Question

Question: At room temperature, monoclinic sulfur has the unit-cell dimensions $\mathbf{a}=\mathbf{11}.\mathbf{04}\mathbf{\backslash }\mathbf{AA},\mathbf{b}=\mathbf{10}.\mathbf{98}\mathbf{\backslash }\mathbf{AA},\mathbf{c}=\mathbf{10}.\mathbf{92}\mathbf{\backslash }\mathbf{AA},and\mathbf{\beta }\mathbf{=}\mathbf{96}\mathbf{.}\mathbf{73}\mathbf{°}$ . Each cell contains 48 atoms of sulfur.

(a) Explain why it is not necessary to give the values of the angles $\mathit{\alpha }\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{\gamma }$ in this cell.

(b) Compute the density of monoclinic sulfur (in gcm^-3 ).

Short answer

Answer

1. $\mathbf{a}\ne \mathbf{b}\ne \mathbf{c}\mathbf{,} \mathbf{\alpha }\mathbf{=}\mathbf{\gamma }\mathbf{=}\mathbf{90}\mathbf{°}\ne \mathbf{\beta }$
2. Density is $1.94 g.c{m}^{-3}$

Step-by-step solution

Monoclinic crystal

It is one of the seven crystal systems. In this crystal, $\mathbf{a}\ne \mathbf{b}\ne \mathbf{c}\mathbf{,} \mathbf{\alpha }\mathbf{=}\mathbf{\gamma }\mathbf{=}\mathbf{90}\mathbf{°}\ne \mathbf{\beta }$

Where, a,b,c represent the edge lengths and are the angles.

Subpart (a)

In monoclinic crystal,

$\mathbf{\alpha }\mathbf{=}\mathbf{\gamma }\mathbf{=}\mathbf{90}\mathbf{°}\ne \mathbf{\beta }$

Hence, $\mathbf{\alpha } \mathbf{and} \mathbf{\gamma }$ values are not required to provide and they both are equal to 90⁰ .

Subpart (b)

Density is the ratio of mass to volume.

Volume of monoclinic crystal is:

$$\begin{gathered} V=abc \sin \beta \\ =\left(11.04\times {10}^{-8} cm\right)\times \left(10.98\times {10}^{-8} cm\right)\times \left(10.92\times {10}^{-8} cm\right)\times \sin \left(96.73°\right) \\ =1314.6\times {10}^{-24} c{m}^3 \end{gathered}$$

Mass is calculated as shown below:

$$\begin{gathered} mass=\frac{n\times Molar mass}{N} \\ =\frac{48 atoms\times 32.0 g/mol}{6.023\times {10}^{23} mol/atoms} \\ =255\times {10}^{-23} g \end{gathered}$$

Density calculation:

$$\begin{gathered} density=\frac{mass}{volume} \\ =\frac{255\times {10}^{-23} g}{1314.6\times {10}^{-24}c{m}^3} \\ =1.94 g.c{m}^{-3} \end{gathered}$$