How many \(Cr-O\) bonds in dichromate ions are of the same bond length and are in resonance?

6

The correct answer is option 2. 6.

Let us delve into the structure and bonding of the dichromate ion \(\text{Cr}_2\text{O}_7^{2-}\) to understand why there are 6 \( \text{Cr-O} \) bonds with the same bond length due to resonance.

Structure of the Dichromate Ion (\(\text{Cr}_2\text{O}_7^{2-}\)):

Composition: The dichromate ion consists of two chromium atoms (\(\text{Cr}\)). There are a total of 7 oxygen atoms. The ion has a structure where the two \(\text{Cr}\) atoms are connected by a common oxygen atom (a bridging oxygen).

Bonding: Each chromium atom is bonded to four oxygen atoms:

Two terminal oxygen atoms with double bonds (\( \text{Cr=O} \)). One oxygen atom shared between the two chromium atoms (bridging oxygen). One oxygen atom connected by a single bond to each chromium atom, but these are involved in resonance.

Resonance and Bond Lengths:

Terminal \( \text{Cr=O} \) Bonds: These are double bonds and do not participate in resonance, so their bond lengths are shorter and distinct.

Bridging \( \text{Cr-O} \) Bonds: The oxygen atom between the two chromium atoms is shared, and this bond is involved in resonance. This bond is identical for both chromium atoms due to resonance, making the bond length equal.

Remaining \( \text{Cr-O} \) Bonds: The other oxygen atoms (not terminal) are also involved in resonance, distributing the bonding characteristics between single and partial double bonds. These bonds also become equivalent in length due to resonance.

Counting Equivalent Bonds:

Bridging Bonds: There are 2 Cr-O bonds that are bridging between the two chromium atoms.

Remaining Non-Terminal \( \text{Cr-O} \) Bonds: Each chromium atom has 2 non-terminal Cr-O bonds that are identical due to resonance. Since there are two \(\text{Cr}\) atoms, these account for 4 bonds in total.

Total Equivalent Bonds:

Bridging Cr-O bonds: 2

Non-terminal Cr-O bonds: 4

Total number of equivalent Cr-O bonds due to resonance = 2 (bridging) + 4 (non-terminal) = 6.

Conclusion:

In the dichromate ion, 6 of the Cr-O bonds are of the same length and are in resonance.

Thus, the correct answer is 6.