Understanding Isomers

Isomers are different compounds that have the same molecular formula but different structural formulas. To find the isomer of a given compound, we first need to determine its molecular formula and then find which of the options has the same molecular formula but a different arrangement of atoms.

Molecular Formula of Diethyl Ether

Diethyl ether has the structure $(C_2H_5)_2O$. Let's calculate its molecular formula:

• Carbon atoms: $2 \times 2 = 4$
• Hydrogen atoms: $5 \times 2 = 10$
• Oxygen atoms: $1$

The molecular formula of diethyl ether is $C_4H_{10}O$. Now, we need to check the molecular formula of each given option.

Analyzing Options for Isomers

Let's analyze the molecular formula for each of the provided options:

1. $(CH_3)_3C-OH$ (tert-butyl alcohol or 2-methylpropan-2-ol)

   This molecule has a central carbon atom bonded to three methyl ($CH_3$) groups and one hydroxyl ($OH$) group.

   • Carbon atoms: $1$ (central) $+ 3 \times 1$ ($CH_3$ groups) $= 4$
   • Hydrogen atoms: $3 \times 3$ ($CH_3$ groups) $+ 1$ ($OH$ group) $= 9 + 1 = 10$
   • Oxygen atoms: $1$ ($OH$ group) $= 1$

   Molecular formula: $C_4H_{10}O$. This molecular formula is the same as diethyl ether.

2. $(CH_3)_2CHOH$ (isopropyl alcohol or propan-2-ol)

   This molecule has a central carbon atom bonded to two methyl ($CH_3$) groups, one hydrogen ($H$) atom, and one hydroxyl ($OH$) group.

   • Carbon atoms: $1$ (central) $+ 2 \times 1$ ($CH_3$ groups) $= 3$
   • Hydrogen atoms: $3 \times 2$ ($CH_3$ groups) $+ 1$ ($CH$ group) $+ 1$ ($OH$ group) $= 6 + 1 + 1 = 8$
   • Oxygen atoms: $1$ ($OH$ group) $= 1$

   Molecular formula: $C_3H_8O$. This is different from $C_4H_{10}O$.

3. $(C_2H_5)_2CHOH$

   This structural representation is unusual in standard chemistry notation and implies a carbon atom bonded to two ethyl groups and one hydroxyl group, requiring 5 bonds around the central carbon, which is not possible. However, if interpreted literally based on the sum of atoms:

   • Carbon atoms: $2 \times 2$ ($C_2H_5$ groups) $+ 1$ ($CH$ group) $= 4 + 1 = 5$
   • Hydrogen atoms: $5 \times 2$ ($C_2H_5$ groups) $+ 1$ ($CH$ group) $+ 1$ ($OH$ group) $= 10 + 1 + 1 = 12$
   • Oxygen atoms: $1$ ($OH$ group) $= 1$

   Molecular formula: $C_5H_{12}O$. This is different from $C_4H_{10}O$.

Identifying the Correct Isomer

We found that $(CH_3)_3C-OH$ has the same molecular formula ($C_4H_{10}O$) as diethyl ether but a different structure. Therefore, $(CH_3)_3C-OH$ is an isomer of diethyl ether.

The other options, $(CH_3)_2CHOH$ and $(C_2H_5)_2CHOH$, have different molecular formulas ($C_3H_8O$ and $C_5H_{12}O$ respectively), so they are not isomers of diethyl ether.

Thus, the isomer of diethyl ether among the given options is $(CH_3)_3C-OH$.