Question

Determine the relative size of \(\mathrm{Cs}\) atom compared to a \(\mathrm{Li}\) atom, if their densities are \(1.87\mathrm{\ g/cc}\) and \(0.53\mathrm{\ g/cc}\) respectively.

• A. \(r(\mathrm{Cs}) = 1.753\ r(\mathrm{Li})\)
• B. \(r(\mathrm{Cs}) = 1.936\ r(\mathrm{Li})\)
• C. \(r(\mathrm{Cs}) = 2.753\ r(\mathrm{Li})\)
• D. \(r(\mathrm{Cs}) = 2.936\ r(\mathrm{Li})\)

Hint:

Density \(\rho = \frac{\text{Atomic Mass}}{N_A \times \text{Volume}}\). Ratio accounts for both mass and volume.

Answer 1

The Correct Option is A

Step 1: Formula / Definition}
\[ \rho = \frac{M}{N_A \cdot \frac{4}{3}\pi r^3} \Rightarrow \rho \propto \frac{1}{r^3} \]
Step 2: Calculation / Simplification}
\(\frac{\rho_{\mathrm{Cs}}}{\rho_{\mathrm{Li}}} = \left(\frac{r_{\mathrm{Li}}}{r_{\mathrm{Cs}}}\right)^3 \cdot \frac{M_{\mathrm{Cs}}}{M_{\mathrm{Li}}}\)
\(\frac{1.87}{0.53} = \left(\frac{r_{\mathrm{Li}}}{r_{\mathrm{Cs}}}\right)^3 \times \frac{133}{7}\)
\(3.528 = \left(\frac{r_{\mathrm{Li}}}{r_{\mathrm{Cs}}}\right)^3 \times 19 \Rightarrow \left(\frac{r_{\mathrm{Li}}}{r_{\mathrm{Cs}}}\right)^3 = 0.1857\)
\(\frac{r_{\mathrm{Cs}}}{r_{\mathrm{Li}}} = (0.1857)^{-1/3} = 1.753\)
Step 3: Final Answer
\[ r(\mathrm{Cs}) = 1.753\ r(\mathrm{Li}) \]