Draw line-angle formulas for the three constitutional isomers with the molecular formula \(\mathrm{C}_{5} \mathrm{H}_{12}\).

Constitutional isomers, also known as structural isomers, are compounds that have the same molecular formula but differ in the way their atoms are connected to each other. This variation in structure can result in different physical and chemical properties even if the number of each type of atom is the same.

For example, let's consider the molecular formula \( \mathrm{C}_5 \mathrm{H}_{12} \). All constitutional isomers of this molecular formula will contain five carbon atoms and twelve hydrogen atoms, but the arrangement of these atoms can vary. The constitutional isomers of \( \mathrm{C}_5 \mathrm{H}_{12} \) are n-pentane with a straight chain, isopentane with a branched chain having a methyl group at the second carbon, and neopentane with a more compact structure where methyl groups are attached to both ends of a three-carbon chain and the central carbon.

Understanding constitutional isomers is important in organic chemistry because each isomer can have very different reactivity and properties. For example, they may have differing boiling points, solubilities, or chemical reactivities, which make them behave differently in biological systems or industrial processes.

The molecular formula \( \mathrm{C}_5 \mathrm{H}_{12} \) represents a class of organic compounds known collectively as pentanes. These are simple hydrocarbons, meaning they consist only of carbon (C) and hydrogen (H) atoms. The subscripted numbers indicate that there are five carbon atoms and twelve hydrogen atoms in each molecule of these compounds.

The significance of having a specific molecular formula like \( \mathrm{C}_5 \mathrm{H}_{12} \) lies in the fact that it allows chemists to determine potential structures for the compound. This formula indicates that they belong to the alkane family, which is characterized by single carbon-carbon bonds and the maximum number of hydrogen atoms per carbon. Despite having the same molecular formula, different structural arrangements of the atoms (constitutional isomers) can significantly influence the physical and chemical properties of a substance.

In the case of \( \mathrm{C}_5 \mathrm{H}_{12} \), the isomers include n-pentane, isopentane, and neopentane, which despite sharing the same formula, have different structural formulas hence applying the concept of constitutional isomers.

Organic chemistry is the branch of chemistry that studies the structure, properties, composition, reactions, and synthesis of carbon-containing compounds. This field of science is vital as it is the foundation for understanding the chemistry of life, since carbon is the building block of biological molecules.

The importance of organic chemistry is reflected in the fact that it covers the study of a vast number of compounds, including natural and synthetic substances. It spans the pharmaceuticals we take, the plastics we use, the fuels that power our vehicles, and the DNA in our cells. In the study of organic chemistry, understanding how different atoms can be arranged and rearranged to form molecules with different properties is fundamental. For example, the exploration of constitutional isomers in organic compounds, such as the ones seen with the molecular formula \( \mathrm{C}_5 \mathrm{H}_{12} \) helps in predicting physical and chemical behaviors of substances, which is crucial for applications in industries and research. Line-angle formulas are a visual and simplistic way to represent these molecules, offering an easier way to interpret and understand their structures.