Step 1: Understand the concept of cellular respiration. Cellular respiration is a metabolic process where cells convert glucose (\(\mathrm{C_6H_{12}O_6}\)) and oxygen (\(\mathrm{O_2}\)) into carbon dioxide (\(\mathrm{CO_2}\)), water (\(\mathrm{H_2O}\)), and energy (in the form of ATP). This process occurs in the mitochondria of eukaryotic cells.
Step 2: Analyze the given equations. The first equation represents the process of cellular respiration: \(\mathrm{C_6H_{12}O_6} + 6\,\mathrm{O_2} \rightarrow 6\,\mathrm{CO_2} + 6\,\mathrm{H_2O} + \text{energy}\). The second equation represents photosynthesis, which is the reverse of cellular respiration. The third equation represents the breakdown of hydrogen peroxide (\(\mathrm{H_2O_2}\)), and the fourth equation represents fermentation, a process that occurs in the absence of oxygen.
Step 3: Identify the correct equation for cellular respiration. Cellular respiration specifically involves the breakdown of glucose (\(\mathrm{C_6H_{12}O_6}\)) in the presence of oxygen (\(\mathrm{O_2}\)) to produce carbon dioxide (\(\mathrm{CO_2}\)), water (\(\mathrm{H_2O}\)), and energy. This matches the first equation: \(\mathrm{C_6H_{12}O_6} + 6\,\mathrm{O_2} \rightarrow 6\,\mathrm{CO_2} + 6\,\mathrm{H_2O} + \text{energy}\).
Step 4: Compare the other equations to cellular respiration. The second equation represents photosynthesis, which is not cellular respiration but rather its reverse. The third equation involves hydrogen peroxide decomposition, which is unrelated to cellular respiration. The fourth equation represents fermentation, which is an anaerobic process and does not involve oxygen.
Step 5: Conclude that the correct equation for cellular respiration is \(\mathrm{C_6H_{12}O_6} + 6\,\mathrm{O_2} \rightarrow 6\,\mathrm{CO_2} + 6\,\mathrm{H_2O} + \text{energy}\). This equation accurately describes the process of cellular respiration, where glucose and oxygen are converted into carbon dioxide, water, and energy.