Chapter 12, Problem 12B.6P

(a)

Interpretation Introduction

Interpretation:

Whether the result of a three bond coupling between the nuclei of metallic elements supports Karplus type equation for tin nuclei has to be stated.

Concept introduction:

The coupling constant is a way of measuring the interaction between protons.  It has the units of frequency.  Karplus equation is a relation between the coupling constant and dihedral angle which is the angle between two planes.

Answer to Problem 12B.6P

The results support a Karplus type equation for tin nuclei.

Explanation of Solution

The Karplus equation for ${}^3{J}_{\text{HH}}$ is a linear equation in $\cos \varphi$ and $\cos 2\varphi$.  The experimentally determined equation for ${}^3{J}_{\text{SnSn}}$ is a linear equation in ${}^3{J}_{\text{HH}}$.  In general, if $F\left(f\right)$ is linear in $f$ and if $f\left(x\right)$ is linear in $x$, the $F\left(x\right)$ is also linear.

Therefore, ${}^3{J}_{\text{SnSn}}$ is linear in $\cos \varphi$ and $\cos 2\varphi$.

The value of ${}^3{J}_{\text{SnSn}}$ is shown below.

    ${}^3{J}_{\text{SnSn}}/\text{Hz}=78.86\left({}^3{J}_{\text{HH}}/\text{Hz}\right)+27.84$        (1)

The expression for ${}^3{J}_{\text{HH}}$ is given below.

    ${}^3{J}_{\text{HH}}=A+B\cos \varphi +C\cos 2\varphi$

Where,

• ${}^3{J}_{\text{HH}}$ is the coupling between two hydrogen nuclei.
• ${}^3{J}_{\text{SnSn}}$ is the coupling between two tin nuclei.
• $\varphi$ is the dihedral angle.

Substitute the expression of ${}^3{J}_{\text{HH}}$ in equation (1).

    ${}^3{J}_{\text{SnSn}}/\text{Hz}=78.86\left(A+B\cos \varphi +C\cos 2\varphi \right)+27.84$        (2)

Where,

• $A,B,$ and $C$ are empirical constants.

The value of $A$ is $7$.

The value of $B$ is $-1$.

The value of $C$ is $5$.

Substitute the value of $A,B,$ and $C$ in equation (2).

    $\begin{array}{c}{}^3{J}_{\text{SnSn}}/\text{Hz}=78.86\left(7+\left(-1\right)\cos \varphi +5\cos 2\varphi \right)+27.84\\ =552.02-78.86\cos \varphi +394.3\cos 2\varphi +27.84\\ =579.86-78.86\cos \varphi +394.3\cos 2\varphi \end{array}$

Therefore, the results support a Karplus type equation for tin nuclei.

(b)

Interpretation Introduction

Interpretation:

The Karplus equation for ${}^3{J}_{\text{SnSn}}$ has to be obtained and has to be plotted as a function of the dihedral angle.

Concept introduction:

Same as mention in concept introduction of part (a).

Answer to Problem 12B.6P

The Karplus equation for ${}^3{J}_{\text{SnSn}}$ is shown below.

    ${}^3{J}_{\text{SnSn}}/\text{Hz}=579.86-78.86\cos \varphi +394.3\cos 2\varphi$

The plot of ${}^3{J}_{\text{SnSn}}$ as a function of the dihedral angle is shown below.

Figure 1

Explanation of Solution

The value of ${}^3{J}_{\text{SnSn}}$ is shown below.

    ${}^3{J}_{\text{SnSn}}/\text{Hz}=78.86\left({}^3{J}_{\text{HH}}/\text{Hz}\right)+27.84$        (1)

The expression for ${}^3{J}_{\text{HH}}$ is given below.

    ${}^3{J}_{\text{HH}}=A+B\cos \varphi +C\cos 2\varphi$

Where,

• ${}^3{J}_{\text{HH}}$ is the coupling between two hydrogen nuclei.
• ${}^3{J}_{\text{SnSn}}$ is the coupling between two tin nuclei.
• $\varphi$ is the dihedral angle.

Substitute the expression of ${}^3{J}_{\text{HH}}$ in equation (1).

    ${}^3{J}_{\text{SnSn}}/\text{Hz}=78.86\left(A+B\cos \varphi +C\cos 2\varphi \right)+27.84$        (2)

Where,

• $A,B,$ and $C$ are empirical constants.

The value of $A$ is $7$.

The value of $B$ is $-1$.

The value of $C$ is $5$.

Substitute the value of $A,B,$ and $C$ in equation (2).

    $\begin{array}{c}{}^3{J}_{\text{SnSn}}/\text{Hz}=78.86\left(7+\left(-1\right)\cos \varphi +5\cos 2\varphi \right)+27.84\\ =552.02-78.86\cos \varphi +394.3\cos 2\varphi +27.84\\ =579.86-78.86\cos \varphi +394.3\cos 2\varphi \end{array}$

Therefore, the Karplus equation for ${}^3{J}_{\text{SnSn}}$ is $579.86-78.86\cos \varphi +394.3\cos 2\varphi$.

The plot of ${}^3{J}_{\text{SnSn}}$ against the dihedral angle, $\varphi$ is shown below.

Figure 1

(c)

Interpretation Introduction

Interpretation:.

The preferred conformation of ${\text{Me}}_{\text{3}}{\text{SnCH}}_{\text{2}}{\text{CHRSnMe}}_{\text{3}}$ has to be stated.

Concept introduction:

Newman projection is a way of representing the groups attached to carbon atoms in the $\text{C-C}$ bonds.  This is used to determine the interaction between the groups attached to carbon atoms and the dihedral angle.

Answer to Problem 12B.6P

The most preferred conformation of ${\text{Me}}_{\text{3}}{\text{SnCH}}_{\text{2}}{\text{CHRSnMe}}_{\text{3}}$ is shown below.

Figure 2

Explanation of Solution

Newman conformations are of three types, staggered, eclipsed and gauche.  Out of these, the staggered conformation is the most stable as the bulky groups are present trans to each other.

The molecules, ${\text{Me}}_{\text{3}}{\text{SnCH}}_{\text{2}}{\text{CHRSnMe}}_{\text{3}}$, is preferred in a staggered conformation as the bulky ${\text{SnMe}}_{\text{3}}$ groups will be anti to each other.

The staggered conformation of ${\text{Me}}_{\text{3}}{\text{SnCH}}_{\text{2}}{\text{CHRSnMe}}_{\text{3}}$ is shown below.

Figure 2