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Copper crystallises in fcc with a unit c...

Copper crystallises in fcc with a unit cell length of 361 pm. What is the radius of copper atom?

A

108 pm

B

128 pm

C

157 pm

D

181 pm

Text Solution

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To find the radius of a copper atom given that it crystallizes in a face-centered cubic (FCC) structure with a unit cell length of 361 pm, we can follow these steps: ### Step 1: Understand the FCC Structure In an FCC unit cell, atoms are located at each corner and the center of each face of the cube. Each corner atom is shared by eight unit cells, and each face-centered atom is shared by two unit cells. ### Step 2: Relate the Edge Length to the Atomic Radius In an FCC structure, the relationship between the edge length (a) and the atomic radius (r) can be derived from the geometry of the cube. The face diagonal of the cube can be expressed in terms of the atomic radius: - The face diagonal of the cube is equal to the sum of the diameters of the atoms along that diagonal. Since there are two atomic radii at the corners and one at the face center, we can write: \[ \text{Face diagonal} = 4r \] - The face diagonal can also be expressed in terms of the edge length (a) as: \[ \text{Face diagonal} = a\sqrt{2} \] ### Step 3: Set Up the Equation Equating the two expressions for the face diagonal gives us: \[ 4r = a\sqrt{2} \] ### Step 4: Solve for the Atomic Radius Rearranging the equation to solve for the radius (r): \[ r = \frac{a\sqrt{2}}{4} \] ### Step 5: Substitute the Given Edge Length Now, substitute the given edge length (a = 361 pm) into the equation: \[ r = \frac{361 \, \text{pm} \cdot \sqrt{2}}{4} \] ### Step 6: Calculate the Radius Calculating the value: 1. Calculate \(\sqrt{2} \approx 1.414\). 2. Substitute this value into the equation: \[ r = \frac{361 \, \text{pm} \cdot 1.414}{4} \] 3. Perform the multiplication: \[ r \approx \frac{511.074 \, \text{pm}}{4} \approx 127.769 \, \text{pm} \] Rounding this to three significant figures gives: \[ r \approx 128 \, \text{pm} \] ### Final Answer The radius of the copper atom is approximately **128 pm**. ---
To find the radius of a copper atom given that it crystallizes in a face-centered cubic (FCC) structure with a unit cell length of 361 pm, we can follow these steps: ### Step 1: Understand the FCC Structure In an FCC unit cell, atoms are located at each corner and the center of each face of the cube. Each corner atom is shared by eight unit cells, and each face-centered atom is shared by two unit cells. ### Step 2: Relate the Edge Length to the Atomic Radius In an FCC structure, the relationship between the edge length (a) and the atomic radius (r) can be derived from the geometry of the cube. The face diagonal of the cube can be expressed in terms of the atomic radius: ...
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