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
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Chapter 12, Problem 12.3IA
Interpretation Introduction

Interpretation:

The exchange rate constant at 280K and 300K has to be calculated and the activation energy of the interconversion has to be determined.

Concept introduction:

The NMR spectra appear differently if the magnetic nuclei jump rapidly between different environments.  When the rate is fast, the spectrum shows a single line at the mean of the two chemical shifts.  When the rate is intermediate, the lines broaden and coalesce into a single broad line.

Expert Solution & Answer
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Answer to Problem 12.3IA

The exchange rate constant at 280K is 0.452×103s1_.  The exchange rate constant at 300K is 0.226×104s1_.  The activation energy is 56219.268Jmol1_.

Explanation of Solution

The two lines collapse into one when the lifetime (τ) is given by the expression below.

    τ=1ν°(δ'δ)×106        (1)

Where,

  • ν° is the operating frequency of the spectrometer.
  • δ' and δ are the chemical shifts.

The operating frequency is 60MHz.

The conversion of MHz to Hz is shown below.

    1MHz=106Hz

Therefore, the conversion of 60MHz to Hz is shown below.

    60MHz=60×106Hz

The value of δ' is 5.2.

The value of δ is 4.0.

Substitute the value of ν°, δ, and δ' in equation (1).

    τ60=12×3.14×60×106Hz(5.24.0)×106=1452.16Hz×1Hz1s1=2.212×103s

The rate constant (k) needed for the interconversion process is calculated by the expression shown below.

    k=1τ        (2)

Substitute the value of τ60 in equation (2).

    k=12.212×103s=0.452×103s1_

Therefore, the rate constant is 0.452×103s1_.

The operating frequency is 300MHz

The conversion of MHz to Hz is shown below.

    1MHz=106Hz

Therefore, the conversion of 300MHz to Hz is shown below.

    300MHz=300×106Hz

The value of δ' is 5.2.

The value of δ is 4.0.

Substitute the value of ν°, δ, and δ' in equation (1).

    τ300=12×3.14×300×106Hz(5.24.0)×106=12260.8Hz×1Hz1s1=4.4232×104s

Substitute the value of τ300 in equation (2).

    k=14.4232×104s=0.226×104s1_

Therefore, the rate constant is 0.226×104s1_.

The relation between the rate (r) and the activation energy (Ea) is given below.

    ln(r60r300)=EaR(1T60T300)        (3)

The lifetimes are directly proportional to the rate.  Therefore equation (3) can be written as shown below.

    ln(τ60τ300)=EaR(1T601T300)        (4)

Where,

  • Ea is the activation energy.
  • R is the gas constant. (8.314JK1mol1)
  • T60 is the temperature when the operating frequency is 60MHz.
  • T300 is the temperature when the operating frequency is 300MHz.

The value of T60 is 280K.

The value of T300 is 300K.

The value of τ60 is 2.212×103s.

The value of τ300 is 4.4232×104s.

Substitute the value of R, T60, T300, τ60, and τ300 in equation (4).

    ln(2.212×103s4.4232×104s)=Ea8.314JK1mol1(1280K1300K)l.61=Ea8.314JK1mol1×2084000KEa=1.61×8.314JK1mol1×84000K20=56219.268Jmol1_

Therefore, the activation energy is 56219.268Jmol1_.

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Chapter 12 Solutions

Atkins' Physical Chemistry

Ch. 12 - Prob. 12A.2BECh. 12 - Prob. 12A.3AECh. 12 - Prob. 12A.3BECh. 12 - Prob. 12A.4AECh. 12 - Prob. 12A.4BECh. 12 - Prob. 12A.5AECh. 12 - Prob. 12A.5BECh. 12 - Prob. 12A.6AECh. 12 - Prob. 12A.6BECh. 12 - Prob. 12A.7AECh. 12 - Prob. 12A.7BECh. 12 - Prob. 12A.8AECh. 12 - Prob. 12A.8BECh. 12 - Prob. 12A.9AECh. 12 - Prob. 12A.9BECh. 12 - Prob. 12A.1PCh. 12 - Prob. 12A.3PCh. 12 - Prob. 12B.1DQCh. 12 - Prob. 12B.2DQCh. 12 - Prob. 12B.3DQCh. 12 - Prob. 12B.4DQCh. 12 - Prob. 12B.5DQCh. 12 - Prob. 12B.1AECh. 12 - Prob. 12B.1BECh. 12 - Prob. 12B.2AECh. 12 - Prob. 12B.2BECh. 12 - Prob. 12B.3AECh. 12 - Prob. 12B.3BECh. 12 - Prob. 12B.4AECh. 12 - Prob. 12B.4BECh. 12 - Prob. 12B.5AECh. 12 - Prob. 12B.5BECh. 12 - Prob. 12B.6AECh. 12 - Prob. 12B.6BECh. 12 - Prob. 12B.7AECh. 12 - Prob. 12B.7BECh. 12 - Prob. 12B.8AECh. 12 - Prob. 12B.8BECh. 12 - Prob. 12B.9AECh. 12 - Prob. 12B.9BECh. 12 - Prob. 12B.10AECh. 12 - Prob. 12B.10BECh. 12 - Prob. 12B.11AECh. 12 - Prob. 12B.11BECh. 12 - Prob. 12B.12AECh. 12 - Prob. 12B.12BECh. 12 - Prob. 12B.13AECh. 12 - Prob. 12B.13BECh. 12 - Prob. 12B.14AECh. 12 - Prob. 12B.14BECh. 12 - Prob. 12B.1PCh. 12 - Prob. 12B.2PCh. 12 - Prob. 12B.3PCh. 12 - Prob. 12B.5PCh. 12 - Prob. 12B.6PCh. 12 - Prob. 12B.7PCh. 12 - Prob. 12B.8PCh. 12 - Prob. 12B.9PCh. 12 - Prob. 12C.1DQCh. 12 - Prob. 12C.2DQCh. 12 - Prob. 12C.3DQCh. 12 - Prob. 12C.4DQCh. 12 - Prob. 12C.5DQCh. 12 - Prob. 12C.1AECh. 12 - Prob. 12C.1BECh. 12 - Prob. 12C.2AECh. 12 - Prob. 12C.2BECh. 12 - Prob. 12C.3AECh. 12 - Prob. 12C.3BECh. 12 - Prob. 12C.4AECh. 12 - Prob. 12C.4BECh. 12 - Prob. 12C.5AECh. 12 - Prob. 12C.5BECh. 12 - Prob. 12C.4PCh. 12 - Prob. 12C.5PCh. 12 - Prob. 12C.6PCh. 12 - Prob. 12C.10PCh. 12 - Prob. 12D.1DQCh. 12 - Prob. 12D.2DQCh. 12 - Prob. 12D.1AECh. 12 - Prob. 12D.1BECh. 12 - Prob. 12D.2AECh. 12 - Prob. 12D.2BECh. 12 - Prob. 12D.3AECh. 12 - Prob. 12D.3BECh. 12 - Prob. 12D.4AECh. 12 - Prob. 12D.4BECh. 12 - Prob. 12D.5AECh. 12 - Prob. 12D.5BECh. 12 - Prob. 12D.6AECh. 12 - Prob. 12D.6BECh. 12 - Prob. 12D.1PCh. 12 - Prob. 12D.2PCh. 12 - Prob. 12D.4PCh. 12 - Prob. 12D.5PCh. 12 - Prob. 12D.6PCh. 12 - Prob. 12D.7PCh. 12 - Prob. 12D.8PCh. 12 - Prob. 12.3IACh. 12 - Prob. 12.4IA
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