this problem is a bit more difficult than the other problems. the density and associated percent…

this problem is a bit more difficult than the other problems. the density and associated percent crystallinity for two polytetrafluoroethylene materials are as follows: \nρ(g/cm³) crystallinity (%) \n2.10 64.0 \n2.30 77.0 \n(a) compute the density of totally crystalline polytetrafluoroethylene. ρ_c = select g/cm³ \n(b) compute the density of totally amorphous polytetrafluoroethylene. ρ_a = select g/cm³ \n(c) determine the percent crystallinity of a specimen having a density of 2.20 g/cm³. % crystallinity = select

this problem is a bit more difficult than the other problems. the density and associated percent crystallinity for two polytetrafluoroethylene materials are as follows: \nρ(g/cm³) crystallinity (%) \n2.10 64.0 \n2.30 77.0 \n(a) compute the density of totally crystalline polytetrafluoroethylene. ρ_c = select g/cm³ \n(b) compute the density of totally amorphous polytetrafluoroethylene. ρ_a = select g/cm³ \n(c) determine the percent crystallinity of a specimen having a density of 2.20 g/cm³. % crystallinity = select

Answer

Explanation:

Step1: Recall the density - crystallinity relationship formula

The formula for percent crystallinity $X_c$ is $X_c=\frac{\rho_c(\rho-\rho_a)}{\rho(\rho_c - \rho_a)}\times100%$, where $\rho$ is the density of the semi - crystalline polymer, $\rho_c$ is the density of the totally crystalline polymer, and $\rho_a$ is the density of the totally amorphous polymer. We can rewrite it for two data points $(X_{c1},\rho_1)$ and $(X_{c2},\rho_2)$ as two equations: $X_{c1}=\frac{\rho_c(\rho_1-\rho_a)}{\rho_1(\rho_c - \rho_a)}\times100%$ and $X_{c2}=\frac{\rho_c(\rho_2-\rho_a)}{\rho_2(\rho_c - \rho_a)}\times100%$. Let $X_{c1} = 64.0%=0.64$, $\rho_1 = 2.10\ g/cm^3$, $X_{c2}=77.0% = 0.77$, $\rho_2 = 2.30\ g/cm^3$.

Step2: Substitute the values into the equations

$0.64=\frac{\rho_c(2.10-\rho_a)}{2.10(\rho_c - \rho_a)}$ and $0.77=\frac{\rho_c(2.30-\rho_a)}{2.30(\rho_c - \rho_a)}$. From $0.64=\frac{\rho_c(2.10-\rho_a)}{2.10(\rho_c - \rho_a)}$, we get $0.64\times2.10(\rho_c - \rho_a)=\rho_c(2.10-\rho_a)$, which simplifies to $1.344\rho_c-1.344\rho_a = 2.10\rho_c-\rho_c\rho_a$. From $0.77=\frac{\rho_c(2.30-\rho_a)}{2.30(\rho_c - \rho_a)}$, we get $0.77\times2.30(\rho_c - \rho_a)=\rho_c(2.30-\rho_a)$, which simplifies to $1.771\rho_c-1.771\rho_a = 2.30\rho_c-\rho_c\rho_a$. Subtract the first - derived equation from the second: $(1.771\rho_c-1.771\rho_a-(1.344\rho_c - 1.344\rho_a))=(2.30\rho_c-\rho_c\rho_a-(2.10\rho_c-\rho_c\rho_a))$. $1.771\rho_c-1.771\rho_a - 1.344\rho_c + 1.344\rho_a=2.30\rho_c-\rho_c\rho_a - 2.10\rho_c+\rho_c\rho_a$. $0.427\rho_c - 0.427\rho_a=0.20\rho_c$. $0.427\rho_c-0.20\rho_c = 0.427\rho_a$. $0.227\rho_c = 0.427\rho_a$, so $\rho_a=\frac{0.227}{0.427}\rho_c$.

Step3: Substitute $\rho_a$ back into the first percent - crystallinity equation

Substitute $\rho_a=\frac{0.227}{0.427}\rho_c$ into $0.64=\frac{\rho_c(2.10-\frac{0.227}{0.427}\rho_c)}{2.10(\rho_c - \frac{0.227}{0.427}\rho_c)}$. First, simplify the denominator: $\rho_c-\frac{0.227}{0.427}\rho_c=\rho_c(1 - \frac{0.227}{0.427})=\rho_c\frac{0.427 - 0.227}{0.427}=\rho_c\frac{0.2}{0.427}$. The numerator is $\rho_c(2.10-\frac{0.227}{0.427}\rho_c)$. $0.64=\frac{2.10-\frac{0.227}{0.427}\rho_c}{2.10\times\frac{0.2}{0.427}}$. $0.64\times2.10\times\frac{0.2}{0.427}=2.10-\frac{0.227}{0.427}\rho_c$. $0.64\times2.10\times\frac{0.2}{0.427}-2.10=-\frac{0.227}{0.427}\rho_c$. $\rho_c\approx2.40\ g/cm^3$.

Step4: Calculate $\rho_a$

Substitute $\rho_c = 2.40\ g/cm^3$ into $\rho_a=\frac{0.227}{0.427}\rho_c$. $\rho_a=\frac{0.227}{0.427}\times2.40\approx1.28\ g/cm^3$.

Step5: Calculate the percent crystallinity for $\rho = 2.20\ g/cm^3$

Use the formula $X_c=\frac{\rho_c(\rho-\rho_a)}{\rho(\rho_c - \rho_a)}\times100%$. Substitute $\rho_c = 2.40\ g/cm^3$, $\rho_a = 1.28\ g/cm^3$, and $\rho = 2.20\ g/cm^3$ into the formula: $X_c=\frac{2.40\times(2.20 - 1.28)}{2.20\times(2.40 - 1.28)}\times100%=\frac{2.40\times0.92}{2.20\times1.12}\times100%=\frac{2.208}{2.464}\times100%\approx89.6%$.

Answer:

(a) $2.40$ (b) $1.28$ (c) $89.6$