2. This is to certified that this presentation on
“RIGID ROTOR” has prepared by ANURAG
GIRI a student of M.Sc. Previous of KALYAN
PG COLLEGE,BHILAI. He has submitted the
presentation during the academic session
2018-2019 towards partial fulfillment as per
requirement of the DURG UNIVERSITY.
Dr. D. N. Sharma
Head of department
chemistry
3. I would like to express my profound sense of respect
and heartfelt gratitude to our lecturer DEEPA MAM
under whose able guidance and support ,I have my
seminar on topic.
I also express my heartfelt thanks to our
respected sir Dr. D. N. SHARMA HOD of chemistry
for this cooperation and providing facilities available
in the college which helped me in presenting the
project in a nice form.
Deepa mam. Anurag Giri
MSc previous
4. INTRODUCTION
RIGID ROTOR MODEL
MOMENT OF INERTIA
ANGULAR MOMENTUM
ENERGY OF THE SYSTEM
CONCLUSION
REFERENCE
5. The rotation of a diatomic molecule can be
described by the rigid rotor model. To
imagine this model think of a spinning
dumbbell. The dumbbell has two masses set
at a fixed distance from one another and spin
around its centre of mass (COM). This model
can be further simplified using the concept of
reduced mass which allows the problem to be
treated as a single body system.
6. A diatomic molecule
consist of two masses
m1 and m2 bound
together. The
distance between
the masses or bond
length(R).C is
centre of mass and
distance of masses
from this centre is r1
and r2 respectively.
7. The system can be simplified using the concept
of reduced mass which allows it to be treated
as one rotating body. Relationship between
the radii of rotation and bond length are
derived from the COM given by;
m1 r1 = m2 r2 (1)
Where R is the sum of two radii of rotation
R = r1 + r2 (2)
8. let, r2 =R – r1
Put value of r2 in equation (1)
m1 r1= m2( R – r1)
m1 r1= m2 R – m2 r1
m1 r1 + m2 r1 = m2 R
(m1 + m2) r1 = m2 R
r1 = m2 Rm1+m2 (3)
Similarly, r2 = m1 Rm1+m2
As we know total moment of inertia of diatomic
molecule is define as;
I = m1 r1
2 + m2 r2
2 (4)
I = m1 r1 r1 + m2 r2 r2
9. I = m2 r2 r1 + m1 r1 r2
From (m2 r2 = m1r1 )
I = (m1+m2) r1 r2
I =[m1m2 /(m1+m2)]R2 (5)
µ = m1m2 /(m1+m2)
Where µ is reduced mass of the system
From above
I = µR2 (6)
This equation is simplest form of moment of
inertia of diatomic molecule
10. Another important concept when dealing with
rotating systems is the angular momentum
defined by:
L= Iω
Looking back at the kinetic energy:
K = Iω2/2
Multiply by I in denominator and numerator
K = I2 ω2/2I
K = L2/2I
11. The kinetic energy of the system will be
K = m1v1
2/2 + m2v2
2/2 (1)
Where v1 and v2 are the linear velocity of the
particle
The relationship between linear velocity and
angular velocity is define by
v = rω
So v1 = ωr1 , v2 = ωr2 put this value in eq. 1
K = K = Iω2/2
Multiply by I in denominator and numerator
K = I2 ω2/2I
K = L2/2I (2)
Where L is angular momentum
12. Using the equation 2 the operator for energy,
Hamiltonian may be written as;
Ĥ = L2/2I (3)
Where L is total angular momentum operator
As we know L2 operator is equal to
L2 =−ℏ2 [(1/sinθ)∂/∂θ(sinθ ∂/∂θ)+(1/sinθ2)(∂2/∂ϕ2)]
Putting the value of L2 operator on equation 3
Ĥ=1/2I[−ℏ2[(1/sinθ)∂/∂θ(sinθ∂/∂θ)+(1/sinθ2)(∂2/∂ϕ2)]
Ĥ=h2/8∏2I[1/sinθ)∂/∂θ(sinθ∂/∂θ)+(1/sinθ2)(∂2/∂ϕ2)]
This is our energy of the rigid rotor in the form of
Schrödinger wave equation.
13. The rigid rotor is a mechanical model that is
used to explain rotating system.
This model can be used in quantum
mechanics to predict the rotational energy of
a diatomic molecule.
The rotational energy depends on the
moment of inertia for the system.
