Chapter 13, Problem 13B.3E

(a)

Interpretation Introduction

Interpretation:

The coupling constant $\left({}^3{J}_{\text{HH}}\right)$ value has to be calculated between the protons in $\text{HCCH}$ group.

Concept Introduction:

Karplus equation:

The magnitude of ${}^3{J}_{\text{HH}}$ in an $\text{HCCH}$ group depends on the dihedral angle, $\varphi$ between the two $\text{C-H}$ bonds.  The variation is given by Karplus equation as follows,

$\begin{array}{l}{}^3{J}_{\text{HH}}=\text{A}\text{+}\text{B}\text{cos}\varphi \text{+}\text{C}\text{cos}\text{2}\varphi \\ \text{where,}\\ \text{J}\text{-}\text{coupling}\text{constant}\end{array}$

Explanation of Solution

Typical values of A, B and C are given below,

  $\begin{array}{l}\text{A}\text{=}\text{+}\text{7}\text{Hz}\text{;}\text{B}\text{=}\text{-}\text{1}\text{Hz}\text{;}\text{C}\text{=}\text{+}\text{5}\text{Hz}\\ \text{dihedral}\text{angle}\left(\varphi \right)\text{=}{\text{0}}^{\text{0}}\end{array}$

Calculate the coupling constant value using Karplus equation,

$\begin{array}{l}{}^{\text{3}}{\text{J}}_{\text{HH}}\text{=}\text{A}\text{+}\text{B}\text{cos}\varphi \text{+}\text{C}\text{cos}\text{2}\varphi \\ \\ \text{=}\text{7}\text{Hz}\text{+}\left(\text{-1}\text{Hz}\text{×}\text{cos}\left({\text{0}}^{\text{0}}\right)\right)\text{+}\left(\text{5}\text{Hz}\text{×}\text{cos}\text{2}\left({\text{0}}^{\text{0}}\right)\right)\\ \\ \text{=}\text{7}\text{Hz}\text{+}\left(\text{-1}\text{Hz}\text{×}\text{1}\right)\text{+}\left(\text{5}\text{Hz}\text{×}\text{1}\right)\\ \\ \text{=}\left(\text{7}\text{+}\text{1}\text{+}\text{5}\right)\text{Hz}\text{=}\text{13}\text{Hz}\end{array}$

Therefore, the coupling constant $\left({}^3{J}_{\text{HH}}\right)$ value is $\text{13}\text{Hz}$.

(b)

Interpretation Introduction

Interpretation:

The coupling constant $\left({}^3{J}_{\text{HH}}\right)$ value has to be calculated between the protons in $\text{HCCH}$ group.

Concept Introduction:

Karplus equation:

The magnitude of ${}^3{J}_{\text{HH}}$ in an $\text{HCCH}$ group depends on the dihedral angle, $\varphi$ between the two $\text{C-H}$ bonds.  The variation is given by Karplus equation as follows,

$\begin{array}{l}{}^3{J}_{\text{HH}}=\text{A}\text{+}\text{B}\text{cos}\varphi \text{+}\text{C}\text{cos}\text{2}\varphi \\ \text{where,}\\ \text{J}\text{-}\text{coupling}\text{constant}\end{array}$

Explanation of Solution

Typical values of A, B and C are given below,

  $\begin{array}{l}\text{A}\text{=}\text{+}\text{7}\text{Hz}\text{;}\text{B}\text{=}\text{-}\text{1}\text{Hz}\text{;}\text{C}\text{=}\text{+}\text{5}\text{Hz}\\ \text{dihedral}\text{angle}\left(\varphi \right)\text{=}12{\text{0}}^{\text{0}}\end{array}$

Calculate the coupling constant value using Karplus equation,

$\begin{array}{l}{}^{\text{3}}{\text{J}}_{\text{HH}}\text{=}\text{A}\text{+}\text{B}\text{cos}\varphi \text{+}\text{C}\text{cos}\text{2}\varphi \\ \\ \text{=}\text{7}\text{Hz}\text{+}\left(\text{-1}\text{Hz}\text{×}\text{cos}\left({\text{120}}^{\text{0}}\right)\right)\text{+}\left(\text{5}\text{Hz}\text{×}\text{cos}\text{2}\left({\text{120}}^{\text{0}}\right)\right)\\ \\ \text{=}\text{7}\text{Hz}\text{+}\left(\text{-1}\text{Hz}\text{×}\left(\text{-}\frac{\text{1}}{\text{2}}\right)\right)\text{+}\left(\text{5}\text{Hz}\text{×}\left(\text{-}\frac{\text{1}}{\text{2}}\right)\right)\\ \\ \text{=}\left(\text{7}\text{+}\text{0}\text{.5}-\text{2}\text{.5}\right)\text{Hz}\text{=}\text{10}\text{Hz}\end{array}$

Therefore, the coupling constant $\left({}^3{J}_{\text{HH}}\right)$ value is $\text{10}\text{Hz}$.