Q12.116P - Chemistry: Molecular Nature Of Matter And Change

Question

The polymer that is used to make 2-L soda bottles [poly (ethylene terephthalate)] has a repeat unit with molecular formula  ${\mathbf{C}}_{10}{\mathbf{H}}_8{\mathbf{O}}_4$ and length 1.075 nm. Calculate the radius of gyration of a chain with molar mass of  $\mathbf{2}\mathbf{.}\mathbf{30}\mathbf{\times }{\mathbf{10}}^4$ g/mol.

 

Step-by-Step Solution

Answer

The molar mass of the molecule ${\mathbf{C}}_{10}{\mathbf{H}}_8{\mathbf{O}}_4$ is 192 g/mole and the molar mass of the polymer is 119.79 repeating units. The radius of the gyration of the polymer is 4.80pm.

Step 1: Definition

A polymer may be defined as a complex molecule that is formed by the addition or condensation of the small units called monomers. Polymer mass is the mass of the complex molecule which contained many types of similar or different monomers.

 The mass of the polymer is very high.

 

Polymer mass can be calculated by the formula:

Polymer mass can be calculated by the formula:

  ${\mathbf{{\rm M}}}_{\mathrm{polymer}}\mathbf{=}{\mathbf{M}}_{\mathrm{repeat}}\mathbf{\times }\mathbf{n}$

 

${\mathbf{{\rm M}}}_{\mathrm{polymer}}$   = Polymer mass,

    

$\begin{array}{c}{\mathbf{M}}_{\mathrm{repeat}}=\mathrm{Repeating}\mathrm{unit}’\mathrm{s}\mathrm{mass} \\ \mathrm{n} =\mathrm{degree}\mathrm{of}\mathrm{polymerization}\end{array}$.

The radius of gyration can be defined as the radius of the whole polymer from the centre to every repeating unit. The radius of gyration can be calculated by the formula:

 

 ${\mathbf{R}}_g\mathbf{=}{\left(\frac{{\mathbf{nl}}_0}{6}\right)}^{\frac{1}{2}}$

 ${\mathbf{R}}_g$ is the radius of gyration,

n is the degree of polymerization and

${\mathrm{I}}_0$ is the length of each repeating unit.

 

Step 2: Explanation

Molar Mass of the molecule ${\mathbf{C}}_{10}{\mathbf{H}}_8{\mathbf{O}}_4$ :

$\begin{array}{c}\mathrm{Molar} \mathrm{Mass} \mathrm{of} \mathrm{molecule}=10\times \mathrm{C}+8\times \mathrm{H}+4\times \mathrm{O}\\ \mathrm{Molar} \mathrm{Mass} \mathrm{of} \mathrm{molecule}=10\times 12+8\times 1+4\times 16\\ \mathrm{Molar} \mathrm{Mass} \mathrm{of} \mathrm{molecule}=120+8+64\\ \mathrm{Molar} \mathrm{Mass} \mathrm{of} \mathrm{molecule}=192\mathrm{g}/\mathrm{mole}\end{array}$

The repeating units of the molecule  ${\mathbf{C}}_{10}{\mathbf{H}}_8{\mathbf{O}}_4$:

 $\begin{array}{c}{\mathbf{{\rm M}}}_{\mathrm{polymer}}\mathbf{=}{\mathbf{M}}_{\mathrm{repeat}}\mathbf{\times }\mathbf{n}\\ \mathbf{2}\mathbf{.}\mathbf{3}\mathbf{\times }{\mathbf{10}}^4\mathbf{g}\mathbf{/}\mathbf{mole}\mathbf{=}\mathbf{192}\mathbf{g}\mathbf{/}\mathbf{mole}\mathbf{\times }\mathbf{n}\\ \mathbf{n}\mathbf{=}\frac{\mathbf{2}\mathbf{.}\mathbf{3}\mathbf{\times }{\mathbf{10}}^4\mathbf{g}\mathbf{/}\mathbf{mole}}{192g/\mathrm{mole}}\\ \mathbf{n}\mathbf{=}\mathbf{119}\mathbf{.}\mathbf{79}\mathbf{repeatunits}\end{array}$

 

The radius of gyration of the polymer ${\mathbf{C}}_{10}{\mathbf{H}}_8{\mathbf{O}}_4$ is:

$\begin{array}{l}{\mathbf{R}}_g\mathbf{=}{\left(\frac{{{\mathbf{nl}}_0}^2}{6}\right)}^{\frac{1}{2}}\\ {\mathbf{R}}_g\mathbf{=}{\mathbf{\left(}\frac{\mathbf{119}\mathbf{.}\mathbf{79}\mathbf{\times }\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{075}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{6}}\mathbf{\right)}}^{\frac{1}{2}}\\ {\mathbf{R}}_g\mathbf{=}{\mathbf{\left(}\frac{\mathbf{119}\mathbf{.}\mathbf{79}\mathbf{\times }\mathbf{1}\mathbf{.}\mathbf{156}}{\mathbf{6}}\mathbf{\right)}}^{\frac{1}{2}}\\ {\mathbf{R}}_g\mathbf{=}{\mathbf{\left(}\frac{\mathbf{138}\mathbf{.}\mathbf{48}}{\mathbf{6}}\mathbf{\right)}}^{\frac{1}{2}}\\ {\mathbf{R}}_g\mathbf{=}{(\mathbf{23}\mathbf{.}\mathbf{08})}^{\frac{1}{2}}\\ {\mathbf{R}}_g\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{80}\mathbf{pm}\end{array}$