Question

The tensile strength and number-average molecular weight for two poly(methyl methacrylate) materials are as follows: Tensile Strength (MPa) Number-Average Molecular Weight (g/mol) 36 30,000 150 50,000 Estimate the tensile strength at a number-average molecular weight of 46,000 g/mol. MPa

Answer 1

Answer:

The required tensile strength is 175.13 MPa.

Explanation:

Given that,

Tensile strength(MPa)                Number Average Molecular weight(g/mol)

       36                                                               30,000

      150                                                               50,000

First we need to find out  $TS_\infty$tensile strength at infinity molecular weight.

The formula of tensile strength

$TS=TS_\infty-\frac{A}{ M_n}$

TS is tensile strength,$TS_\infty$ at infinity molecular weight, $M_n$ is molecular weight and A is constant.

Now substituting $TS=36$ and $M_n=30,000$

$36=TS_\infty-\frac{A}{30,000}$

$\Rightarrow TS_\infty=36+\frac{A}{30,000}$  ........(1)

Now substituting $TS=150$ and $M_n=50,000$

$150=TS_\infty-\frac{A}{50,000}$

$\Rightarrow TS_\infty=150+\frac{A}{50,000}$ ......(2)

Subtract (1) from (2)

$TS_\infty-TS_\infty=150+\frac{A}{50,000}-36-\frac{A}{30,000}$

$\Rightarrow150+\frac{A}{50,000}-36-\frac{A}{30,000}=0$

$\Rightarrow \frac{A}{30,000}-\frac{A}{50,000}=150-36$

$\Rightarrow \frac{5A-3A}{1,50,000}=114$

$\Rightarrow \frac{2A}{1,50,000}=114$

$\Rightarrow A=\frac{1,50,000 \times114}{2}$

$\Rightarrow A=8,550,000$

Putting the value of A in equation (1)

$TS_\infty=36+\frac{8,550,000}{30,000}$

        =361 MPa

Now to find the tensile strength at a number average molecular weight of 46,000 g/mol , Putting the value $TS_\infty$, TS and A

$\therefore TS=361-\frac{8,550,000}{ 46,000}$

         = 175.13 MPa

The required tensile strength is 175.13 MPa.