Let's address your questions one by one: --- ### **Question 1: Skeletal Structure** **Molecule Given:** ``` CH3 | CH3—N—C—CH3 || O ``` #### **How to Draw the Skeletal Structure ("Line" Structure):** - In skeletal structures, **hydrogens attached to carbons are not shown**, and **carbon atoms are represented by the ends or bends of lines**. - **Heteroatoms** (like N, O) and their directly attached hydrogens are shown. - The molecule is **N,N-dimethylacetamide** (DMA). **Skeletal structure:** ``` N / | \ C | C=O \ / (CH3) ``` Or, drawn simply (lines for C-C and C-N bonds, explicit N and O): ``` N / \ CH3 C=O \ / CH3 ``` Where the lines represent C-N and C-C bonds, and the O is double-bonded to the central carbon. --- ### **Question 2: Mg Atom Radius Calculation** Given: - Atoms are spherical in shape. - Only **74.0%** of the space inside the cube is taken up by Mg atoms (this is the packing efficiency for **face-centered cubic (FCC)** unit cell). - Need to **calculate the radius (r) of an Mg atom in picometers**. #### **Step-by-Step Solution:** 1. **Let the edge length of the cube = a** 2. **Volume of cube = \( a^3 \)** 3. **Volume occupied by atoms = 74.0% of \( a^3 \)** \[ \text{Volume occupied by atoms} = 0.740 \times a^3 \] 4. **In FCC, 4 atoms per unit cell**. 5. **Volume of 4 atoms (spheres):** \[ 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3 \] 6. **Equate occupied volume to actual atom volume:** \[ 0.740 \, a^3 = \frac{16}{3} \pi r^3 \] 7. **Relationship between r and a in FCC:** - In FCC, \( a = 2\sqrt{2}r \), so \( r = \frac{a}{2\sqrt{2}} \) 8. **Plug \( r \) in terms of \( a \) into volume equation:** \[ 0.740 \, a^3 = \frac{16}{3} \pi \left( \frac{a}{2\sqrt{2}} \right)^3 \] \[ 0.740 \, a^3 = \frac{16}{3} \pi \left( \frac{a^3}{8 \cdot 2\sqrt{2}} \right) \] \[ 0.740 \, a^3 = \frac{16}{3} \pi \cdot \frac{a^3}{16\sqrt{2}} \] \[ 0.740 \, a^3 = \frac{1}{3} \pi \cdot \frac{a^3}{\sqrt{2}} \] \[ 0.740 = \frac{\pi}{3\sqrt{2}} \] (This confirms the 74% packing.) **If you have the value of 'a' (unit cell edge length),** you can directly calculate the atomic radius: \[ r = \frac{a}{2\sqrt{2}} \] If no value for \( a \) is given, this is the formula you would use **once you have the unit cell edge length** (usually given in pm or Å). --- **Summary:** - **Skeletal structure**: see drawing above. - **Mg atom radius**: \( r = \frac{a}{2\sqrt{2}} \) (where \( a \) is the unit cell length). Let me know if you have the value for \( a \), and I can plug in the numbers for you!