs \( Z \) plot for a constant \( n \)?
[\( E \): Energy of the stationary state,
\( Z \): atomic number, \( n \) = principal quantum number]

Consider an imaginary ion \( ^{48}_{22}X^{3-} \). The nucleus contains '\( a \)'% more neutrons than the number of electrons in the ion. The value of 'a' is ______ (nearest integer)

Given below are two statements:
Statement I : According to Bohr’s model of an atom, qualitatively the magnitude of velocity of electron increases with decrease in positive charges on the nucleus as there is no strong hold on the electron by the nucleus.
Statement II : According to Bohr’s model of an atom, qualitatively the magnitude of velocity of electron increases with decrease in principal quantum number.
In the light of the above statements, chose the most appropriate answer from the options given below:

The energy required by electrons, present in the first Bohr orbit of hydrogen atom to be excited to second Bohr orbit is __________ \( \text{J mol}^{-1} \). Given: \( R_H = 2.18 \times 10^{-11} \, \text{ergs} \).

A certain orbital has \( n = 4 \) and \( m_l = -3 \). The number of radial nodes in this orbital is ______. (Round off to the Nearest Integer).

Given, 
(A)  $n = 5, m_{\ell} = -1$
(B)  $n = 3, \ell = 2, m_{\ell} = -1, m_s = +\frac{1}{2}$ 
The maximum number of electron(s) in an atom that can have the quantum numbers as given in (A) and (B) respectively are: 

The number of radial node/s for \( 3p \) orbital is:

The number of following statement/s which is/are incorrect is ________.
(A) Line emission spectra are used to study the electronic structure.
(B) The emission spectra of atoms in the gas phase show a continuous spread of wavelength from red to violet.
(C) An absorption spectrum is like the photographic negative of an emission spectrum.
(D) The element helium was discovered in the sun by spectroscopic method.

Arrange the following atomic orbitals of multi electron atoms in order of increasing energy.
A. \( n = 3, \, l = 2, \, m = +1 \)
B. \( n = 4, \, l = 0, \, m = 0 \)
C. \( n = 6, \, l = 1, \, m = 0 \)
D. \( n = 5, \, l = 1, \, m = +1 \)
E. \( n = 2, \, l = 1, \, m = +1 \)
Choose the correct answer from the options given below:

Which of the following is the correct plot for the probability density \( \psi^2(r) \) as a functions of distance \( r \) of the electron from the nucleus for \( 2s \) orbitals?

Given below are two statements:
Statement I : When an electric discharge is passed through gaseous hydrogen, the hydrogen molecules dissociate and the energetically excited hydrogen atoms produce electromagnetic radiation of discrete frequencies.
Statement II : The frequency of second line of Balmer series obtained from \( \mathrm{He^+} \) is equal to that of first line of Lyman series obtained from hydrogen atom.
In the light of the above statements, choose the correct answer from the options given below:

In a multielectron atom, which of the following orbitals described by three quantum numbers will have same energy in absence of electric and magnetic fields?
A. \( n = 1, \, \ell = 0, \, m_1 = 0 \)                                    B. \( n = 2, \, \ell = 0, \, m_1 = 0 \)
C. \( n = 2, \, \ell = 1, \, m_1 = 1 \)                                    D. \( n = 3, \, \ell = 2, \, m_1 = 1 \)
E. \( n = 3, \, \ell = 2, \, m_1 = 0 \)
Choose the correct answer from the options given below:

If the Thompson model of the atom was correct, then the result of Rutherford’s foil experiment would have been:

The minimum uncertainty in the speed of an electron in an one dimensional region of length \( 2a_{0} \) (Where \( a_{0} = \) Bohr radius \( 52.9\,\mathrm{pm} \)) is _________ \( \mathrm{km\,s^{-1}} \).
(Given: Mass of electron \( = 9.1 \times 10^{-31}\,\mathrm{kg} \), Planck’s constant \( h = 6.63 \times 10^{-34}\,\mathrm{J\,s} \))

The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is equal to \( \dfrac{h^{2}}{x m a_{0}^{2}} \). The value of \( 10x \) is __________. (\( a_{0} \) is radius of Bohr’s orbit) (Nearest integer) [Given: \( \pi = 3.14 \)]

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R:
Assertion A: In the photoelectric effect, the electrons are ejected from the metal surface as soon as the beam of light of frequency greater than threshold frequency strikes the surface.
Reason R: When the photon of any energy strikes an electron in the atom, transfer of energy from the photon to the electron takes place.
In the light of the above statements, choose the most appropriate answer from the options given below:

The surface of sodium metal is irradiated with radiation of wavelength \( x \, \text{nm} \). The kinetic energy of ejected electrons is \( 2.8 \times 10^{-20} \, \text{J} \). The work function of sodium is \( 2.3 \, \text{eV} \). The value of \( x \) is ______ \( \times 10^{2} \, \text{nm} \). (Nearest integer)

(Given : \( 6.6 \times 10^{-34} \, \text{J s} \); \( 1e = 1.6 \times 10^{-19} \, \text{J} \); \( c = 3.0 \times 10^{8} \, \text{m s}^{-1} \) )

Given below are two statements:
Statement (I): For a given shell, the total number of allowed orbitals is given by \( n^2 \).
Statement (II): For any subshell, the spatial orientation of the orbitals is given by \( -l \) to \( +l \) values including zero.
In the light of the above statements, choose the correct answer from the options given below:

The correct decreasing order of energy for the orbital having, following set of quantum numbers:
(A) \( n = 3, \, l = 0, \, m = 0 \)
(B) \( n = 4, \, l = 0, \, m = 0 \)
(C) \( n = 3, \, l = 1, \, m = 0 \)
(D) \( n = 3, \, l = 2, \, m = 1 \)
is:

Identify the INCORRECT statements from the following:
A. Notation \( ^{24}_{12}\mathrm{Mg} \) represents \( 24 \) protons and \( 12 \) neutrons.
B. Wavelength of a radiation of frequency \( 4.5 \times 10^{15} \, \text{s}^{-1} \) is \( 6.7 \times 10^{-8} \, \text{m} \).
C. One radiation has wavelength \( \lambda_1 = 900 \, \text{nm} \) and energy \( = E_1 \); other radiation has wavelength = \( \lambda_2  (300 \, \text{nm}) \) and energy \( = E_2 \). \( E_1 : E_2 = 3 : 1 \).
D. Number of photons of light of wavelength \( 2000 \, \text{pm} \) that provides \( 1 \, \text{J} \) of energy is \( 1.006 \times 10^{16} \).
Choose the correct answer from the options given below:

Which of the following sets of quantum numbers is not allowed?

According to the wave-particle duality of matter by de-Broglie, which of the following graph plot presents most appropriate relationship between wavelength of electron \( \lambda \) and momentum of electron \( p \)?

What is the ratio of wave number of first line (lowest energy line) of Balmer series of \( \text{H} \) atomic spectrum to first line of its Brackett series?

If the work function of a metal is \( 6.63 \times 10^{-19} \, \text{J} \), the maximum wavelength of the photon required to remove a photoelectron from the metal is ______ \( \text{nm} \). (nearest integer) [Given: \( h = 6.63 \times 10^{-34} \, \text{J s} \), and \( c = 3 \times 10^{8} \, \text{m s}^{-1} \)]

The metal ion (in gaseous state) with lowest spin-only magnetic moment value is

Which of the following statements are true?
(A) The subsidiary quantum number \( l \) describes the shape of the orbital occupied by the electron.
(B) is the boundary surface diagram of the \( 2p_x \) orbital.
(C) The \( + \) and \( - \) signs in the wave function of the \( 2p_x \) orbital refer to charge.
(D) The wave function of \( 2p_x \) orbital is zero everywhere in the \( xy \) plane.
Choose the correct answer from the options given below:

The energy of one mole of photons of radiation of wavelength \( 300 \) \( \text{nm} \) is (Given : \( h = 6.63 \times 10^{-34} \) \( \text{J s} \), \( N_{A} = 6.02 \times 10^{23} \) \( \text{mol}^{-1} \), \( c = 3 \times 10^{8} \) \( \text{m s}^{-1} \) )

The minimum energy that must be possessed by photons in order to produce the photoelectric effect with platinum metal is:
[Given: The threshold frequency of platinum is \( 1.3 \times 10^{15} \, \mathrm{s}^{-1} \) and \( h = 6.6 \times 10^{-34} \, \text{J s} \).]

According to Bohr model of hydrogen atom, which of the following statement is incorrect?

If the radius of the first orbit of hydrogen atom is \( a_0 \), then de Broglie’s wavelength of electron in \(3^{rd} \)orbit is:

What is the spin-only magnetic moment value (BM) of a divalent metal ion with atomic number \( 25 \), in it’s aqueous solutions?

The number of incorrect statements/s about the black body from the following is _______
(A) Emit or absorb energy in the form of electromagnetic radiation.
(B) Frequency distribution of the emitted radiation depends on temperature.
(C) At a given temperature, intensity vas frequency curve passes trough a maximum value.
(D) The maximum of the intensity vas frequency curve is at a higher frequency at higher temperature compared to that at lower temperature.

Which of the following electronic configuration would be associated with the highest magnetic moment ?

Given below are two statements about X-ray spectra of elements:
Statement (I): A plot of \( \sqrt{\nu} \) (\( \nu = \) frequency of X-rays emitted) vs atomic mass is a straight line.
Statement (II): A plot of \( \nu \) (\( \nu = \) frequency of X-rays emitted) vs atomic number is a straight line.
In the light of the above statements, choose the correct answer from the options given below:

For a metal ion, the calculated magnetic moment is \( 4.90 \, \text{BM} \). This metal ion has ______ number of unpaired electrons.

A metal surface is exposed to \( 500 \, \text{nm} \) radiation. The threshold frequency of the metal for photoelectric current is \( 4.3 \times 10^{14} \, \text{Hz} \). The velocity of ejected electron is ________ \( \times 10^{5} \, \text{m s}^{-1} \). (Nearest integer)
[Use : \( h = 6.63 \times 10^{-34} \, \text{J s} \), \( m_{e} = 9.0 \times 10^{-31} \, \text{kg} \) ]

The wavelength of spectral line obtained in the spectrum of \( \mathrm{Li^{2+}} \) ion, when the transition takes place between two levels whose sum is \( 4 \) and difference is \( 2 \), is:

Based on Heisenberg’s uncertainty principle, the uncertainty in the velocity of the electron to the found within an atomic nucleus of diameter \( 10^{-15} \, \text{m} \) is _____ \( \times 10^{9} \, \text{ms}^{-1} \) (nearest integer)
[Given : mass of electron \( = 9.1 \times 10^{-31} \, \text{kg} \). Plank's constant \( (h) = 6.626 \times 10^{-34} \, \text{Js} \)] (Value of \( \pi = 3.14 \))

\( \mathrm{Ge}\ (Z = 32) \) in its ground state electronic configuration has \( x \) completely filled orbitals with \( m_l = 0 \). The value of \( x \) is __________.

The value of magnetic quantum number of the outermost electron of \( \mathrm{Zn}^{+} \) ion is _____ . (Integer answer)

The de-Broglie’s wavelength of an electron in the \( 4^{\text{th}} \) orbit is \( \_\_\_\_\_\_ \, \pi a_0 \). \( (a_0 = \) Bohr’s radius\() \)

The orbital angular momentum of an electron in \(3s\) orbital is \( \dfrac{x h}{2\pi} \). The value of \( x \) is __________. \((\text{Nearest integer})\)

Electrons in a cathode ray tube have been emitted with a velocity of \( 1000 \, \mathrm{m\, s^{-1}} \). The number of following statements which is/are true about the emitted radiation is _________.
Given: \( h = 6 \times 10^{-34} \, \mathrm{J\, s} \), \( m_e = 9 \times 10^{-31} \, \mathrm{kg} \)
\( (\mathrm{A}) \) The deBroglie wavelength of the electron emitted is \( 666.67 \, \mathrm{nm} \).
\( (\mathrm{B}) \) The characteristic of electrons emitted depend upon the material of the electrodes of the cathode ray tube.
\( (\mathrm{C}) \) The cathode rays start from cathode and move towards anode.
\( (\mathrm{D}) \) The nature of the emitted electrons depends on the nature of the gas present in cathode ray tube.

For electrons in ‘\( 2s \)’ and ‘\( 2p \)’ orbitals, the orbital angular momentum values, respectively are:

For hydrogen atom, the orbital/s with lowest energy is/are:
(A) \( 4s \)    (B) \( 3p_x \)    (C) \( 3d_{x^2-y^2} \)    (D) \( 3d_{z^2} \)    (E) \( 4p_z \)
Choose the correct answer from the options given below:

Given below are two statements:
Statement I: According to Bohr’s Model of hydrogen atom, the angular momentum of an electron in a given stationary state is quantized.
Statement II: The concept of electron in Bohr’s orbit, violates the Heisenberg uncertainty principle.
In the light of the above statements, choose the most appropriate answer from the options given below:

If the wavelength for an electron emitted from H-atom is \( 3.3 \times 10^{-10} \, \text{m} \), then energy absorbed by the electron in its ground state compared to minimum energy required for its escape from the atom, is ________ times. (Nearest integer)
[Given: \( h = 6.626 \times 10^{-34} \, \text{J s} \)]
Mass of electron \( = 9.1 \times 10^{-31} \, \text{kg} \)

The radius of the \( 2 \)nd orbit of \( \mathrm{Li^{2+}} \) is \( x \). The expected radius of the \( 3 \)rd orbit of \( \mathrm{Be^{3+}} \) is:

It is observed that characteristic X-ray spectra of elements show regularity. When frequency to the power “n” i.e., \( \nu^n \) of X-rays emitted is plotted against atomic number “Z”, following graph is:

The value of “n” is

Given below are two statements:
Statement (I): The orbitals having same energy are called as degenerate orbitals.
Statement (II): In hydrogen atom, \( 3p \) and \( 3d \) orbitals are not degenerate orbitals.
In the light of the above statements, choose the most appropriate answer from the options given below:

The number of electrons present in all the completely filled subshells having \( n = 4 \) and \( s = +\frac{1}{2} \) is __________. (Where \( n \) = principal quantum number and \( s \) = spin quantum number)

If $a_0$ is denoted as the Bohr radius of hydrogen atom, then what is the de-Broglie wavelength ($\lambda$) of the electron present in the second orbit of hydrogen atom? [$n$ : any integer]

Which transition in the hydrogen spectrum would have the same wavelength as the Balmer type transition from \( n = 4 \) to \( n = 2 \) of \( \mathrm{He^+} \) spectrum.

Which one of the following about an electron occupying the \( 1\text{s} \) orbital in a hydrogen atom is incorrect? \( \text{(Bohr’s radius is represented by } a_0 \text{)} \)

According to Bohr’s atomic theory:
(A) Kinetic energy of electron is \( \propto \dfrac{Z^2}{n^2} \).
(B) The product of velocity \( (v) \) of electron and principal quantum number \( (n) \), ‘\( vn \)’ \( \propto Z^2 \).
(C) Frequency of revolution of electron in an orbit is \( \propto \dfrac{Z^3}{n^3} \).
(D) Coulombic force of attraction on the electron is \( \propto \dfrac{Z^3}{n^4} \).
Choose the most appropriate answer from the options given below:

Maximum number of electrons that can be accommodated in shell with \( n = 4 \) are :

The longest wavelength of light that can be used for the ionisation of lithium atom (Li) in its ground state is \( x \times 10^{-8} \, \text{m} \). The value of \( x \) is __________. (Nearest Integer)
(Given : Energy of the electron in the first shell of the hydrogen atom is \( -2.2 \times 10^{-18} \, \text{J} \) ; \( h = 6.63 \times 10^{-34} \, \text{J s} \) and \( c = 3 \times 10^{8} \, \text{m s}^{-1} \))

The maximum number of orbitals which can be identified with \( n = 4 \) and \( m_{1} = 0 \) is ____________ .

The total number of isoelectronic species from the given set is ________.
\( \text{O}^{2-},\ \text{F}^-,\ \text{Al},\ \text{Mg}^{2+},\ \text{Na}^+,\ \text{O}^+,\ \text{Mg},\ \text{Al}^{3+},\ \text{F} \)

If the radius of the \( 3^{\text{rd}} \) Bohr’s orbit of hydrogen atom is \( r_3 \) and the radius of \( 4^{\text{th}} \) Bohr’s orbit is \( r_4 \). Then:

The number of photons emitted by a monochromatic (single frequency) infrared range finder of power \( 1 \,\mathrm{mW} \) and wavelength of \( 1000 \,\mathrm{nm} \), in \( 0.1 \) second is \( x \times 10^{13} \). The value of \( x \) is _______. (Nearest integer)
\( (h = 6.63 \times 10^{-34} \,\mathrm{J\,s}, \; c = 3.00 \times 10^{8} \,\mathrm{m\,s^{-1}}) \)

Choose the incorrect statement about Dalton’s Atomic Theory:

The number of correct statements from the following is ________.
A. For \( 1s \) orbital, the probability density is maximum at the nucleus
B. For \( 2s \) orbital, the probability density first increases to maximum and then decreases sharply to zero
C. Boundary surface diagrams of the orbitals encloses a region of \( 100\% \) probability of finding the electron
D. \( p \) and \( d \)-orbitals have \( 1 \) and \( 2 \) angular nodes respectively
E. Probability density of \( p \)-orbital is zero at the nucleus

Identify the correct statements from the following:
A. Heisenberg uncertainty principle is applicable to electrons.
B. The size of \( 2p_{x} \) orbital is less than the size of \( 3p_{x} \) orbital.
C. The energy of \( 2s \) orbital of H atom is equal to the energy of \( 2s \) orbital of Li.
D. The electronic configuration of Cr is \( [\mathrm{Ar}] \, 3d^{5} \, 4s^{1} \)
Choose the correct answer from the options given below:

A \( 50 \) watt bulb emits monochromatic red light of wavelength of \( 795 \, \text{nm} \). The number of photons emitted per second by the bulb is \( x \times 10^{20} \). The value of \( x \) is:
[Given: \( h = 6.63 \times 10^{-34} \, \text{J s} \) and \( c = 3.0 \times 10^{8} \, \text{m s}^{-1} \)]

The number of atomic orbitals from the following having \( 5 \) radial nodes is __________.
\( 7s \), \( 7p \), \( 6s \), \( 8p \), \( 8d \)

Which of the following statements are correct, if the threshold frequency of caesium is \( 5.16 \times 10^{14} \, \mathrm{Hz} \)?
A. When \( \mathrm{Cs} \) is placed inside a vacuum chamber with an ammeter connected to it and yellow light is focused on \( \mathrm{Cs} \), the ammeter shows the presence of current
B. When the brightness of the yellow light is dimmed, the value of the current in the ammeter is reduced
C. When a red light is used instead of the yellow light, the current produced is higher with respect to the yellow light
D. When a blue light is used, the ammeter shows the formation of current
E. when a white light is used, the ammeter shows formation of current
Choose the correct answer from the options given below:

The hydrogen spectrum consists of several spectral lines in Lyman series \( (L_1, L_2, L_3, \ldots; \, L_1 \) has lowest energy among Lyman series). Similarly it consists of several spectral lines in Balmer series \( (B_1, B_2, B_3, \ldots; \, B_1 \) has lowest energy among Balmer lines). The energy of \( L_1 \) is \( x \) times the energy of \( B_1 \). The value of \( x \) is _______ \( \times 10^{-1} \). (Nearest integer)

Arrange the following orbitals in decreasing order of energy:
A. \( n = 3,\, l = 0,\, m = 0 \)
B. \( n = 4,\, l = 0,\, m = 0 \)
C. \( n = 3,\, l = 1,\, m = 0 \)
D. \( n = 3,\, l = 2,\, m = 1 \)

Consider the following set of quantum numbers.

The number of correct sets of quantum numbers is ________.

The energy of one mole of photons of radiation of frequency \( 2 \times 10^{12} \, \text{Hz} \) in \( \text{J mol}^{-1} \) is ________. (Nearest integer) [Given : \( h = 6.626 \times 10^{-34} \, \text{Js} \), \( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \)]

Given below are two statements :
Statement-I : Rutherford’s gold foil experiment cannot explain the line spectrum of hydrogen atom.
Statement-II : Bohr’s model of hydrogen atom contradicts Heisenberg’s uncertainly principle.
In the light of the above statements, choose the most appropriate answer from the options given below :

Value of work function \( (W_0) \) for a few metals are given below:

The number of metals which will show photoelectric effect when light of wavelength is \( 400 \, \mathrm{nm} \) falls on it is _____.
Given \( h = 6.6 \times 10^{-34} \, \mathrm{J\,s} \) 
\( c = 3 \times 10^{8} \, \mathrm{m\,s^{-1}} \) 
\( e = 1.6 \times 10^{-19} \, \mathrm{C} \)

The spin only magnetic moment of \( \mathrm{Cu^+} \) ion is: [Given: Atomic number of copper (\( \mathrm{Cu} \)) \( = 29 \)]

When a certain metal was irradiated with light of frequency \( 4.0 \times 10^{16} \, \text{s}^{-1} \), the photoelectrons emitted has three times the kinetic energy as the kinetic energy of photoelectrons emitted when the metal was irradiated with light of frequency \( 2.0 \times 10^{16} \, \text{s}^{-1} \). The threshold frequency \( (v_0 \)) of the metal is:

The ionization energy of sodium in \( \text{kJ mol}^{-1} \), if electromagnetic radiation of wavelength \( 242 \, \text{nm} \) is just sufficient to ionize sodium atom is ______.

Following figure shows spectrum of an ideal black body at four different temperatures. The number of correct statement/s from the following is __________ .


(A)    $T_4 > T_3 > T_2 > T_1$
(B)    The black body consists of particles performing simple harmonic motion.
(C)    The peak of the spectrum shifts to shorter wavelength as temperature increases.
(D)    $\frac{T_1}{v_1} = \frac{T_2}{v_2} = \frac{T_3}{v_3} \neq \text{constant}$
(E)    The given spectrum could be explained using quantisation of energy.

Given below are two statements. One is labeled as Assertion A and the other is labeled as Reason R.
Assertion A: Energy of \( 2s \) orbital of hydrogen atom is greater than that of \( 2s \) orbital of lithium.
Reason R: Energies of the orbitals in the same subshell decrease with increase in the atomic number.
In the light of the above statements, choose the correct answer from the options given below:

Which of the following statements are correct?
(a) The electronic configuration of Cr is \( \mathrm{[Ar]\, 3d^5\, 4s^1} \)
(b) The magnetic quantum number may have a negative value
(c) In the ground state of an atom, the orbitals are filled in order of their increasing energies
(d) The total number of nodes are given by \( n - 2 \)
Choose the most appropriate answer from the options given below:

The correct stability order of the following species/molecules is:

Consider two radiations of wavelength.
\(1\). \(\lambda_1=2000\mathring A\)    \(2\).  \(\lambda_2=6000 \mathring A\)
The ratio of the energies of these two radiations \( \left( \dfrac{E_1}{E_2} \right) \) is ______. (Nearest integer)

What is the energy (in \( \text{J atom}^{-1} \)) required for the following process?
\( \mathrm{Li^{2+}(g)} \rightarrow \mathrm{Li^{3+}(g)} + e^{-} \)
(Take the ionization energy for the H atom in the ground state as \( 2.18 \times 10^{-18} \, \text{J atom}^{-1} \))

Which of the following is/are not correct with respect to energy of atomic orbitals of hydrogen atom?
(A) \( 1s < 2p < 3d < 4s \)        (B) \( 1s < 2s = 2p < 3s = 3p \)
(C) \( 1s < 2s < 2p < 3s < 3p \)   (D) \( 1s < 2s < 4s < 3d \)
Choose the correct answer from the options given below:

The figure shows a sample of \( \mathrm{H} \)-atoms having electron revolving in higher orbit ‘\( n \)’.

If this electron makes transition from this orbit ‘\( n \)’ to ground state, number of lines emitted in Balmer series are:

When the excited electron of a H-atom from \( n = 5 \) drops to the ground state, the maximum number of emission lines observed are ________.

The wavelength of an electron and a neutron will become equal when the velocity of the electron is \( x \) times the velocity of neutron. The value of \( x \) is __________. (Nearest Integer)
(Mass of electron is \( 9.1 \times 10^{-31} \, \text{kg} \) and mass of neutron is \( 1.6 \times 10^{-27} \, \text{kg} \))

Given below are two statements:
Statement (I): It is impossible to specify simultaneously with arbitrary precision, both the linear momentum and the position of a particle.
Statement (II): If the uncertainty in the measurement of position and uncertainty in measurement of momentum are equal for an electron, then the uncertainty in the measurement of velocity is \(\displaystyle \ge \sqrt{\frac{h}{\pi}} \times \frac{1}{2m} \).
In the light of the above statements, choose the correct answer from the options given below:

Consider a hydrogen like ion with an unknown atomic number \( Z \). An electron transition from the \( n = 3 \) state to the \( n = 2 \) state, emitting a photon of wavelength \( 70 \, \text{nm} \). If the same transition in a hydrogen atom (\( Z = 1 \)) occurs with the emission of a photon with a wavelength of \( 630 \, \text{nm} \). The value of \( Z \) and the ion is:

Wave number for a radiation having \( 5800 \mathring{\text{A}} \) wavelength is \( x \times 10 \, \text{cm}^{-1} \). The value of \( x \) is ____________.(Integer answer)

Q. 53
Match the List-I with List-II.

An accelerated electron has a speed of \( 5 \times 10^{6} \, \mathrm{m}\,\mathrm{s}^{-1} \) with an uncertainty of \( 0.02\% \). The uncertainty in finding its location while in motion is \( x \times 10^{-9} \, \mathrm{m} \). The value of \( x \) is _______. (Nearest integer)
[Use mass of electron \( = 9.1 \times 10^{-31} \, \mathrm{kg} \), \( h = 6.63 \times 10^{-34} \, \mathrm{J\,s} \), \( \pi = 3.14 \) ]

The Azimuthal quantum number for the valence electrons of \( \mathrm{Ga^+} \) ion is ______.
(Atomic number of \( \mathrm{Ga} = 31 \))

The extra stability of half-filled subshell is due to:
(A) Symmetrical distribution of electrons
(B) Smaller coulombic repulsion energy
(C) The presence of electrons with the same spin in non-degenerate orbitals
(D) Larger exchange energy
(E) Relatively smaller shielding of electrons by one another
Indentify the correct statements;

The wave numbers of three spectral lines of H atom are considered. Identify the set of spectral lines belonging to Balmer series.

(R = Rydberg constant)

The species having identical radii according to the Bohr's theory are:
A. H (first orbit)     B. \( \mathrm{He^{+}} \) (first orbit)
C. \( \mathrm{He^{+}} \) (Second orbit)     D. \( \mathrm{Li^{2+}} \) (first orbit)
E. \( \mathrm{Be^{3+}} \) (Second orbit)
Choose the correct answer from the options given below:

For hydrogen atom energy of an electron in first excited state is \( -3.4 \, \text{eV} \), K.E. of the same electron of hydrogen atom is \( x \, \text{eV} \). Value of \( x \) is __________ \( \times 10^{-1} \, \text{eV} \) (Nearest integer)

The energy of first (lowest) Balmer line of \( \mathrm{H} \) atom is \( x \, \text{J} \). The energy (in \( \text{J} \)) of second Balmer line of \( \mathrm{H} \) atom is:

If wavelength of the first line of the Paschen series of hydrogen atom is \( 270 \, \text{nm} \), then the wavelength of the second line of this series is _______ \( \text{nm} \). (Nearest integer).

Given below are two statements:
Statement I: Bohr’s theory accounts for the stability and line spectrum of \( \text{Li}^+ \) ion.
Statement II: Bohr’s theory was unable to explain the splitting of spectral lines in the presence of a magnetic field.
In the light of the above statements, choose the most appropriate answer from the options given below:

Compare the energies of following sets of quantum numbers for multielectron system.

(A) \( n = 4, \, l = 1 \)    (B) \( n = 4, \, l = 2 \)    (C) \( n = 3, \, l = 1 \)

(D) \( n = 3, \, l = 2 \)    (E) \( n = 4, \, l = 0 \)

Choose the correct answer from the options given below:

Which of the following is correct set of \( 4 \) quantum numbers of \( 19^{\\text{th}} \) electron in Chromium (Atomic number \( = 24 \)) in accordance with Aufbau principle?

The four quantum numbers for the electron in the outer most orbital of potassium (atomic no. \( 19 \)) are:

In an atom, total number of electrons having quantum numbers \( n = 4 \), \( |m_\ell| = 1 \) and \( m_s = -\dfrac{1}{2} \) is \( \_ \_ \_ \_ \_ \).

The wavelength of electrons accelerated from rest through a potential difference of \( 40 \, \text{kV} \) is \( x \times 10^{-12} \, \text{m} \). The value of \( x \) is ____________. (Nearest integer)
Given: Mass of electron \( = 9.1 \times 10^{-31} \, \text{kg} \)
Charge on an electron \( = 1.6 \times 10^{-19} \, \text{C} \)
Plank’s constant \( = 6.63 \times 10^{-34} \, \text{Js} \)

Identify the incorrect statement form the following.

The spin only magnetic moment of the complex present in Fehling’s reagent is ______ B.M. (Nearest integer).

Given below are two statements:
Statement (I) : A spectral line will be observed for a \( 2p_x \to 2p_y \) transition.
Statement (II) : \( 2p_x \) and \( 2p_y \) (are degenerate orbitals.
In the light of the above statements, choose the correct answer from the options given below:

A certain orbital has no angular nodes and two radial nodes. The orbital is:

Figure \( 1 \). Electron parabability density for \( 2s \) orbital

Figure \( 2 \). wave function for \( 2s \) orbital
Which of the following point in Figure \( 2 \) most accurately represents the nodal surface as shown in Figure \( 1 \)?

Consider the ground state of chromium atom (\( Z = 24 \)). How many electrons are with Azimuthal quantum number \( l = 1 \) and \( l = 2 \) respectively?

Three particle \( A \), \( B \) and \( C \) have masses \( m \), \( 2m \) and \( 4m \) while their kinetic energies are \( 2K \), \( 4K \) and \( K \) respectively. If \( \lambda_A \), \( \lambda_B \) and \( \lambda_C \) denote their de Broglie wavelengths, then:

The Bohr radius of a hydrogen like species is \( 70.53 \, \text{pm} \). The species and the stationary state \( (n) \) are respectively.
(Given: Hydrogen atom Bohr radius is \( 52.9 \, \text{pm} \))

The \( \mathrm{Eu^{2+}} \) ion is a strong reducing agent in spite of its ground state electronic configuration (outermost) : [Atomic number of Eu = \( 63 \)]

Assume that the radius of the first Bohr orbit of hydrogen atom is \( 0.6 \, \text{Å} \). The radius of the third Bohr orbit of \( \mathrm{He^+} \) is __________ picometer. (Nearest Integer)

The work functions of two metals (\( M_A \) and \( M_B )\) are in the \( 1:2 \) ratio. When these metals are exposed to photons of energy \( 6 \, \text{eV} \), the kinetic energy of liberated electrons of \( M_A : M_B \) is in the ratio of \( 2.642 : 1 \). The work function (in eV) of \( M_A \) and \( M_B \) are respectively,

Given below are the quantum numbers for \( 4 \) electrons.
(a) \( n = 3, \, l = 2, \, m_1 = 1, \, m_s = +{1}/{2} \)
(b) \( n = 4, \, l = 1, \, m_1 = 0, \, m_s = +{1}/{2} \)
(c) \( n = 4, \, l = 2, \, m_1 = -2, \, m_s = -{1}/{2} \)
(d) \( n = 3, \, l = 1, \, m_1 = -1, \, m_s = +{1}/{2} \)
The correct order of increasing energy is

Which of the following statement(s) is/are true?
A. If two orbitals have the same value of \( (n + l) \), the orbital with lower value of \( n \) will have lower energy.
B. Energies of the orbitals in the same subshell increase with increase in atomic number.
C. The size of \( 2p_{x} \) orbital is less than the size of \( 3p_{x} \) orbital.
D. Among \( 5f, 6s, 4d, 5p \) and \( 5d \) orbitals, none of the orbitals have \( 2 \) radial nodes.
Choose the correct answer from the options given below:

Consider the following statements:
(a) The principal quantum number ' \( n \) ' is a positive integer with values of ' \( n \) ' \( = 1, 2, 3, \ldots \).
(b) The azimuthal quantum number ' \( l \) ' for a given ' \( n \) ' (principal quantum number) can have values as ' \( l \) ' \( = 0, 1, 2, \ldots \, n \).
(c) Magnetic orbital quantum number ' \( m_l \) ' for a particular ' \( l \) ' (azimuthal quantum number) has \( (2l + 1) \) values
.(d) \( \pm 1/2 \) are the two possible orientations of electron spin.
(e) For \( l = 5 \), there will be a total of \( 9 \) orbital
Which of the above statements are correct?

The number of given orbitals which have electron density along the axis is _______.
\( p_x,\; p_y,\; p_z,\; d_{xy},\; d_{yz},\; d_{xz},\; d_{z^2},\; d_{x^2 - y^2} \)

First and second ionization enthalpies of lithium are \( 520 \, \text{kJ} \, \text{mol}^{-1} \) and \( 7297 \, \text{kJ} \, \text{mol}^{-1} \) respectively. Energy required to convert \( 3.5 \, \text{mg} \) lithium (g) into \( \text{Li}^{2+}(g) \) [\( \text{Li}(g) \rightarrow \text{Li}^{2+}(g) \)] is ________ \( \text{kJ} \, \text{mol}^{-1} \). (Nearest integer)
[Molar mass of Li = \( 7 \, \text{g} \, \text{mol}^{-1} \)]

Consider the following pairs of electrons
(A) (a) \( n = 3, \ l = 1, \ m_{l} = 1, \ m_{s} = +\frac{1}{2} \) (b) \( n = 3, \ l = 2, \ m_{l} = 1, \ m_{s} = +\frac{1}{2} \)
(B) (a) \( n = 3, \ l = 1, \ m_{l} = -2, \ m_{s} = -\frac{1}{2} \) (b) \( n = 3, \ l = 2, \ m_{l} = -1, \ m_{s} = -\frac{1}{2} \)
(C) (a) \( n = 4, \ l = 2, \ m_{l} = 2, \ m_{s} = +\frac{1}{2} \) (b) \( n = 3, \ l = 2, \ m_{l} = 2, \ m_{s} = +\frac{1}{2} \)

The pairs of electrons present in degenerate orbitals is/are:

A hypothetical electromagnetic wave is show below:

The frequency of the wave is \( x \times 10^{19} \) Hz, \( x = \) ______ (Nearest integer)

The correct set of four quantum numbers for the valence electron of rubidium atom \( (Z = 37) \) is:

Which of the following statements regarding the energy of the stationary state is true in the following one-electron systems?

The number of radial and angular nodes in \( 4d \) orbital are, respectively:

Number of electrons that Vanadium \( (Z = 23) \) has in \( p \)-orbitals is equal to _______.

Which of the following postulate of Bohr’s model of hydrogen atom is not in agreement with quantum mechanical model of an atom?

The energy of an electron in first Bohr orbit of H-atom is \( -13.6 \, \text{eV} \). The magnitude of energy value of electron in the first excited state of \( \mathrm{Be}^{3+} \) is ______ eV. (Nearest integer value)