(a)
Interpretation:
The number of Raman-active vibrations for the
Concept introduction:
The characters of the irreducible representations of the given point group can be multiplied by each other. The only condition is the characters of the same symmetry operations are multiplied together. The multiplication of the characters is commutative.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
Answer to Problem 14.94E
The number of Raman-active vibrations for the
Explanation of Solution
The symmetry of
The character table for point group
operations | |||||
This reducible representation reduced using great orthogonality theorem as shown below.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
The order of the group is
The great orthogonality theorem orthogonality of the irreducible representation of
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
The character of
Therefore,
Therefore, there are four Raman-active vibrations and two IR-active vibrations would be observed by
Therefore, the number of Raman-active vibrations for the
The number of Raman-active vibrations for the
(b)
Interpretation:
The number of Raman-active vibrations for the
Concept introduction:
The characters of the irreducible representations of the given point group can be multiplied by each other. The only condition is the characters of the same symmetry operations are multiplied together. The multiplication of the characters is commutative.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
Answer to Problem 14.94E
The number of Raman-active vibrations for the
Explanation of Solution
The symmetry of
The character table for point group
operations | |||
This reducible representation reduced using great orthogonality theorem as shown below.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
The order of the group is
The great orthogonality theorem orthogonality of the irreducible representation of
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
The character of
Therefore,
The
Therefore, there are six Raman-active vibrations and six IR-active vibrations would be observed by
Therefore, the number of Raman-active vibrations for the
The number of Raman-active vibrations for the
(c)
Interpretation:
The number of Raman-active vibrations for the
Concept introduction:
The characters of the irreducible representations of the given point group can be multiplied by each other. The only condition is the characters of the same symmetry operations are multiplied together. The multiplication of the characters is commutative.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
Answer to Problem 14.94E
The number of Raman-active vibrations for the
Explanation of Solution
The symmetry of
The character table for point group
operations | ||||
This reducible representation reduced using great orthogonality theorem as shown below.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
The order of the group is
The great orthogonality theorem orthogonality of the irreducible representation of
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Therefore,
The
The
The
Therefore, there are nine Raman-active vibrations and eight IR-active vibrations would be observed by
Therefore, the number of Raman-active vibrations for the
The number of Raman-active vibrations for the
(d)
Interpretation:
The number of Raman-active vibrations for the
Concept introduction:
The characters of the irreducible representations of the given point group can be multiplied by each other. The only condition is the characters of the same symmetry operations are multiplied together. The multiplication of the characters is commutative.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
Answer to Problem 14.94E
The number of Raman-active vibrations for the
Explanation of Solution
The symmetry of
The character table for point group
operations | |||
This reducible representation reduced using great orthogonality theorem as shown below.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
The order of the group is
The great orthogonality theorem orthogonality of the irreducible representation of
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
The character of
Therefore,
The
Therefore, there are six Raman-active vibrations and six IR-active vibrations would be observed by
Therefore, the number of Raman-active vibrations for the
The number of Raman-active vibrations for the
(e)
Interpretation:
The number of Raman-active vibrations for the
Concept introduction:
The characters of the irreducible representations of the given point group can be multiplied by each other. The only condition is the characters of the same symmetry operations are multiplied together. The multiplication of the characters is commutative.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
Answer to Problem 14.94E
The number of Raman-active vibrations for the
Explanation of Solution
The symmetry of
The character table for point group
operations | |||||
This reducible representation reduced using great orthogonality theorem as shown below.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
The order of the group is
The great orthogonality theorem orthogonality of the irreducible representation of
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
The character of
Therefore,
Therefore, there are four Raman-active vibrations and two IR-active vibrations would be observed by
Therefore, the number of Raman-active vibrations for the
The number of Raman-active vibrations for the
Want to see more full solutions like this?
Chapter 14 Solutions
Bundle: Physical Chemistry, 2nd + Student Solutions Manual
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- Rotational spectra are affected slightly by the fact that different isotopes have different masses. Suppose a sample of the common isotope 1H35Cl is changed to 1H37Cl. (a) By what fraction is the molecule’s rotational inertia different? (The bond length is 0.127 nm in each case.) (b) What is the change in energy of theℓ = 1 to theℓ = 0 transition if the isotope is changed?arrow_forwardA molecule in a gas undergoes about 1.0 × 109 collisions in each second. Suppose that (a) every collision is effective in deactivating the molecule rotationally and (b) that one collision in 10 is effective. Calculate the width (in cm³¹) of rotational transitions in the molecule.arrow_forwardConsider the rotational spectrum of a linear molecule at 298 K with a moment of inertia of 1.23×10−461.23\times10^{-46}1.23×10−46 kg m2 . (a) What is the frequency for the transition from J = 2 to J = 3? (b) What is the most populated rotational level for this molecule? Would the transition in (a) give the most intense signal in the rotational spectrum?arrow_forward
- 3. ^14N^16O (the superscripts represent the atomic mass number) (a) NO molecules rotate at an angular velocity of 2.01x10^12 rev/s, at the quantized rotational state with the rotational quantum number J of 3. Calculate the bond length of NO molecules. (b) Can NO molecules rotate under light irradiation? Explain your answer. (c) Calculate the effective force constant of the vibrational mode of NO at a frequency of 5.63x10^13 Hz measured by the infrared absorption spectrum. (d) NO has a bond energy of 6.29 eV. Applying the parabolic approximation to estimate the longest distance in which N and O atoms can be stretched before the dissociation of the molecular bondarrow_forwardHow many normal modes of vibration are there for the following molecules and, in each case, briefly explain why this is so: (i) C6H6, (ii) C6H5CH3, and (iii) HC≡C−C≡CH?arrow_forwardExplain why the lines in the spectrum for H35Cl and H37Cl give rise to different rotational constants for the two molecules.arrow_forward
- Which of the following molecules may show a pure rotational Raman spectrum: (i) CH2Cl2, (ii) CH3CH3, (iii) SF6, (iv) N2O?arrow_forwardConsider H35Cl. (a) Calculate the reduced mass. Compare your value to the mass of a H atom and explain the comparison physically. (b) Given that the zero-point energy of this oscillator is 1495 cm¹, determine the force constant for the H-Cl bond. the vibrational constant for D35 Cl. (c) Determinearrow_forwardThe wavenumber of the fundamental vibrational transition of Cl 2 is 565 em·'. Calculate the force constant of thebond.arrow_forward
- Physical ChemistryChemistryISBN:9781133958437Author:Ball, David W. (david Warren), BAER, TomasPublisher:Wadsworth Cengage Learning,Principles of Modern ChemistryChemistryISBN:9781305079113Author:David W. Oxtoby, H. Pat Gillis, Laurie J. ButlerPublisher:Cengage Learning