Step 1: Understand the concept of cellular respiration. Cellular respiration is the process by which 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.
Step 2: Recall the general chemical equation for cellular respiration. The correct equation 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 breakdown of glucose in the presence of oxygen.
Step 3: Compare the given options to the correct equation. Look for the equation that matches the inputs (glucose and oxygen) and outputs (carbon dioxide, water, and energy).
Step 4: Eliminate incorrect options. For example, the equation $\mathrm{C_6H_{12}O_6} \rightarrow 2\,\mathrm{C_2H_5OH} + 2\,\mathrm{CO_2}$ represents fermentation, not cellular respiration. Similarly, $\mathrm{C_6H_{12}O_6} + 6\,\mathrm{CO_2} \rightarrow 6\,\mathrm{O_2} + 6\,\mathrm{H_2O}$ is incorrect because it reverses the process.
Step 5: Identify the correct equation. 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 represents the process of glucose oxidation in the presence of oxygen.