Step 1: Understand the concept of cellular respiration. Cellular respiration is a metabolic process where glucose ($\mathrm{C_6H_{12}O_6}$) is broken down in the presence of oxygen ($\mathrm{O_2}$) to produce carbon dioxide ($\mathrm{CO_2}$), water ($\mathrm{H_2O}$), and energy in the form of ATP.
Step 2: Recall the overall chemical equation for cellular respiration. The general formula is: $\mathrm{C_6H_{12}O_6} + 6\ \mathrm{O_2} \rightarrow 6\ \mathrm{CO_2} + 6\ \mathrm{H_2O} + \text{energy}$. This represents the complete oxidation of glucose.
Step 3: Analyze the given options. Compare each option to the correct chemical equation for cellular respiration. Look for the presence of glucose ($\mathrm{C_6H_{12}O_6}$), oxygen ($\mathrm{O_2}$), carbon dioxide ($\mathrm{CO_2}$), water ($\mathrm{H_2O}$), and energy.
Step 4: Eliminate incorrect options. For example, the second option ($\mathrm{C_6H_{12}O_6} \rightarrow 2\ \mathrm{C_2H_5OH} + 2\ \mathrm{CO_2}$) represents fermentation, not cellular respiration. Similarly, the third option ($6\ \mathrm{CO_2} + 6\ \mathrm{H_2O} \rightarrow \mathrm{C_6H_{12}O_6} + 6\ \mathrm{O_2}$) is the reverse process, photosynthesis.
Step 5: Identify the correct representation. The correct equation for cellular respiration is $\mathrm{C_6H_{12}O_6} + 6\ \mathrm{O_2} \rightarrow 6\ \mathrm{CO_2} + 6\ \mathrm{H_2O} + \text{energy}$, as it accurately describes the breakdown of glucose in the presence of oxygen to produce carbon dioxide, water, and energy.