Two ionic solids `AB` and `CB` crystallize in the same lattice. If `r_(A^(o+))//r_(B^(ɵ))` and `r_(C^(o+))//r_(B^(ɵ))` are `0.50` and `0.70`, respectively, then the ratio of edge length of `AB` and `CD` is

Options

A. `0.68`
B. `0.78`
C. `0.88`
D. `0.98`

Explanation

To solve the problem, we need to find the ratio of the edge lengths of the ionic solids \(AB\) and \(CB\) given the ratios of the ionic radii. Let's break down the solution step by step.

### Step 1: Write down the given ratios
We have the following ratios provided in the question:
1. \(\frac{r_{A^{+}}}{r_{B^{-}}} = 0.50\)
2. \(\frac{r_{C^{+}}}{r_{B^{-}}} = 0.70\)

### Step 2: Express the ratios in terms of \(r_{B^{-}}\)
From the first ratio, we can express \(r_{A^{+}}\):
\[ r_{A^{+}} = 0.50 \cdot r_{B^{-}} \]
From the second ratio, we can express \(r_{C^{+}}\):
\[ r_{C^{+}} = 0.70 \cdot r_{B^{-}} \]

### Step 3: Add 1 to both sides of the ratios
Now, we will add 1 to both sides of each ratio:
1. For \(AB\):
\[ \frac{r_{A^{+}} + r_{B^{-}}}{r_{B^{-}}} = 1 + 0.50 = 1.50 \]
This gives us:
\[ r_{A^{+}} + r_{B^{-}} = 1.50 \cdot r_{B^{-}} \]

2. For \(CB\):
\[ \frac{r_{C^{+}} + r_{B^{-}}}{r_{B^{-}}} = 1 + 0.70 = 1.70 \]
This gives us:
\[ r_{C^{+}} + r_{B^{-}} = 1.70 \cdot r_{B^{-}} \]

### Step 4: Set up the equations
From the above steps, we have:
1. \(r_{A^{+}} + r_{B^{-}} = 1.50 \cdot r_{B^{-}}\) (Equation 1)
2. \(r_{C^{+}} + r_{B^{-}} = 1.70 \cdot r_{B^{-}}\) (Equation 2)

### Step 5: Divide the two equations
To find the ratio of the edge lengths, we can divide Equation 1 by Equation 2:
\[ \frac{r_{A^{+}} + r_{B^{-}}}{r_{C^{+}} + r_{B^{-}}} = \frac{1.50 \cdot r_{B^{-}}}{1.70 \cdot r_{B^{-}}} \]
This simplifies to:
\[ \frac{r_{A^{+}} + r_{B^{-}}}{r_{C^{+}} + r_{B^{-}}} = \frac{1.50}{1.70} \]

### Step 6: Simplify the ratio
Now we can simplify \(\frac{1.50}{1.70}\):
\[ \frac{1.50}{1.70} = \frac{15}{17} \approx 0.882 \]

### Step 7: Conclusion
Thus, the ratio of the edge lengths of \(AB\) to \(CB\) is approximately:
\[ \frac{l_{AB}}{l_{CB}} = \frac{15}{17} \approx 0.882 \]

### Final Answer
The ratio of the edge lengths of \(AB\) and \(CB\) is \( \frac{15}{17} \) or approximately \(0.88\).

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