The question asks about the number of Cr-O bonds in the dichromate ion that have the same bond length and are involved in resonance. To answer this, we need to understand the structure of the dichromate ion, $\text{Cr}_2\text{O}_7^{2-}$.

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

The dichromate ion consists of two chromium atoms, seven oxygen atoms, and carries a $-2$ charge. Its structure can be viewed as two $\text{CrO}_4$ tetrahedral units sharing one oxygen atom. The two chromium atoms are linked through a bridging oxygen atom.

• Each chromium atom is bonded to four oxygen atoms.
• One oxygen atom acts as a bridge between the two chromium atoms (bridging oxygen).
• The remaining six oxygen atoms are bonded to only one chromium atom each (terminal oxygen atoms).

The structure can be represented as $\text{O}_3\text{Cr} - \text{O} - \text{CrO}_3^{2-}$. There are two Cr-O-Cr bridging bonds and six Cr-O terminal bonds.

Resonance in Dichromate Ion

The negative charge in the dichromate ion is delocalized over the terminal oxygen atoms. This delocalization occurs through resonance involving the $\pi$ system. Resonance structures can be drawn where the double bond character shifts between the chromium atom and the terminal oxygen atoms.

Consider one of the $\text{CrO}_3$ parts attached to the bridging oxygen. The chromium atom is bonded to three terminal oxygen atoms. Due to resonance, the double bond character is shared among these three terminal Cr-O bonds. This happens for both chromium atoms.

Bond Lengths in Dichromate Ion

Because of resonance, the terminal Cr-O bonds in the dichromate ion become equivalent in terms of bond length and bond order. The negative charges and $\pi$ bonding are delocalized across the chromium and the terminal oxygen atoms.

• There are six terminal Cr-O bonds.
• These six terminal bonds participate in resonance.
• Due to resonance, these six terminal Cr-O bonds have the same bond length.

The two bridging Cr-O bonds (connecting the two chromium atoms) are different. They do not participate in the same type of resonance as the terminal bonds and typically have a longer bond length than the terminal Cr-O bonds.

Therefore, the Cr-O bonds that are of the same bond length and are in resonance are the six terminal Cr-O bonds.

Summary of Cr-O Bonds

Based on the structure and resonance, there are 6 terminal Cr-O bonds that are equivalent in length and are involved in resonance stabilization of the negative charge.

Revision Table: Dichromate Ion Structure and Bonds

Additional Information: Chromium Oxoanions

Chromium forms several oxoanions, with chromate ($\text{CrO}_4^{2-}$) and dichromate ($\text{Cr}_2\text{O}_7^{2-}$) being the most common ones. These ions are interconvertible depending on the pH of the solution.

• In acidic solution, chromate ions ($\text{CrO}_4^{2-}$) are converted to dichromate ions ($\text{Cr}_2\text{O}_7^{2-}$). The reaction is $2\text{CrO}_4^{2-} \text{(aq)} + 2\text{H}^+ \text{(aq)} \rightleftharpoons \text{Cr}_2\text{O}_7^{2-} \text{(aq)} + \text{H}_2\text{O (l)}$.
• In basic solution, dichromate ions ($\text{Cr}_2\text{O}_7^{2-}$) are converted back to chromate ions ($\text{CrO}_4^{2-}$). The reaction is $\text{Cr}_2\text{O}_7^{2-} \text{(aq)} + 2\text{OH}^- \text{(aq)} \rightleftharpoons 2\text{CrO}_4^{2-} \text{(aq)} + \text{H}_2\text{O (l)}$.
• In the chromate ion ($\text{CrO}_4^{2-}$), all four Cr-O bonds are equivalent due to resonance and have the same bond length.
• Dichromate ions are strong oxidizing agents, commonly used in titrations.