This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Identify the given radii and determine the crystal structure. The radius of the cation A^+ (r_c) is 1.0 \, pm. The radius of the anion X^- (r_a) is 2.0 \, pm. Calculate the radius ratio (r_c)/(r_a): (r_c)/(r_a) = 1.0 \, pm2.0 \, pm = 0.5 For ionic compounds with a radius ratio in the range 0.414 (r_c)/(r_a) < 0.732, the coordination number is 6. This typically corresponds to an NaCl-type crystal structure, where the anions form a Face-Centered Cubic (FCC) lattice and the cations occupy the octahedral voids. Step 2: Determine the relationship between the unit cell edge length and the ionic radii for an NaCl-type structure. In an NaCl-type structure, the cation and anion touch along the edge of the unit cell. The edge length (a) is twice the sum of the cation and anion radii. a = 2(r_c + r_a) Step 3: Calculate the edge length (a) of the unit cell. a = 2(1.0 \, pm + 2.0 \, pm) a = 2(3.0 \, pm) a = 6.0 \, pm Step 4: Calculate the volume of the unit cell (V). The volume of a cubic unit cell is given by V = a^3. V = (6.0 \, pm)^3 V = 216 \, pm^3 Step 5: Compare the calculated volume with the given options. The calculated volume is 216 \, pm^3, which matches option (d). The final answer is d) 216 \, pm^3.