   Chapter 14, Problem 14.39E

Chapter
Section
Textbook Problem

Verify equation 14.26.

Interpretation Introduction

Interpretation:

The equation EJ+1EJ=2B(J+1)4DJ(J+1)3 is to be verified.

Concept introduction:

The rotational energy of a molecule is given by an expression shown below.

Erot=BJ(J+1)DJJ2(J+1)2

Where,

B is the rotational constant.

DJ is the centrifugal distortion constant.

J is the quantum number of lower energy rotational state.

Explanation

The equation 14.26 is given below.

EJ+1EJ=2B(J+1)4DJ(J+1)3

The rotational energy of a molecule is given by an expression shown below.

Erot=BJ(J+1)DJJ2(J+1)2 …(1)

Where,

B is the rotational constant.

DJ is the centrifugal distortion constant.

J is the quantum number of lower energy rotational state.

The equation (1) can be written in terms of J as shown below.

EJ=BJ(J+1)DJJ2(J+1)2 …(2)

Substitute J=J+1 in equation (2).

EJ+1=B(J+1)[(J+1)+1]DJ(J+1)2[(J+1)+1]2=B(J+1)(J+2)DJ(J+1)2(J+2)2

EJ+1=B(J2+3J+2)DJ(J+1)2(J2+4J+4) …(3)

The equation (3) is subtracted from the equation (1)

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