Chapter 12.15, Problem 40AAP

A unidirectional carbon fiber–epoxy resin composite contains 68% by volume of carbon fiber and 32% epoxy resin. The density of the carbon fiber is 1.79 g/cm³ and that of the epoxy resin is 1.20 g/cm³. (a) What are the weight percentages of carbon fibers and epoxy resin in the composite? (b) What is the average density of the composite?

To determine

The weight percentage of carbon fibers in the composite.

The weight percentage of epoxy resin in the composite.

Answer to Problem 40AAP

The weight percentage of carbon fibers in the composite is $76\%$.

The weight percentage of epoxy resin in the composite is $24\%$.

Explanation of Solution

Write the expression to calculate mass of carbon fiber in the composite $\left(m_{cf}\right)$.

 $m_{cf}={\rho }_{cf}{V}_{cf}$                                                                                                               (I)

Here, density of carbon fiber is ${\rho }_{cf}$ and volume fraction of carbon fiber is $V_{cf}$.

Write the expression to calculate mass of epoxy resin in the composite $\left(m_{er}\right)$.

 $m_{er}={\rho }_{er}{V}_{er}$                                                                                                             (II)

Here, density of epoxy resin is ${\rho }_{er}$ and volume fraction of epoxy resin is $V_{er}$.

Write the expression to calculate total mass of the composite $\left(m\right)$.

 $m=m_{cf}+m_{er}$                                                                                                          (III)

Here, density of epoxy resin is ${\rho }_{er}$ and volume fraction of epoxy resin is $V_{er}$.

Write the expression to calculate weight percentage of carbon fiber in the composite $\left(\%W_{cf}\right)$.

 $\%W_{cf}=\frac{m_{cf}}{m}\times 100\%$                                                                                               (IV)

Write the expression to calculate weight percentage of epoxy resin in the composite $\left(\%W_{er}\right)$.

 $\%W_{er}=\frac{m_{er}}{m}\times 100\%$                                                                                                 (V)

Conclusion:

Substitute $1.79\text{g}/{\text{cm}}^{\text{3}}$ for ${\rho }_{cf}$ and $0.68$ for $V_{cf}$ in Equation (I).

 $\begin{array}{c}{m}_{cf}=\left(1.79\text{g}/{\text{cm}}^{\text{3}}\right)\left(0.68\right)\\ =1.217\text{g}\end{array}$

Substitute $1.20\text{g}/{\text{cm}}^{\text{3}}$ for ${\rho }_{er}$ and $0.32$ for $V_{er}$ in Equation (II).

 $\begin{array}{c}{m}_{er}=\left(1.20\text{g}/{\text{cm}}^{\text{3}}\right)\left(0.32\right)\\ =0.384\text{g}\end{array}$

Substitute $1.217\text{g}$ for $m_{cf}$ and $0.384\text{g}$ for $m_{er}$ in Equation (III).

 $\begin{array}{c}m=1.217\text{g}+0.384\text{g}\\ =1.601\text{g}\end{array}$

Substitute $1.217\text{g}$ for $m_{cf}$ and $1.601\text{g}$ for $m$ in Equation (IV).

 $\begin{array}{c}\%W_{cf}=\frac{1.217\text{g}}{1.601\text{g}}\times 100\%\\ =76\%\end{array}$

Substitute $0.384\text{g}$ for $m_{er}$ and $1.601\text{g}$ for $m$ in Equation (V).

 $\begin{array}{c}\%W_{er}=\frac{0.384\text{g}}{1.601\text{g}}\times 100\%\\ =24\%\end{array}$

Thus, the weight percentage of carbon fibers in the composite is $76\%$ and the weight percentage of epoxy resin in the composite is $24\%$.

To determine

The average density of the composite.

Answer to Problem 40AAP

The average density of the composite is $1.601\text{g}/{\text{cm}}^{\text{3}}$.

Explanation of Solution

Write the expression to calculate average density of the composite $\left({\rho }_c\right)$.

 ${\rho }_c=\frac{m}{V}$                                                                                                                   (VI)

Here, total volume of composite is $V$.

Conclusion:

Substitute $1.601\text{g}$ for $m$ and $1{\text{cm}}^{\text{3}}$ for $V$ in Equation (I).

 $\begin{array}{c}{\rho }_c=\frac{1.601\text{g}}{1{\text{cm}}^{\text{3}}}\\ =1.601\text{g}/{\text{cm}}^{\text{3}}\end{array}$

Thus, the average density of the composite is $1.601\text{g}/{\text{cm}}^{\text{3}}$.