Question: An organic chemist synthesizes a compound with molecular formula $ C_nH_{2n} $, where the ratio of carbon to hydrogen atoms satisfies $ \frac{a}{b} = 3 $ and their sum is $ a + b = 16 $. Find the value of $ a^2 - b^2 $.
["Title: Solving a Molecular Puzzle: Organic Chemist Determines Key Structural Ratios and Reacts to Compute $ a^2 - b^2 $", "In the intricate world of organic chemistry, precise molecular formulas and precise atomic ratios are essential for understanding molecular identity and reactivity. A recent scenario challenges an organic chemist to analyze a hypothetical compound with molecular formula $ C_nH_{2n} $, where the ratio of carbon atoms $ a $ to hydrogen atoms $ b $ satisfies two critical conditions:", "$$\n\frac{a}{b} = 3 \quad \ ext{and} \quad a + b = 16\n$$", "This seemingly straightforward mathematical problem unlocks deeper insight into molecular structure and bonding—particularly useful when computing derived quantities such as $ a^2 - b^2 $.", "---", "Step 1: Set up equations from given ratios", "We are given:\n$$\n\frac{a}{b} = 3 \Rightarrow a = 3b\n$$\n$$\na + b = 16\n$$", "Substitute $ a = 3b $ into the sum equation:\n$$\n3b + b = 16 \Rightarrow 4b = 16 \Rightarrow b = 4\n$$", "Then:\n$$\na = 3 \cdot 4 = 12\n$$", "So, the number of carbon atoms is $ a = 12 $, and hydrogen atoms is $ b = 4 $. This ratio $ 12:4 = 3:1 $ confirms the given condition.", "---", "Step 2: Compute $ a^2 - b^2 $", "Using the identity for the difference of squares:\n$$\na^2 - b^2 = (a - b)(a + b)\n$$", "We already know:\n- $ a + b = 16 $\n- $ a - b = 12 - 4 = 8 $", "Thus:\n$$\na^2 - b^2 = 8 \cdot 16 = 128\n$$", "---", "Step 3: Interpret the result in a chemical context", "While $ a^2 - b^2 $ is a purely mathematical expression, it holds relevance in organic synthesis when comparing degrees of unsaturation or assessing functional group ratios. Here, the difference $ a - b = 8 $ reflects a balance between carbon backbone length and hydrogen richness—key in predicting reactivity, stability, and synthesis pathways.", "For example, in hydrocarbons with formula $ C_nH_{2n} $, such a high $ a/b $ ratio indicates a nearly saturated linear chain with possible aromatic or conjugated systems—information critical for designing targeted reactions.", "---", "Conclusion", "Through algebraic precision and molecular reasoning, the organic chemist successfully determines that:\n$$\na = 12, \quad b = 4 \quad \Rightarrow \quad a^2 - b^2 = 128\n$$", "This elegant result bridges computational reasoning and chemical intuition, demonstrating how fundamental math underpins molecular discovery.", "---", "Keywords: organic chemistry, molecular formula $ C_nH_{2n} $, $ a/b = 3 $, $ a + b = 16 $, $ a^2 - b^2 $, stoichiometry, structural analysis, chemical synthesis."]
