1. Identify the reactants and initial reaction. The starting material is an acidified solution of potassium chromate ($K_2CrO_4$). In an acidic medium, chromate ions ($CrO_4^{2-}$) are in equilibrium with dichromate ions ($Cr_2O_7^{2-}$). The solution will predominantly contain the orange-colored dichromate ion. $2 CrO_4^{2-} (yellow) + 2 H^+ \rightleftharpoons Cr_2O_7^{2-} (orange) + H_2O$

2. Reaction with Hydrogen Peroxide. When hydrogen peroxide ($H_2O_2$) is added to the acidified dichromate solution, a reaction occurs to form a deep blue colored compound, which is designated as 'X'. This is a characteristic test for chromium. The reaction is as follows: $Cr_2O_7^{2-} + 4 H_2O_2 + 2 H^+ \rightarrow 2 CrO_5 + 5 H_2O$ The compound 'X' is chromium(VI) oxide peroxide, with the chemical formula $CrO_5$.

3. Role of Amyl Alcohol. The blue compound $CrO_5$ is unstable in aqueous solution and decomposes rapidly. However, it is soluble in organic solvents like amyl alcohol or ether. Shaking the mixture causes the $CrO_5$ to be extracted into the amyl alcohol layer, where it is more stable, resulting in the observed blue alcohol layer.

4. Determine the structure of X ($CrO_5$). The question asks for the number of oxygen atoms bonded to chromium through single bonds in a molecule of X. To answer this, we need to know the structure of $CrO_5$.

• The oxidation state of chromium in $CrO_5$ is +6, the same as in chromate and dichromate.
• If all five oxygen atoms were oxide ions ($O^{2-}$), the oxidation state of chromium would have to be +10, which is not possible. This implies that some of the oxygen atoms are present as peroxide ions ($O_2^{2-}$), where each oxygen has an oxidation state of -1.
• The structure of $CrO_5$ is a butterfly-like (or peroxo) structure. The central chromium atom is bonded to one oxygen atom through a double bond (an oxo ligand) and to four other oxygen atoms through single bonds. These four oxygen atoms are part of two peroxide linkages ($–O–O–$).

The structure can be visualized as:

      O     //    Cr   /  \  O -- O  |    |  O -- O

More accurately depicted as:

       O      //    Cr   / \   O---O /     \ O-------O

This shows a central Cr atom, one doubly bonded O atom, and two peroxide groups ($O_2^{2-}$) chelating the Cr atom.

5. Count the required bonds. Let's analyze the bonds from the central chromium atom:

• There is one double bond between Chromium and one Oxygen atom ($Cr=O$).
• There are four single bonds between Chromium and the four other Oxygen atoms ($Cr-O$). These four oxygen atoms belong to the two peroxide groups.

The question asks for the number of oxygen atoms bonded to chromium through only single bonds. Based on the structure, there are 4 such oxygen atoms.

Therefore, the number of oxygen atoms bonded to chromium through only single bonds in a molecule of X ($CrO_5$) is 4.