Question

{"given":"Copper crystallizes in face-centered cubic (fcc) structure with unit cell edge length a = 361 pm. We need to find the atomic radius of copper. ","key_observation":"In an fcc structure, atoms touch each other along the face diagonal of the unit cell. The face diagonal equals $\\sqrt{2}a$ and contains exactly 4 atomic radii (2 radii from corner atoms and 2 radii from the face-centered atom). Therefore, the relationship is $\\sqrt{2}a = 4r$, where r is the atomic radius.","option_analysis":[{"label":"(A)","text":"108 pm","verdict":"incorrect","explanation":"This value is too small. Using the fcc relationship $r = \\frac{\\sqrt{2}a}{4} = \\frac{\\sqrt{2} \\times 361}{4} = \\frac{510.45}{4} = 127.6$ pm, not 108 pm."},{"label":"(B)","text":"127 pm","verdict":"correct","explanation":"This is correct. Using the formula $r = \\frac{\\sqrt{2}a}{4} = \\frac{\\sqrt{2} \\times 361}{4} = \\frac{361 \\times 1.414}{4} = \\frac{510.45}{4} = 127.6$ pm, which rounds to 127 pm."},{"label":"(C)","text":"157 pm","verdict":"incorrect","explanation":"This value is too large. The correct calculation gives 127 pm, not 157 pm. This might result from using an incorrect crystal structure relationship."},{"label":"(D)","text":"181 pm","verdict":"incorrect","explanation":"This value is significantly too large. It's almost 1.5 times the correct answer of 127 pm and doesn't follow from the fcc structure relationship."}],"answer":"(B)","formula_steps":[]}