Quantum Chemistry

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

Solution:

Given energy level E =

So, the given energy level corresponds to n = 3
No, of nodes in ψ₃= 3 - 1 = 2

Probabilty density

QUESTION: 2

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

A.
1.53 Å
B.
1.39 Å
C.
1.33 Å
D.
1.23 Å

Solution:

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,
R_μv= 1.52Å - 0.186 Å x 0.66 = 1.39 Å

QUESTION: 3

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

A.
6α +8β
B.
6α +6β
C.
8α +6β
D.
8α +8β

Solution:

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

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_________

Solution:

First excited state wavefimction

First order eneigy correction to the first excited state is

QUESTION: 5

σ_x, σy and σ_z are the Pauli m atrices. The expression is equal to 2σ_xσ_y+ σ_yσ_xis equal to

A.
-3iσ_z
B.
-iσ_z
C.
iσ_z
D.
3iσ_z

Solution:

2σ_xσ_y+ σ_yσ_x
=σ_xσ_y+_xσ_y + σ_yσ_x
= σ_xσ_y, as σ_xσ_y+ σ_yσ_x= 0
= iσz

QUESTION: 6

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

A.
0
B.
C.
D.

Solution:

*Answer can only contain numeric values

QUESTION: 7

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

Solution:

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

QUESTION: 8

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

A.
0
B.
C.
- xf (x)
D.
xf(x)

Solution:

*Answer can only contain numeric values

QUESTION: 9

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

Solution:

Energy eigenvalue of a particle in a cubic box.

Groimd state energy corresponds to n_x = n_y = n_z= 1

Three times the ground state energy =

*Answer can only contain numeric values

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)

Solution:

Given:
Nonnalization condition:

*Answer can only contain numeric values

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)

Solution:

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:

Solution:

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

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

Solution:

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

Collected ground state wave function,

*Answer can only contain numeric values

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)

Solution:

Maximum kinetic energy o qf the electrons,

QUESTION: 15

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

Solution:

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

Solution:

The radial probability density is given by

Only option (d) satisfies this condition.

*Answer can only contain numeric values

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)

Solution:

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

QUESTION: 18

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

A.
one maximum, one minimum only
B.
Two maxima, two minima only
C.
one maximum only
D.
Two maxima, one minimum only

Solution:

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

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)

Solution:

For minimmn energy,

The minimum energy will be

QUESTION: 20

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

A.
B.
C.
3a₀
D.

Solution:

Quantum Chemistry - 2

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

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

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

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

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