The radius of an atom of an element is 55 pm. What is the edge length of the unit cell if it is body-centred cubic?

Options

A. 144.6 pm
B. 163.4 pm
C. 127.0 pm
D. 123.5pm

Answer

To find the edge length of a body-centered cubic (BCC) unit cell given the radius of an atom, we can follow these steps:

### Step 1: Identify the relationship between the radius and edge length for BCC
In a body-centered cubic structure, the relationship between the atomic radius (R) and the edge length (a) is given by the formula:

\[ R = \frac{\sqrt{3}}{4} a \]

### Step 2: Rearrange the formula to solve for edge length (a)
To find the edge length (a), we can rearrange the formula:

\[ a = \frac{4R}{\sqrt{3}} \]

### Step 3: Substitute the given radius into the formula
We are given the radius (R) of the atom as 55 picometers (pm). Now we can substitute this value into the rearranged formula:

\[ a = \frac{4 \times 55 \, \text{pm}}{\sqrt{3}} \]

### Step 4: Calculate the edge length
Now we perform the calculation:

1. Calculate \(4 \times 55\):
\[ 4 \times 55 = 220 \]

2. Calculate \(\sqrt{3}\) (approximately 1.732):
\[ a = \frac{220}{1.732} \]

3. Finally, perform the division:
\[ a \approx 127.02 \, \text{pm} \]

### Step 5: Round the answer
Since we are dealing with picometers, we can round the answer to:

\[ a \approx 127 \, \text{pm} \]

### Conclusion
The edge length of the unit cell for the body-centered cubic structure is approximately **127 pm**.

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