The density of totally crystalline nylon 6,6 at room temperature is \(1.213 \mathrm{~g} / \mathrm{cm}^{3} .\) Also, at room temperature the unit cell for this material is triclinic with the following lattice parameters: $$ \begin{array}{ll} If the volume of a triclinic unit cell, is a function of these lattice parameters as $$ V_{\mathrm{tri}}=a b c \sqrt{1-\cos ^{2} \alpha-\cos ^{2} \beta-\cos ^{2} \gamma+}{2 \cos \alpha \cos \beta \cos \gamma} $$ determine the number of repeat units per unit cell. a=0.497 \mathrm{~nm} & \alpha=48.4^{\circ} \\ b=0.547 \mathrm{~nm} & \beta=76.6^{\circ} \\ c=1.729 \mathrm{~nm} & \gamma=62.5^{\circ} \end{array} $$

First, we need to find the volume of the triclinic unit cell using the given formula and lattice parameters: $$ V_{\mathrm{tri}}=a b c \sqrt{1-\cos ^{2} \alpha-\cos ^{2} \beta-\cos ^{2} \gamma+2 \cos \alpha \cos \beta \cos \gamma} $$ Plug in the given values for the lattice parameters: $$ V_{\mathrm{tri}}=(0.497)(0.547)(1.729) \sqrt{1-\cos ^{2} 48.4^{\circ}-\cos ^{2} 76.6^{\circ}-\cos ^{2} 62.5^{\circ} +2 \cos 48.4^{\circ} \cos 76.6^{\circ} \cos 62.5^{\circ}} $$ Compute the volume: $$ V_{\mathrm{tri}} \approx 0.474 \mathrm{~nm}^{3} $$

To find the volume of a single nylon 6,6 repeat unit, we will use the given density and the molar mass of nylon 6,6. The molar mass of nylon 6,6 is \(226.32 \mathrm{~g/mol}\) (12 carbon atoms, 22 hydrogen atoms, 2 nitrogen atoms, and 2 oxygen atoms). First, convert the density to \(\mathrm{g/nm}^3\): $$ 1.213 \frac{\mathrm{g}}{\mathrm{cm}^3} \times \frac{1 \mathrm{~nm}^3}{1\times10^{-21} \mathrm{~cm}^3} = 1.213 \times 10^3 \frac{\mathrm{g}}{\mathrm{~nm}^3} $$ Now determine the volume of one nylon 6,6 repeat unit: $$ \frac{1 \mathrm{mol}}{N_A} \times \frac{226.32 \mathrm{~g}}{1 \mathrm{mol}} \times \frac{1 \mathrm{~nm}^3}{1.213 \times 10^3 \mathrm{~g}} \approx 1.839 \times 10^{-2} \mathrm{~nm}^3 $$ Where \(N_A\) is Avogadro's number.

Now, we can find the number of repeat units per unit cell by dividing the total unit cell volume by the volume of a single repeat unit: $$ \frac{0.474 \mathrm{~nm}^{3}}{1.839\times10^{-2} \mathrm{~nm}^3} \approx 25.79 $$ However, the number of repeat units must be an integer. Therefore, we round the result to the nearest whole number: $$ \mathrm{Number~of~repeat~units} \approx 26 $$ Hence, there are approximately 26 repeat units per unit cell of totally crystalline nylon 6,6 at room temperature.