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 11C.17P

(a)

Interpretation Introduction

Interpretation:

The vibrational wavenumber of CO has to be stated.

Concept introduction:

Each vibrational spectrum at high resolution exhibits closely spaced components.  There is very less spacing between these components signifying that these components are due to rotational transitions.  During simultaneous vibrational and rotational changes, the rotational quantum number changes by a factor of one during vibrational transition of a molecule.  On combining the vibration and rotation terms, the vibration-rotation spectra gives the equation as shown below.

    S˜(ν,J)=(ν+12)ν˜+B˜J(J+1)

(a)

Expert Solution
Check Mark

Answer to Problem 11C.17P

The vibrational wavenumber of CO is at 2150cm1_.

Explanation of Solution

The vibrational wavenumber is given by the position of band centre. In case of carbon monoxide, a low resolution spectrum shows that the band centre occurs at 2150cm1. Therefore, the vibrational wavenumber of Carbon monoxide occurs at 2150cm1_.

(b)

Interpretation Introduction

Interpretation:

The molar zero-point vibrational energy of CO has to be stated.

Concept introduction:

The same concept introduction as discussed in part (a).

(b)

Expert Solution
Check Mark

Answer to Problem 11C.17P

The molar zero-point energy is 12869.014Jmol1_.

Explanation of Solution

The zero point energy is calculated by the formula shown below.

  E=(v+12)hcν˜+B˜J(J+1)        (1)

Where,

  • ν is vibrational quantum number
  • h is the reduced Planck’s constant
  • c is the speed of light
  • ν˜ is the wavenumber
  • B˜ is the rotational constant.
  • J is the rotational quantum number,.

Substitute ν˜ as 2150cm1, h as 6.626×1034Js , c as 2.998×1010cms1 ν as 0, J as 0 in equation (1).

  E=(0+12)(6.626×1034Js)(2.998×1010cms1)(2150cm1)+B˜×0(0+1)=(12)(6.626×1034Js)(2.998×1010cms1)(2150cm1)=4.274×1020J2=2.137×1020J

For molar zero energy, multiplying the above value by Avogadro’s number as shown below.

  E(J/mol)=E(J)×NA        (2)

Substitute E(J)        (2)

E(J/mol)=2.137×1020J×6.022×1023mol1=12869.014Jmol1_

Therefore, the molar zero-point energy is 12869.014Jmol1_.

(c)

Interpretation Introduction

Interpretation:

The force constant of CO has to be stated.

Concept introduction:

The strength of a bond in a bond is measured by force constant. The force constant of a molecule is related to the effective mass and vibrational frequency.  The vibrational energy of a molecule is given by the equation as shown below.

Eν=(ν+12)ω.

(c)

Expert Solution
Check Mark

Answer to Problem 11C.17P

The force constant for CO is 1863.01 N m-1_.

Explanation of Solution

The wavenumber of a molecule is given by the equation as shown below.

  ν˜=12πc(kfmeff)12        (3)

Where,

  • kf is the force constant.
  • meff is the effective mass.
  • c is the speed of light

Rearrange the above equation for force constant as shown below.

  ν˜=12πc(kfmeff)124ν˜2π2c2=kfmeff

Thus the equation reduces to

  kf=4π2c2ν˜2meff        (4)

The effective mass of CO is calculated as shown below.

  meff=mambma+mb        (5)

Where,

  • ma is the mass of first atom
  • mb is the mass of second atom

For CO, substitute the value of ma as 12 g/mol and mb as 16 g/mol and in equation (5).

  meff=12 g/mol×16 g/mol12+16 g/mol=19228 g/mol=6.85g/mol

Convert the effective mass in kg as shown below.

  1 g/mol =103 kg6.022×1023/mol6.85 g/mol=6.85 g/mol×103 kg6.022×1023 /mol=1.138×1026 kg

Substitute ν˜ as 2150 cm1, meff as 1.138×1026 kg and c as 2.998×1010 cm s1 in equation (4).

  kf=4×(3.14)2×(2.998×1010 cms1)2×(2150 cm1)2×(1.138×1026kg)=4×9.8596×8.98×1020cm2s2×4622500cm2×1.138×1026kg=1863.01 N m-1_

Therefore, the force constant for CO is 1863.01 N m-1_.

(d)

Interpretation Introduction

Interpretation:

The rotational constant of CO has to be stated.

Concept introduction:

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.

(d)

Expert Solution
Check Mark

Answer to Problem 11C.17P

The rotational constant is 3.8275cm1_.

Explanation of Solution

In a high resolution spectrum, it is observed that the band centre is split into two peaks. As the vibrational spectrum splits to give two peaks with a separation of 7.655cm1.  This separation between two peaks is given by 2B˜.  Therefore, rotational constant is calculated as shown below.

  2B˜=7.655cm1B˜=7.6552cm1=3.8275cm1_

Therefore, the rotational constant is 3.8275cm1_.

(e)

Interpretation Introduction

Interpretation:

The bond length of CO has to be stated.

Concept introduction:

The same concept introduction as mentioned in part (d)

(e)

Expert Solution
Check Mark

Answer to Problem 11C.17P

The value of bond length of CO is 8.02×1011 m_.

Explanation of Solution

The value of rotational constant helps in determining the moment of inertia of a molecule as shown below.

  B˜=4πIc        (6)

Where,

  • is the reduced Planck’s constant.
  • I is the moment of inertia
  • c is the speed of light.

Substitute the value of B˜        (6)

  3.8275 cm1=1.05457×1034 Js4×3.14×I×2.998×1010 cm s1I=1.05457×1034 Js4×3.14×3.8275 cm1×2.998×1010 cm s1I=1.05457×1034 Js1.44×1012I=7.32×1047 kg m2

The moment of inertia is given by the equation as shown below.

  I=μR2        (7)

Where,

  • μ is the reduced mass
  • R is the bond length of the molecule.

Substitute μ as 1.138×1026 kg and I as 7.32×1047 kg m2 in equation (7) as shown below.

  7.32×1047 kg m2=1.138×1026 kg×R2R2=6.43×1021 m2R=8.02×1011 m_

Therefore, the value of bond length of CO is 8.02×1011 m_.

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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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