Use shorthand line notation (see Exercise 125\()\) to illustrate the structural isomers of \(\mathrm{C}_{7} \mathrm{H}_{16}\) .

First, we need to draw the longest possible continuous chain of carbon atoms for C7H16. We can start with a straight chain of 7 carbon atoms with 16 hydrogen atoms attached.

Now, we will create isomers with one less carbon atom in the main chain by branching one carbon: 1st isomer: -Remove a carbon atom from the straight chain and attach it as a side chain to the second carbon atom from one end, creating a 6-membered main chain with a 1-carbon side chain. 2nd isomer: -Attach the side chain to the third carbon atom from one end, forming a different isomer with a 6-membered main chain and a 1-carbon side chain. We can't create more isomers with a 6-membered main chain, as attaching the side chain to the fourth carbon atom would create a mirror image of the second isomer we drew.

Now, we will create isomers with 5 carbon atoms in the main chain by branching more carbon atoms: 3rd isomer: -Remove two carbon atoms from the straight chain and attach them individually as side chains to the second carbon atom from one end, creating a 5-membered main chain with two 1-carbon side chains. 4th isomer: -Attach one of the side chains to the third carbon atom, forming another isomer with a 5-membered main chain and two 1-carbon side chains. 5th isomer: -Remove another carbon atom from the main chain and attach both of the side chains to the same carbon atom, creating a 5-membered main chain with a 2-carbon side chain and a 1-carbon side chain. We can't create more isomers while maintaining a 5-membered main chain without replicating a previous structure.

Finally, we will create isomers with 4 carbon atoms in the main chain by branching even more carbon atoms: 6th isomer: -Remove three carbon atoms from the straight chain and attach them individually as side chains, creating a 4-membered main chain with three 1-carbon side chains. We can't create more isomers while maintaining a 4-membered main chain without replicating a previous structure.

In total, we have found 6 structural isomers for C7H16: 1. Straight chain with 7 carbon atoms 2. 6-membered main chain with a 1-carbon side chain on the 2nd carbon atom 3. 6-membered main chain with a 1-carbon side chain on the 3rd carbon atom 4. 5-membered main chain with two 1-carbon side chains on the 2nd carbon atom 5. 5-membered main chain with 1-carbon side chains on the 2nd and 3rd carbon atoms 6. 4-membered main chain with three 1-carbon side chains These are all the structural isomers for C7H16 that can be represented using shorthand line notation.