The density of totally crystalline polypropylene at room temperature is \(0.946 \mathrm{~g} / \mathrm{cm}^{3}\). Also, at room temperature the unit cell for this material is monoclinic with the following lattice parameters: $$ \begin{array}{ll} a=0.666 \mathrm{~nm} & \alpha=90^{\circ} \\ b=2.078 \mathrm{~nm} & \beta=99.62^{\circ} \\ c=0.650 \mathrm{~nm} & \gamma=90^{\circ} \end{array} $$ If the volume of a monoclinic unit cell, \(V_{\text {mono }}\) is a function of these lattice parameters as $$ V_{\text {mono }}=a b c \sin \beta $$ determine the number of repeat units per unit cell.

By using the given formula for the volume of a monoclinic unit cell, \(V_{\text {mono }} = a b c \sin \beta\), we can calculate the volume. First, we need to convert the given lattice parameters from degrees to radians: $$\beta = 99.62^{\circ} = 99.62 \cdot \frac{\pi}{180} = 1.73814 \thinspace \mathrm{rad}$$The conversion is necessary since the sine function in most calculators uses radians. Next, plug in the values of \(a\), \(b\), \(c\), and \(\beta\) from the given lattice parameters:$$V_{\text {mono } } = (0.666 \mathrm {~nm}) \cdot (2.078 \mathrm {~nm}) \cdot (0.650 \mathrm {~nm}) \cdot \sin(1.73814 \mathrm {~rad})$$)

Since the density is given in \(g/cm^{3}\), it would be convenient to convert the volume from \(nm^{3}\) to \(cm^{3}\). One nanometer is equal to \(10^{-7} cm\). Therefore, we can multiply each dimension of the lattice by \(10^{-7}\) and proceed with the calculation:$$V_{\text {mono }} = (6.66 \cdot 10^{-8}\mathrm{~cm}) \cdot (2.078 \cdot 10^{-7}\mathrm{~cm}) \cdot (6.50 \cdot 10^{-8}\mathrm{~cm}) \cdot \sin(1.73814 \mathrm {~rad})$$$$V_{\text {mono }} = 1.012330 \cdot 10^{-21} \mathrm {~cm^{3}}$$Now, we have the calculated volume of the unit cell in \(cm^{3}\). )

Now, using the given density, calculate the mass of a unit cell:$$\rho = \frac{m}{V}$$$$m = \rho V$$$$m = (0.946\mathrm{~g/ cm^{3}})(1.012330 \cdot 10^{-21} \mathrm {~cm^{3}})$$$$m = 9.577788 \cdot 10^{-22} \mathrm{~g}$$)

The molecular formula of polypropylene is C\(_{3}\)H\(_{6}\). The molecular mass of polypropylene is \(3(12.01\mathrm {~g / mol}) + 6(1.008\mathrm { ~g / mol})= 42.08\mathrm {~g / mol}\). We can use Avogadro's number to find the number of molecules of polypropylene in a unit cell. To do that, we need to calculate the mass per repeat unit in grams:$$\frac{\text {mass per repeat unit}}{\text{one repeat unit}} = \frac{42.08 \mathrm { ~g / mol}}{6.022 \cdot 10^{23}\mathrm { ~units / mol}} = 6.98 \cdot 10^{-23} \mathrm{ ~g / unit}$$Now, to find the number of repeat units per unit cell, divide the mass of the unit cell calculated in Step 3 by the mass per repeat unit:$$\text {Repeat units per unit cell} = \frac{9.577788 \cdot 10^{-22} \mathrm { ~g}}{6.98 \cdot 10^{-23} \mathrm { ~g / unit}}$$$${\text {Repeat units per unit cell}} = 13.71$$Since there cannot be a fraction of a repeat unit, we round this number to the nearest whole number:$$\text {Repeat units per unit cell} = 14$$) Thus, there are 14 repeat units per unit cell for totally crystalline polypropylene at room temperature.