ATKINS' PHYSICAL CHEMISTRY
ATKINS' PHYSICAL CHEMISTRY
11th Edition
ISBN: 9780190053956
Author: ATKINS
Publisher: Oxford University Press
Question
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Chapter 11, Problem 11B.14P
Interpretation Introduction

Interpretation:

The CC and CH bond lengths have to be determined.

Concept introduction:

The moment of inertia of a molecule about an axis passing through the centre of mass is directly proportional to the reduced mass of the molecule and the bond length of the molecule.  The rotational energy of a molecule is usually expressed in terms of its rotational constant.  The rotational constant of a molecule depends upon the moment of inertia of the molecule.

Expert Solution & Answer
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Answer to Problem 11B.14P

The CC bond length is 1.396×1010m_.  The CH bond length is 1.085×1010m_.

Explanation of Solution

Rotation about any axis perpendicular to C6 may be represented by the rotation of a pseudo-linear molecule about the x axis as shown in the figure given below.

ATKINS' PHYSICAL CHEMISTRY, Chapter 11, Problem 11B.14P , additional homework tip  1

Figure 1

The expression for moment of inertia of C6H6 (IC6H6) is given below.

    IC6H6=4mHRH2+4mCRC2        (1)

Where,

  • mH is the mass of hydrogen atom.
  • mC is the mass of carbon atom.
  • RH is the distance of the hydrogen atom from the x axis.
  • RC is the distance of the carbon atom from the x axis.

The expression for moment of inertia of C6D6 (IC6D6) is given below.

    IC6D6=4mDRD2+4mCRC2        (2)

Where,

  • mD is the mass of the deuterium atom.
  • mC is the mass of carbon atom.
  • RH is the distance of the deuterium atom from the x axis.
  • RC is the distance of the carbon atom from the x axis.

The value of IC6H6 is 1.4759×1045kgm2.

Substitute the value of IC6H6 in equation (1).

    4mHRH2+4mCRC2=1.4759×1045kgm2        (3)

The value of IC6D6 is 1.7845×1045kgm2.

Substitute the value of IC6D6 in equation (2).

    4mDRD2+4mCRC2=1.7845×1045kgm2        (4)

Substract equation (3) from equation (4).

    4mDRD2+4mCRC2(4mHRH2+4mCRC2)=1.7845×1045kgm21.4759×1045kgm24(mDRD2mHRH2)=30.86×1047kgm2

It is assumed that RH is equal to RD.

Therefore, the above equation can be written as shown below.

    4(mDRH2mHRH2)=30.86×1047kgm24(mDmH)RH2=30.86×1047kgm2        (5)

The value of mD is 2.01410mu.

The conversion of 1mu to kg is shown below.

    1mu=1.66×1027kg

Therefore, the conversion of 2.01410mu to kg is shown below.

    2.01410mu=2.01410×1.6605×1027kg=3.344×1027kg

The value of mH is 1.0078mu.

The conversion of 1mu to kg is shown below.

    1mu=1.66×1027kg

Therefore, the conversion of 1.0078mu to kg is shown below.

    1.0078mu=1.0078×1.6605×1027kg=1.6735×1027kg

Substitute the value of mD and mH in equation (5).

    4(3.344×1027kg(1.6735×1027kg))RH2=30.86×1047kgm2RH=(30.86×1047kgm26.682×1027kg)1/2RH=(4.6183×1020m2)1/2=2.149×1010m

The value of RC can be determined by rearranging equation (1) as shown below.

    IC6H6=4mHRH2+4mCRC24mCRC2=IC6H64mHRH2RC2=IC6H64mHRH24mCRC=(IC6H64mHRH24mC)1/2        (6)

The value of IC6H6 is 1.4759×1045kgm2.

The value of mH is 1.6735×1027kg.

The value of RH is 2.149×1010m.

The value of mC is 12.011mu.

The conversion of 1mu to kg is shown below.

    1mu=1.66×1027kg

Therefore, the conversion of 12.011mu to kg is shown below.

    12.011mu=12.011×1.6605×1027kg=19.944×1027kg

Substitute the values of IC6H6, mH, RH, and mC in equation (6).

    RC=(1.4759×1045kgm2(4×1.6735×1027kg×(2.149×1010m)2)4×19.944×1027kg)1/2=(1.4759×1045kgm20.3091×1045kgm279.776×1027kg)1/2=(1.4626×1020m2)1/2=1.209×1010m

The relation between RH, RC, RCH, and RCC is shown in the figure given below.

ATKINS' PHYSICAL CHEMISTRY, Chapter 11, Problem 11B.14P , additional homework tip  2

Figure 2

Where,

  • RCC is the CC bond length.
  • RCH is the CH bond length.

From the figure shown above, the relation between RCC and RC given below holds true.

    RCC=RCcos30°        (7)

The value of RC is 1.209×1010m.

Substitute the value of RC in equation (7).

    RCC=1.209×1010m0.866=1.396×1010m_

Therefore, the CC bond length is 1.396×1010m_.

From the figure shown above, the relation between RCH, RC and RH given below holds true.

    RCH=RHRCcos30°        (8)

The value of RC is 1.209×1010m.

The value of RH is 2.149×1010m.

Substitute the value of RC and RH in equation (8).

    RCH=2.149×1010m1.209×1010m0.866=0.94×1010m0.866=1.085×1010m_

Therefore, the CH bond length is 1.085×1010m_.

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Chapter 11 Solutions

ATKINS' PHYSICAL CHEMISTRY

Ch. 11 - Prob. 11E.2STCh. 11 - Prob. 11F.2STCh. 11 - Prob. 11A.1DQCh. 11 - Prob. 11A.2DQCh. 11 - Prob. 11A.3DQCh. 11 - Prob. 11A.1AECh. 11 - Prob. 11A.1BECh. 11 - Prob. 11A.2AECh. 11 - Prob. 11A.2BECh. 11 - Prob. 11A.3AECh. 11 - Prob. 11A.3BECh. 11 - Prob. 11A.4AECh. 11 - Prob. 11A.4BECh. 11 - Prob. 11A.5AECh. 11 - Prob. 11A.5BECh. 11 - Prob. 11A.6AECh. 11 - Prob. 11A.6BECh. 11 - Prob. 11A.7AECh. 11 - Prob. 11A.7BECh. 11 - Prob. 11A.8AECh. 11 - Prob. 11A.8BECh. 11 - Prob. 11A.9AECh. 11 - Prob. 11A.9BECh. 11 - Prob. 11A.10AECh. 11 - Prob. 11A.10BECh. 11 - Prob. 11A.11AECh. 11 - Prob. 11A.11BECh. 11 - Prob. 11A.12AECh. 11 - Prob. 11A.12BECh. 11 - Prob. 11A.1PCh. 11 - Prob. 11A.2PCh. 11 - Prob. 11A.3PCh. 11 - Prob. 11A.4PCh. 11 - Prob. 11A.5PCh. 11 - Prob. 11A.7PCh. 11 - Prob. 11A.8PCh. 11 - Prob. 11A.9PCh. 11 - Prob. 11A.11PCh. 11 - Prob. 11B.1DQCh. 11 - Prob. 11B.2DQCh. 11 - Prob. 11B.3DQCh. 11 - Prob. 11B.4DQCh. 11 - Prob. 11B.5DQCh. 11 - Prob. 11B.6DQCh. 11 - Prob. 11B.7DQCh. 11 - Prob. 11B.8DQCh. 11 - Prob. 11B.1AECh. 11 - Prob. 11B.1BECh. 11 - Prob. 11B.3AECh. 11 - Prob. 11B.3BECh. 11 - Prob. 11B.4BECh. 11 - Prob. 11B.5AECh. 11 - Prob. 11B.5BECh. 11 - Prob. 11B.6AECh. 11 - Prob. 11B.6BECh. 11 - Prob. 11B.7AECh. 11 - Prob. 11B.7BECh. 11 - Prob. 11B.8AECh. 11 - Prob. 11B.8BECh. 11 - Prob. 11B.9AECh. 11 - Prob. 11B.9BECh. 11 - Prob. 11B.10AECh. 11 - Prob. 11B.10BECh. 11 - Prob. 11B.11AECh. 11 - Prob. 11B.11BECh. 11 - Prob. 11B.12AECh. 11 - Prob. 11B.12BECh. 11 - Prob. 11B.13AECh. 11 - Prob. 11B.13BECh. 11 - Prob. 11B.14AECh. 11 - Prob. 11B.14BECh. 11 - Prob. 11B.1PCh. 11 - Prob. 11B.2PCh. 11 - Prob. 11B.3PCh. 11 - Prob. 11B.4PCh. 11 - Prob. 11B.5PCh. 11 - Prob. 11B.6PCh. 11 - Prob. 11B.7PCh. 11 - Prob. 11B.8PCh. 11 - Prob. 11B.9PCh. 11 - Prob. 11B.10PCh. 11 - Prob. 11B.11PCh. 11 - Prob. 11B.12PCh. 11 - Prob. 11B.13PCh. 11 - Prob. 11B.14PCh. 11 - Prob. 11C.1DQCh. 11 - Prob. 11C.2DQCh. 11 - Prob. 11C.3DQCh. 11 - Prob. 11C.4DQCh. 11 - Prob. 11C.1AECh. 11 - Prob. 11C.1BECh. 11 - Prob. 11C.2AECh. 11 - Prob. 11C.2BECh. 11 - Prob. 11C.3AECh. 11 - Prob. 11C.3BECh. 11 - Prob. 11C.4AECh. 11 - Prob. 11C.4BECh. 11 - Prob. 11C.5AECh. 11 - Prob. 11C.5BECh. 11 - Prob. 11C.6AECh. 11 - Prob. 11C.6BECh. 11 - Prob. 11C.7AECh. 11 - Prob. 11C.7BECh. 11 - Prob. 11C.8AECh. 11 - Prob. 11C.8BECh. 11 - Prob. 11C.2PCh. 11 - Prob. 11C.3PCh. 11 - Prob. 11C.4PCh. 11 - Prob. 11C.5PCh. 11 - Prob. 11C.6PCh. 11 - Prob. 11C.7PCh. 11 - Prob. 11C.8PCh. 11 - Prob. 11C.9PCh. 11 - Prob. 11C.10PCh. 11 - Prob. 11C.11PCh. 11 - Prob. 11C.12PCh. 11 - Prob. 11C.13PCh. 11 - Prob. 11C.15PCh. 11 - Prob. 11C.17PCh. 11 - Prob. 11C.18PCh. 11 - Prob. 11C.19PCh. 11 - Prob. 11D.1DQCh. 11 - Prob. 11D.2DQCh. 11 - Prob. 11D.3DQCh. 11 - Prob. 11D.1AECh. 11 - Prob. 11D.1BECh. 11 - Prob. 11D.2AECh. 11 - Prob. 11D.2BECh. 11 - Prob. 11D.3AECh. 11 - Prob. 11D.3BECh. 11 - Prob. 11D.4AECh. 11 - Prob. 11D.4BECh. 11 - Prob. 11D.5AECh. 11 - Prob. 11D.5BECh. 11 - Prob. 11D.6AECh. 11 - Prob. 11D.6BECh. 11 - Prob. 11D.7AECh. 11 - Prob. 11D.7BECh. 11 - Prob. 11D.2PCh. 11 - Prob. 11E.1DQCh. 11 - Prob. 11E.1AECh. 11 - Prob. 11E.1BECh. 11 - Prob. 11E.2AECh. 11 - Prob. 11E.2BECh. 11 - Prob. 11E.3AECh. 11 - Prob. 11E.3BECh. 11 - Prob. 11E.1PCh. 11 - Prob. 11E.2PCh. 11 - Prob. 11F.1DQCh. 11 - Prob. 11F.2DQCh. 11 - Prob. 11F.3DQCh. 11 - Prob. 11F.4DQCh. 11 - Prob. 11F.5DQCh. 11 - Prob. 11F.6DQCh. 11 - Prob. 11F.1AECh. 11 - Prob. 11F.1BECh. 11 - Prob. 11F.2AECh. 11 - Prob. 11F.2BECh. 11 - Prob. 11F.3AECh. 11 - Prob. 11F.3BECh. 11 - Prob. 11F.4AECh. 11 - Prob. 11F.4BECh. 11 - Prob. 11F.5AECh. 11 - Prob. 11F.5BECh. 11 - Prob. 11F.6AECh. 11 - Prob. 11F.6BECh. 11 - Prob. 11F.7AECh. 11 - Prob. 11F.7BECh. 11 - Prob. 11F.8AECh. 11 - Prob. 11F.8BECh. 11 - Prob. 11F.9AECh. 11 - Prob. 11F.9BECh. 11 - Prob. 11F.10AECh. 11 - Prob. 11F.10BECh. 11 - Prob. 11F.11AECh. 11 - Prob. 11F.11BECh. 11 - Prob. 11F.12AECh. 11 - Prob. 11F.12BECh. 11 - Prob. 11F.13AECh. 11 - Prob. 11F.13BECh. 11 - Prob. 11F.1PCh. 11 - Prob. 11F.2PCh. 11 - Prob. 11F.3PCh. 11 - Prob. 11F.4PCh. 11 - Prob. 11F.5PCh. 11 - Prob. 11F.6PCh. 11 - Prob. 11F.7PCh. 11 - Prob. 11F.8PCh. 11 - Prob. 11F.9PCh. 11 - Prob. 11F.10PCh. 11 - Prob. 11F.11PCh. 11 - Prob. 11F.12PCh. 11 - Prob. 11G.1DQCh. 11 - Prob. 11G.2DQCh. 11 - Prob. 11G.3DQCh. 11 - Prob. 11G.4DQCh. 11 - Prob. 11G.5DQCh. 11 - Prob. 11G.1AECh. 11 - Prob. 11G.1BECh. 11 - Prob. 11G.2AECh. 11 - Prob. 11G.2BECh. 11 - Prob. 11G.1PCh. 11 - Prob. 11G.2PCh. 11 - Prob. 11G.3PCh. 11 - Prob. 11G.4PCh. 11 - Prob. 11G.5PCh. 11 - Prob. 11G.6PCh. 11 - Prob. 11.1IACh. 11 - Prob. 11.5IACh. 11 - Prob. 11.8IA
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