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Quantum Chemistry - 2


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20 Questions MCQ Test GATE Chemistry Mock Test Series | Quantum Chemistry - 2

Quantum Chemistry - 2 for Chemistry 2022 is part of GATE Chemistry Mock Test Series preparation. The Quantum Chemistry - 2 questions and answers have been prepared according to the Chemistry exam syllabus.The Quantum Chemistry - 2 MCQs are made for Chemistry 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Quantum Chemistry - 2 below.
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Quantum Chemistry - 2 - Question 1

For the energy level  the probability density for a particle of mass m is a one dimensional potential box of width L is given by

Detailed Solution for Quantum Chemistry - 2 - Question 1

Given energy level E = 

So, the given energy level corresponds to n = 3
No, of nodes in ψ= 3 - 1 = 2

Probabilty density 

Quantum Chemistry - 2 - Question 2

π-boncl length between C-C atoms. f.i and v is calculated by using the empirical formula,
Rμv = 1.52Å - 0.186 Å Pμv
where Pμv= is π - bond order between μ and v C -atom .
According to HM O theory. delocalization energy o f Benzene is 2|β|. Then T-bond length in Benzene is

Detailed Solution for Quantum Chemistry - 2 - Question 2

Since, delocalization energy = 2|β|
So, one extra delocalizedT-bond Benzene.
The total energy of π-bond = 4(According to HMO Theory)
Since, there are 6(C- C ) bond in benzene.
So. each (C-C) bond has 4/6 π-bond = 2/s = 0.66
Now. by using formula,
Rμv = 1.52Å - 0.186 Å x 0.66 = 1.39 Å

Quantum Chemistry - 2 - Question 3

According to Huckel molecular orbital theory, the total energy of hypothetical localized Benzene (C6H6) molecule is

Detailed Solution for Quantum Chemistry - 2 - Question 3


So, there are three localized tt-bond. Ech have energy = 2α+ 2β
So, total localized energy= 3(2α + 2β) = 6α +6β

*Answer can only contain numeric values
Quantum Chemistry - 2 - Question 4

A perturbation  is introduced in a box of width 1 unit, where V0 is ( << 1 ) is a positive constant. The first order energy correction to the first exited state is α V0 . The value of α is_________


Detailed Solution for Quantum Chemistry - 2 - Question 4

First excited state wavefimction

First order eneigy correction to the first excited state is

Quantum Chemistry - 2 - Question 5

σx , σy and σz  are the Pauli m atrices. The expression is equal to 2σσ+ σσx is equal to 

Detailed Solution for Quantum Chemistry - 2 - Question 5

σ+ σσ
= σσy + x σy + σσ
= σσy, as σσ+ σσx = 0
= iσz

Quantum Chemistry - 2 - Question 6

For linear momentum vector operator  -and position vector operator   the operator  is equivalent to

Detailed Solution for Quantum Chemistry - 2 - Question 6


*Answer can only contain numeric values
Quantum Chemistry - 2 - Question 7

The degeneracy of the n = 2 level for a three dimensional isotropic oscillator is ________. (an integer)


Detailed Solution for Quantum Chemistry - 2 - Question 7

Degeneracy for a three dimensional isotropic oscillator gn = 
n= 2

Quantum Chemistry - 2 - Question 8

The operation of the commutator  on a function f ( x ) is equal to

Detailed Solution for Quantum Chemistry - 2 - Question 8

*Answer can only contain numeric values
Quantum Chemistry - 2 - Question 9

The degeneracy of a quantum particle in a cubic box having energy three times that of the lowest energy is ________


Detailed Solution for Quantum Chemistry - 2 - Question 9

Energy eigenvalue of a particle in a cubic box.

Groimd state energy corresponds to nx = ny = nz = 1

Three times the ground state energy = 

*Answer can only contain numeric values
Quantum Chemistry - 2 - Question 10

Consider a normalized molecular orbital  constructed from two different atomic orbitals  and  that fonn an orthonormal set. Here i = √-1. The value of |c|2 is ______ (upto one decim al place)


Detailed Solution for Quantum Chemistry - 2 - Question 10

Given: 
Nonnalization condition:

*Answer can only contain numeric values
Quantum Chemistry - 2 - Question 11

The normalized wavefiinctions ψ1 and ψ2 correspond to the ground state and the first excited states of a particle in a potential. The operator A acts on the wavefiinctions as The expectation value of the operator A for the state is ________. (upto two decimal places)


Detailed Solution for Quantum Chemistry - 2 - Question 11


Quantum Chemistry - 2 - Question 12

Three identical non- interacting particles, each of spin 1/2 and mass m. are moving in a one-dimensional infinite potential well given by

the energy of the lowest energy state of the system is:

Detailed Solution for Quantum Chemistry - 2 - Question 12

Figure shows the configuration of the lowest energy state of the system

Quantum Chemistry - 2 - Question 13

A certain 2-level system lias stationary state energy  with normalized wave functions
 respectively. A small perturbation V is applied on the system such that V12 = 2. The wavefunction of the particle in groimd state collected upto first order, will be

Detailed Solution for Quantum Chemistry - 2 - Question 13

Fii'st order collection in the ground state wave function,

Collected ground state wave function,

*Answer can only contain numeric values
Quantum Chemistry - 2 - Question 14

Given that work function of a metal is 5 eV and that photons of 9 eV are incident on it. the maximum velocity of emitted electrons is approximately______ x106 m/s (upto one decimal place)


Detailed Solution for Quantum Chemistry - 2 - Question 14

Maximum kinetic energy o qf the electrons,

Quantum Chemistry - 2 - Question 15

For a system of two spin -1/2 particles, the value 0f 

Detailed Solution for Quantum Chemistry - 2 - Question 15

Quantum Chemistry - 2 - Question 16

The ground state wave function of hydrogen atom is given by where a0 is the Bohr
radius. The plot of the radial probability density P(r) for the hydrogen atom in the ground state is

Detailed Solution for Quantum Chemistry - 2 - Question 16

The radial probability density is given by

Only option (d) satisfies this condition.
 

*Answer can only contain numeric values
Quantum Chemistry - 2 - Question 17

A particle of mass m is constrained to move in a circular ring of radius R . When a perturbation

(where a is a real constant) is added, the first order collection to the ground state energy is The value
of η is ______ (up to one decimal place)


Detailed Solution for Quantum Chemistry - 2 - Question 17

Wavefiinction for a particle in a circular ring of radius R is given by

Ground state wave function corresponds to m = 0, 
First order collection to ground state energy


 

Quantum Chemistry - 2 - Question 18

The wavefiuiction for a linear harmonic oscillator described by N where α and N are constants, has

Detailed Solution for Quantum Chemistry - 2 - Question 18



For small values of 
For large values of 
The wavefunction will cut y-axis aty = -2.
ψ is even function. so it is symetric about y-axis.
From the graph, there are two maxima and one minimum only.

*Answer can only contain numeric values
Quantum Chemistry - 2 - Question 19

The ‘‘trial'’ wavefuction for He atom is taken as where N is the normalization factor and Z ' is the variational parameter. Expression for the energy in terms of Z ' is found to be

Then the lowest value of the energy ( E ') for He atom i s _________________(in a.u .)
(Round off to two decimal places)


Detailed Solution for Quantum Chemistry - 2 - Question 19


For minimmn energy,

The minimum energy will be

Quantum Chemistry - 2 - Question 20

The average distance between the electron and nucleus of a He+ (singly ionised Heliumion) in the 1s state is

Detailed Solution for Quantum Chemistry - 2 - Question 20

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