Question

How many normal vibrational modes does each of following molecules have?

$\begin{array}{l}\left(a\right)N{H}_3\\ \left(b\right)C_2{H}_4\\ \left(c\right)CC{l}_2{F}_2\\ \left(d\right)C{H}_{3}C{H}_{2}OH\end{array}$

Short answer

(a) $N{H}_3$have 6 normal vibrational modes.

(b) $C_2{H}_4$ have 12 normal vibrational modes.

(c) $CC{l}_2{F}_2$ have 9 normal vibrational modes.

(d) $C{H}_{3}C{H}_{2}OH$have 21 normal vibrational modes.

Step-by-step solution

Definition of normal modes

A molecule's atoms are always in motion, and the molecule as a whole is in constant translational and rotational motion. There is only one motion in a diatomic molecule. Normal modes are the several types of vibrations that polyatomic molecules have.

Normal vibrational modes for linear and nonlinear molecules

For a polyatomic molecule, vibrational modes is given by$(3n-5)$ for linear molecule and $(3n-6)$for nonlinear molecule where n is number of atoms in molecule.

Calculations of normal vibrational modes using the formula for  NH3

Calculate the normal vibrational modes using the relations:

$(3n-5)$for linear molecule and$(3n-6)$ for nonlinear molecule where n is number of atoms in molecule.

$N{H}_3$has 6 normal vibrational modes as the molecule is nonlinear.

The number of atoms, $n=4$

Calculate the Normal vibrational modes

$\begin{array}{l}=3n-6\\ =3\times 4-6\\ =12-6\\ =6\end{array}$

$N{H}_3$ have 6 normal vibrational modes.

Calculations of normal vibrational modes using the formula for  C2H4

Calculate the normal vibrational modes using the relations:

$(3n-5)$for linear molecule and $(3n-6)$for nonlinear molecule where n is number of atoms in molecule.

$C_2{H}_4$has 12 normal vibrational modes as the molecule is nonlinear.

The number of atoms in$C_2{H}_4$ is, $n=6$

Normal vibrational modes for nonlinear molecule

$\begin{array}{l}=3n-6\\ =3\times 6-6\\ =18-6\\ =12\end{array}$

$C_2{H}_4$ have 12 normal vibrational modes.

 Step 5: Calculations of normal vibrational modes using the formula for  CCl2F2

Calculate the normal vibrational modes using the relations:

$(3n-5)$for linear molecule and $(3n-6)$for nonlinear molecule where n is number of atoms in molecule.

$CC{l}_2{F}_2$has 9 normal vibrational modes as the molecule is nonlinear.

The number of atoms,$n=5$

Normal vibrational modes for nonlinear molecule is$3n-6$

$\begin{array}{l}=3\times 5-6\\ =15-6\\ =9\end{array}$

$CC{l}_2{F}_2$ have 9 normal vibrational modes.

Calculations of normal vibrational modes using the formula for  CH3CH2OH

Calculate the normal vibrational modes using the relations:

$(3n-5)$for linear molecule and $(3n-6)$for nonlinear molecule where n is number of atoms in molecule.

$C{H}_{3}C{H}_{2}OH$has 21 normal vibrational modes as the molecule is nonlinear.

The number of atoms,$n=9$

Normal vibrational modes for nonlinear moleculeis$3n-6$

$\begin{array}{l}=3\times 9-6\\ =27-6\\ =21\end{array}$

$C{H}_{3}C{H}_{2}OH$ have 21 normal vibrational modes.