Step 1: Understand the process of cellular respiration. Cellular respiration is a metabolic pathway that converts glucose (\(\mathrm{C_6H_{12}O_6}\)) into usable energy in the form of ATP, with oxygen (\(\mathrm{O_2}\)) as the final electron acceptor. The byproducts of this process are carbon dioxide (\(\mathrm{CO_2}\)) and water (\(\mathrm{H_2O}\)).
Step 2: Analyze the given equations. The correct equation for cellular respiration should include glucose and oxygen as reactants, and carbon dioxide, water, and energy as products. This matches the overall reaction of cellular respiration: \(\mathrm{C_6H_{12}O_6} + 6\,\mathrm{O_2} \rightarrow 6\,\mathrm{CO_2} + 6\,\mathrm{H_2O} + \text{energy}\).
Step 3: Compare the other options. The second equation, \(\mathrm{C_6H_{12}O_6} \rightarrow 2\,\mathrm{C_2H_5OH} + 2\,\mathrm{CO_2}\), represents fermentation, not cellular respiration. The third equation, \$6\,\mathrm{CO_2} + 6\,\mathrm{H_2O} \rightarrow \mathrm{C_6H_{12}O_6} + 6\,\mathrm{O_2}\(, describes photosynthesis, which is the reverse of cellular respiration. The fourth equation, \)2\,\mathrm{H_2O} + 2\,\mathrm{CO_2} \rightarrow \mathrm{C_2H_4O_2} + 2\,\mathrm{O_2}$, does not represent a biological process related to cellular respiration.
Step 4: Confirm the correct equation. The first equation, \(\mathrm{C_6H_{12}O_6} + 6\,\mathrm{O_2} \rightarrow 6\,\mathrm{CO_2} + 6\,\mathrm{H_2O} + \text{energy}\), accurately summarizes the overall reaction of cellular respiration, as it includes the correct reactants and products.
Step 5: Conclude the reasoning. Cellular respiration is a catabolic process that breaks down glucose in the presence of oxygen to produce energy, carbon dioxide, and water. This matches the first equation provided in the problem.