If `R` is the radius of the octahedral voids and `r` is the radius of the atom in close packing, then `r//R` is equal to

Options

A. `2.41`
B. `4.76`
C. `3.22`
D. `9.1`

Explanation

To solve the problem of finding the ratio \( \frac{r}{R} \) where \( R \) is the radius of the octahedral voids and \( r \) is the radius of the atom in close packing, we can follow these steps:

### Step 1: Understand the relationship between \( r \) and \( R \)

In a close-packed structure, the radius of the octahedral voids (\( R \)) and the radius of the atoms (\( r \)) have a specific relationship. The ratio \( \frac{R}{r} \) varies between 0.414 and 0.732 depending on the coordination and the type of packing.

### Step 2: Use the lower limit of the ratio

For our calculations, we will use the lower limit of the ratio, which is \( \frac{R}{r} = 0.414 \). This is a common approximation used in solid-state chemistry.

### Step 3: Rearrange the equation

We need to find \( \frac{r}{R} \). To do this, we can rearrange the equation:

\[ \frac{R}{r} = 0.414 \implies \frac{r}{R} = \frac{1}{0.414} \]

### Step 4: Calculate \( \frac{r}{R} \)

Now, we can calculate \( \frac{r}{R} \):

\[ \frac{r}{R} = \frac{1}{0.414} \approx 2.415 \]

### Step 5: Round the answer

Rounding \( 2.415 \) gives us \( 2.41 \).

### Final Answer

Thus, the ratio \( \frac{r}{R} \) is approximately \( 2.41 \).

### Conclusion

The answer to the question is option A: \( 2.41 \).

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