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Chapter 15: Problem 18

The tensile strength and number-average molecular weight for two polyethylene materials are as follows: \begin{tabular}{cc} \hline Tensile Strength (MPa) & Number-Average Molecular Weight \((\mathrm{g} / \mathbf{m o l})\) \\ \hline 85 & 12,700 \\ 150 & 28,500 \\ \hline \end{tabular} Estimate the number-average molecular weight that is required to give a tensile strength of \(195 \mathrm{MPa}\).

Short Answer

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Question: Estimate the number-average molecular weight required for a polyethylene material with a tensile strength of 195 MPa, given the following data points: - Data point 1: Tensile strength = 85 MPa and Number-Average Molecular Weight = 12,700 g/mol - Data point 2: Tensile strength = 150 MPa and Number-Average Molecular Weight = 28,500 g/mol Answer: The estimated number-average molecular weight required for a polyethylene material with a tensile strength of 195 MPa is approximately 39,160 g/mol.

Step by step solution

01

Identify the given data points

We are given two data points: - Data point 1: Tensile strength = \(85\ \mathrm{MPa}\) and Number-Average Molecular Weight = \(12,700\ \mathrm{g/mol}\) - Data point 2: Tensile strength = \(150\ \mathrm{MPa}\) and Number-Average Molecular Weight = \(28,500\ \mathrm{g/mol}\) Our objective is to find the number-average molecular weight that corresponds to a tensile strength of \(195\ \mathrm{MPa}\).
02

Use interpolation to find a relationship between the data points

Let's call the tensile strength 'x' and the number-average molecular weight 'y'. We will form a straight line between the two data points, and then use it to estimate the required molecular weight. To find the relationship between the data points, we can use the interpolation formula: \(\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}\) where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two given data points. In our case, \((x_1, y_1) = (85, 12,700)\) and \((x_2, y_2) = (150, 28,500)\).
03

Plug in the values and solve for the required molecular weight

Applying the interpolation formula with the given data points and the desired tensile strength: \(\frac{y - 12700}{195 - 85} = \frac{28,500 - 12,700}{150 - 85}\) Now we can solve for 'y': \(y - 12700 = \frac{(28,500 - 12,700)(195 - 85)}{150-85}\) \(y - 12700 = \frac{15,800 \cdot 110}{65}\) \(y - 12700 = 26,460\) Now, add 12,700 to both sides: \(y = 26,460 + 12,700\) \(y = 39,160\)
04

Conclusion

The estimated number-average molecular weight required for a polyethylene material with a tensile strength of \(195\ \mathrm{MPa}\) is approximately \(39,160\ \mathrm{g/mol}\).

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Most popular questions from this chapter

Which of the following polyethylene thin films would have the better mechanical characteristics: (1) formed by blowing, or (2) formed by extrusion and then rolled? Why?

Contrast compression, injection, and transfer molding techniques that are used to form plastic materials.

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