Atkins' Physical Chemistry
Atkins' Physical Chemistry
11th Edition
ISBN: 9780198769866
Author: ATKINS, P. W. (peter William), De Paula, Julio, Keeler, JAMES
Publisher: Oxford University Press
Question
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Chapter 11, Problem 11B.7P
Interpretation Introduction

Interpretation:

The bond lengths of CO and CS in OCS molecule have to be stated.

Concept introduction:

A rotational spectrum is a spectrum in which only the rotation of molecule changes.  For a molecule to exhibit pure rotational spectrum, the molecule must possess a permanent electric dipole moment.  The molecules possessing dipole moment is considered as a handle that stirs the electromagnetic field into oscillation.  The energy for a rotational spectrum relates the quantum number and rotational constant to give the expression, E=B˜J(J+1).  The rotational constant further gives a relation with the moment of inertia of the molecule.

Expert Solution & Answer
Check Mark

Answer to Problem 11B.7P

The bond length of CO in OCS molecule is 115.1pm_ and the bond length of CS in OCS molecule is 156.8pm_.

Explanation of Solution

The given data for absorption in the microwave spectrum of 16O12CS in GHz is mentioned in the table below.

J1234
32S24.3259236.4888248.6516460.81408
34S23.7323347.46240  

The expression for the relationship between frequency of absorption line and J is shown below.

    ν=2(J+1)h8π2IνI=2(J+1)h8π2

Where,

  • I is the moment inertia of OCS molecule.
  • h is the Planck’s constant.

The value of h is 6.626×1027ergs.

The value of frequency for 32S and J=1 is v=24.32592GHz or 24.32592×109Hz.

The value of frequency for 34S and J=1 is v=23.73233GHz or 23.73233×109Hz.

If I1 and I2 are the moment of inertia for 32S and 34S respectively, then the expression for νI is given below.

    24.32592GHz×I1=23.73233GHz×I2I2I1=24.32592GHz23.73233GHz=1.025        (1)

The moment of inertia for 32S is calculated by the expression given below.

    I=2(J+1)h8π2ν        (2)

Substitute J as 1, v=24.32592×109Hz and h as 6.626×1027ergs in equation (2).

    I1=2(1+1)×6.626×1027ergs8(3.14)2×24.32592×109Hz=4×6.626×1027gcm2s8×9.8596×24.32592×109s1=1.3813×1038gcm2

Substitute I1=1.3813×1038gcm2 in equation (1).

    I2I1=1.025I21.3813×1038gcm2=1.025I2=1.3813×1038gcm2×1.025I2=1.4158×1038gcm2

The exact atomic mass of 16O is 15.9949mu.

The exact atomic mass of 12C is 12.00mu.

The exact atomic mass of 32S is 31.9721mu.

The exact atomic mass of 34S is 33.9679mu.

The inertia of a triatomic linear molecule (16O12CS) is given by the equation as follows.

  I=mOR12+mSR22(mOR1mSR2)2mO+ms+mC        (3)

Where,

  • mO is the mass of oxygen.
  • mS is the mass of sulfur.
  • R1 is the bond length of CO.
  • R2 is the bond length between CS.
  • mC is the mass of carbon.

For 16O12C32S molecule, substitute the values of mO, mC, and mS in equation (3).

    I1=15.9949mu×R12+31.9721mu×R22(15.9949mu×R131.9721mu×R2)215.9949mu+31.9721mu+12.00mu=15.9949mu×R12+31.9721mu×R22(15.9949×R131.9721×R2)2mu259.967mu=15.9949muR12+31.9721muR224.2663muR1217.0463muR22+17.0557muR1R2=11.7286muR12+14.9258muR22+17.0557muR1R2        (4)

For 16O12C34S molecule, substitute the values of mO, mC, and mS in equation (3).

    I2=15.9949mu×R12+33.9679mu×R22(15.9949mu×R133.9679mu×R2)215.9949mu+33.9679mu+12.00mu=15.9949mu×R12+33.9679mu×R22(15.9949×R133.9679×R2)2mu259.967mu=15.9949muR12+33.9679muR224.1289muR1218.6211muR22+17.5368muR1R2=11.8660muR12+15.3468muR22+17.5368muR1R2        (5)

Substitute the value of I1 and mu as 1.66×1024g in equation (4).

    1.3813×1038gcm2=1.66×1024g(11.7286R12+14.9258R22+17.0557R1R2)1.3813×1038gcm21.66×1024g=(11.7286R12+14.9258R22+17.0557R1R2)11.7286R12+14.9258R22+17.0557R1R2=8.32×1015cm2        (6)

Substitute the value of I2 and mu as 1.66×1024g in equation (4).

    1.415×1038gcm2=1.66×1024g(11.8660R12+15.3468R22+17.5368R1R2)1.415×1038gcm21.66×1024g=(11.8660R12+15.3468R22+17.5368R1R2)11.8660R12+15.3468R22+17.5368R1R2=8.529×1015cm2        (7)

The equations (6) and (7) are simplified into one new equation as follows.

    (11.7286R12+14.9258R22+17.0557R1R2=8.32×1015cm2)×11.8660(11.8660R12+15.3468R22+17.5368R1R2=8.529×1015cm2)×_11.72862.8905R22+3.297R1R2=1.305×1015cm2        (8)

The equations (6) and (7) are again simplified to obtain the second new equation as follows.

    (11.7286R12+14.9258R22+17.0557R1R2=8.32×1015cm2)×15.3468(11.8660R12+15.3468R22+17.5368R1R2=8.529×1015cm2)×_14.92582.89R12=0.383×1015cm2        (9)

The further simplification of equation (9) is give below.

    2.89R12=0.383×1015cm2R1=0.383×1015cm22.89R1=1.151×108cmR1=115.1pm_

Thus, the bond length of CO in OCS is 115.1pm_.

Substitute the value of R1 as 1.151×108cm in equation (8).

    2.8905R22+3.297×1.151×108cm×R2=1.305×1015cm2R22+1.312×108cm×R20.4516×1015cm=0

The further simplified equation for R2 is given below.

    R2=1.312×108cm+(1.312×108cm)2+4×0.4516×1015cm2=1.312×108cm+(1.72×1016cm2)+1.8064×1015cm2=1.5679×10-8cm=156.8pm_

Therefore, the bond length of CS in OCS molecule is 156.8pm_.

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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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