The tensile strength and number-average molecular weight for two polyethylene materials are as follows: Estimate the number-average molecular weight that is required to give a tensile strength of \(140 \mathrm{MPa}\).

We are given the tensile strengths and number-average molecular weights for two polyethylene materials. Let's denote tensile strength as T (in MPa) and number-average molecular weight as M (in g/mol). Material 1: \(T_1 = 120 \mathrm{MPa}\) , \(M_1 = 50000 \mathrm{g/mol}\) Material 2: \(T_2 = 100 \mathrm{MPa}\) , \(M_2 = 40000 \mathrm{g/mol}\) We want to find the molecular weight required for a tensile strength of \(T_3 = 140 \mathrm{MPa}\).

The given data can be plotted as points \((M_1, T_1)\) and \((M_2, T_2)\) on a graph, with M on the x-axis and T on the y-axis. We want to find an equation of a straight line that best fits those points and can be used to predict the tensile strength for any given molecular weight. For a straight line, we can use the equation \(T = aM + b\), where a and b are constants. We can find the line of best fit by using the formula: \(a = \frac{(M_1 - \overline{M})(T_1 - \overline{T}) + (M_2 - \overline{M})(T_2 - \overline{T})}{(M_1 - \overline{M})^2 + (M_2 - \overline{M})^2}\), where \(\overline{M}\) and \(\overline{T}\) denote the average molecular weight and tensile strength, respectively.

First, we need to find the average molecular weight \(\overline{M}\) and the average tensile strength \(\overline{T}\): \(\overline{M} = \frac{M_1 + M_2}{2} = \frac{50000 + 40000}{2} = 45000 \mathrm{g/mol}\) \(\overline{T} = \frac{T_1 + T_2}{2} = \frac{120 + 100}{2} = 110 \mathrm{MPa}\) Now we can calculate the constant a using the formula given above: \(a = \frac{(50000 - 45000)(120 - 110) + (40000 - 45000)(100 - 110)}{(50000 - 45000)^2 + (40000 - 45000)^2} = \frac{5000(10) + (-5000)(-10)}{5000^2 + (-5000)^2} = \frac{1}{2}\) We can find the constant b by substituting any of the points (either Material 1 or Material 2) into the linear equation: \(120 = (50000 \times \frac{1}{2}) + b \Rightarrow b = -24500\) So the equation of the line is: \(T = \frac{1}{2} M - 24500\)

Now that we have the equation of the line, we can use it to estimate the molecular weight required for a tensile strength of \(140 \mathrm{MPa}\). Plug the desired tensile strength into the equation and solve for M: \(140 = \frac{1}{2} M - 24500\) \(M = \frac{140 + 24500}{\frac{1}{2}} = 49000 \mathrm{g/mol}\) The estimated number-average molecular weight required for a tensile strength of \(140 \mathrm{MPa}\) is approximately \(49000 \mathrm{g/mol}\).