Gold (atomic radius = 0.144nm) crystallises in a face centred unit cell. What is the length of the side of the cell ?

Explanation

To find the length of the side of the face-centered cubic (FCC) unit cell for gold, we can follow these steps:

### Step 1: Understand the relationship between atomic radius and edge length in FCC
In a face-centered cubic (FCC) unit cell, the relationship between the atomic radius (R) and the edge length (A) is given by the formula:
\[ R = \frac{A}{2\sqrt{2}} \]

### Step 2: Rearrange the formula to solve for A
To find the edge length (A), we can rearrange the formula:
\[ A = 2R\sqrt{2} \]

### Step 3: Substitute the given value of R
We know the atomic radius of gold (R) is 0.144 nm. Now we can substitute this value into the equation:
\[ A = 2 \times 0.144 \, \text{nm} \times \sqrt{2} \]

### Step 4: Calculate \(\sqrt{2}\)
The value of \(\sqrt{2}\) is approximately 1.414. Now we can substitute this value into the equation:
\[ A = 2 \times 0.144 \, \text{nm} \times 1.414 \]

### Step 5: Perform the multiplication
Now, we can calculate the value:
\[ A = 2 \times 0.144 \times 1.414 \]
\[ A = 2 \times 0.144 \times 1.414 \approx 0.407 \, \text{nm} \]

### Step 6: Final result
Thus, the length of the side of the cell (A) is approximately:
\[ A \approx 0.407 \, \text{nm} \]

### Summary
The length of the side of the face-centered cubic unit cell for gold is approximately 0.407 nm.

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