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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 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 : \( R_H = 2.18 \times 10^{-18} \, \text{J} \))

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.

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:

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:

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.

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

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.

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:

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

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.

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

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

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

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

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. \( n = 3,\, l = 2,\, m = 1 \)

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 : \( h = 6.626 \times 10^{-34} \, \text{Js} \), \( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \)]

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 \( h = 6.6 \times 10^{-34} \, \mathrm{J\,s} \)
\( c = 3 \times 10^{8} \, \mathrm{m\,s^{-1}} \)
\( e = 1.6 \times 10^{-19} \, \mathrm{C} \)

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.

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).

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

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JEE Main 2023 25th Jan Morning Shift

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: