1. We need the number of different chain isomers of $\mathrm{C_7H_{16}}$.

2. $\mathrm{C_7H_{16}}$ is heptane, an alkane with 7 carbon atoms.

3. Chain isomers differ in the arrangement of the carbon skeleton. Let us list all possible distinct carbon chains.

(i) Straight chain

1. n-Heptane

(ii) One methyl branch: hexane backbone

Now place one methyl group on a 6-carbon chain.

2. 2-Methylhexane
3. 3-Methylhexane

(Positions 4 and 5 are same as 3 and 2 by numbering from the other end.)

(iii) Two methyl branches: pentane backbone

Now place two methyl groups on a 5-carbon chain.

4. 2,2-Dimethylpentane
5. 2,3-Dimethylpentane
6. 2,4-Dimethylpentane
7. 3,3-Dimethylpentane

(iv) One ethyl branch: pentane backbone

8. 3-Ethylpentane

(2-Ethylpentane is not distinct; its longest chain becomes 6 carbons, so it is actually 3-methylhexane.)

(v) Three methyl branches: butane backbone

9. 2,2,3-Trimethylbutane

10. Thus, total number of chain isomers of $\mathrm{C_7H_{16}}$ is $9$

11. Therefore, the required integer answer is: $\boxed{9}$

12. Comparison with stored correct answer:

• Stored correct answer = $9$
• Derived answer = $9$
• They agree.