Understanding the radius ratio is crucial when studying the structure of ionic crystals like NaCl. The radius ratio is defined as the ratio of the radius of the cation ($r_{+}$) to the radius of the anion ($r_{-}$). This value helps predict the coordination number and the overall crystal structure adopted by an ionic compound.

In solid state chemistry, the radius ratio rule is a guideline used to determine how ions pack together. Different ranges of radius ratio values correspond to different coordination numbers and arrangements of ions in the crystal lattice.

Radius Ratio and Crystal Structure in NaCl Crystal

The NaCl crystal structure is a face-centered cubic (FCC) lattice for both the cation (Na$^{+}$) and the anion (Cl$^{-}$), with one lattice offset relative to the other. In this structure, each cation is surrounded by six anions, and each anion is surrounded by six cations. This means both ions have a coordination number of 6. This arrangement places the smaller cation in the octahedral voids formed by the larger anions.

The theoretical range of radius ratio values that corresponds to a coordination number of 6 (octahedral coordination) is typically between 0.414 and 0.732. This range arises from geometric considerations of fitting a cation into an octahedral void formed by anions.

Calculating Radius Ratio for NaCl Crystal

To find the specific radius ratio for the NaCl crystal, we need the values of the ionic radii for Na$^{+}$ and Cl$^{-}$. Typical experimental values for ionic radii are approximately:

• Radius of Na$^{+}$ ($r_{+}$): 102 pm (picometers)
• Radius of Cl$^{-}$ ($r_{-}$): 181 pm

Using these values, the radius ratio is calculated as:

$\text{Radius Ratio} = \frac{r_{+}}{r_{-}} = \frac{102 \text{ pm}}{181 \text{ pm}}$

Calculating this value:

$\frac{102}{181} \approx 0.5635$

This calculated radius ratio of approximately 0.5635 falls within the expected radius ratio range for octahedral coordination (0.414 to 0.732), which is consistent with the known crystal structure of NaCl.

Comparing Calculated Radius Ratio to Options

Let's look at the given options and compare them to our calculated value:

• Option 1: 0.4 (Falls below the octahedral range, suggesting tetrahedral or lower coordination)
• Option 2: 0.98 (Falls within the cubic range, suggesting coordination number 8)
• Option 3: 1.0 (Suggests equal radii, possibly simple cubic or other high coordination)
• Option 4: 0.52 (Falls within the octahedral range of 0.414 - 0.732)

Our calculated value of ~0.5635 is closest to 0.52 among the given options and clearly falls within the correct radius ratio range for the NaCl crystal structure and its coordination number of 6. Minor variations in reported ionic radii values can lead to slight differences in the calculated radius ratio, but it will remain within the expected range for a specific structure type. The value 0.52 is often used as a representative value falling comfortably within the octahedral radius ratio range.

Therefore, the radius ratio in NaCl crystal that aligns with its known structure and coordination number is closest to 0.52 among the provided choices. This confirms why NaCl adopts an octahedral structure type in solid state chemistry.