Question

When one molecule of glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\), is metabolized in our bodies, it combines with six \(\mathrm{O}_{2}\) molecules to form six \(\mathrm{CO}_{2}\) molecules and six \(\mathrm{H}_{2} \mathrm{O}\) molecules. (a) Write a balanced equation for this reaction. (b) How many \(\mathrm{CO}_{2}\) molecules are formed when 12 glucose molecules react? (c) If \(30 \mathrm{O}_{2}\) molecules react, how many glucose molecules also react?

Short answer

a) The balanced equation for this reaction is \(C_{6} H_{12} O_{6} + 6O_{2} \rightarrow 6CO_{2} + 6H_{2} O\). b) \(72 CO_{2}\) molecules are formed when 12 glucose molecules react. c) \(5 glucose\) molecules react when 30 \(O_{2}\) molecules react.

Step-by-step solution

Writing the Balanced Equation

The chemical equation for the metabolism of glucose is not balanced initially - unbalanced equation is \(C_{6} H_{12} O_{6} + O_{2} \rightarrow CO_{2} + H_{2} O\). A balanced chemical equation ensures that the same amount of atoms are present on both sides of the equation. This is in accordance with the law of conservation of mass. To achieve this, use the following steps: a) Balance the carbon atoms first. There are 6 carbon atoms in a glucose molecule so put a 6 in front of the \(CO_{2}\) on the right side of the equation. b) Balance the hydrogen atoms. There are 12 hydrogen atoms in a glucose molecule so put a 6 in front of the \(H_{2}O\) on the right side of the equation. c) Finally balance the oxygen atoms. There are already 12 oxygen atoms on the right side and 6 oxygen atoms in a glucose molecule on the left side. Hence, put a 6 in front of the \(O_{2}\) molecule. The balanced chemical equation is \(C_{6} H_{12} O_{6} + 6O_{2} \rightarrow 6CO_{2} + 6H_{2} O\)

Calculate the number of CO2 molecules

Given that 12 glucose molecules react, you are asked to determine how many \(CO_{2}\) molecules are formed. From the balanced chemical equation in Step 1, it is observed that 1 molecule of glucose forms 6 molecules of \(CO_{2}\). Therefore, if 12 glucose molecules react, the amount of \(CO_{2}\) produced would be \(12 \times 6 = 72\) molecules.

Calculate the number of glucose molecules

You are given that 30 \(O_{2}\) molecules react and asked to find how many glucose molecules also react.From the balanced equation in Step 1, it can be observed that 1 molecule of glucose reacts with 6 molecules of \(O_{2}\). Therefore, if 30 \(O_{2}\) molecules reacted, the number of glucose molecules reacted would be \(30/6 = 5\) molecules.

Key concepts

Glucose Metabolism

Glucose metabolism is a vital process in our bodies, where glucose is broken down to release energy. This energy is crucial for cellular functions and bodily activities. Essentially, glucose, which is a simple sugar with the formula \( \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \), reacts with oxygen \( (\mathrm{O}_{2}) \) to produce carbon dioxide \( (\mathrm{CO}_{2}) \), water \( (\mathrm{H}_{2} \mathrm{O}) \), and energy.
This reaction occurs through a series of complex biochemical steps, primarily in the mitochondria of cells.

• This process is central in maintaining the energy currency of the cell, ATP (adenosine triphosphate).
• Glucose metabolism involves pathways like glycolysis, the Krebs cycle, and oxidative phosphorylation.
• Effective metabolism ensures that our muscles, brain, and other organs function properly.

Glucose metabolism is also known as cellular respiration, and it provides the energy needed for various cellular activities and maintains homeostasis. It's fascinating to note that while all cells use glucose, different tissues may use varying amounts or store glucose differently, such as the liver and muscles storing excess glucose as glycogen.

Balancing Chemical Equations

Balancing chemical equations is essential to ensure that the same number of each type of atom appears on both sides of the equation. This respects the law of conservation of mass, stating that matter cannot be created nor destroyed in a chemical reaction.To balance the equation for glucose metabolism, follow these steps:

• Start by balancing the carbon atoms. With 6 carbon atoms in glucose, you need 6 \( \mathrm{CO}_{2} \) to balance the carbon.
• Next, balance the hydrogen atoms. With 12 hydrogen atoms in glucose, you need 6 \( \mathrm{H}_{2} \mathrm{O} \) molecules to provide 12 hydrogen atoms.
• Lastly, count the oxygen atoms to balance them. You have 6 oxygen atoms in glucose, and the 6 \( \mathrm{CO}_{2} \) and 6 \( \mathrm{H}_{2} \mathrm{O} \) provide a total of 18 oxygen atoms. Therefore, you need 6 molecules of \( \mathrm{O}_{2} \) to account for the 12 additional oxygen need.

The balanced equation is: \( \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} + 6 \mathrm{O}_{2} \rightarrow 6 \mathrm{CO}_{2} + 6 \mathrm{H}_{2} \mathrm{O} \). Balancing is crucial because it ensures that reactions conserve atoms and reflects the true stoichiometric relationships in a reaction.

Stoichiometry

Stoichiometry is a branch of chemistry that deals with the quantitative relationships of the reactants and products in a chemical reaction. Using stoichiometry, one can predict how much product will form from given amounts of reactants, or vice versa.To illustrate stoichiometry in the context of glucose metabolism:Consider the balanced equation \( \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} + 6 \mathrm{O}_{2} \rightarrow 6 \mathrm{CO}_{2} + 6 \mathrm{H}_{2} \mathrm{O} \). This equation tells us the stoichiometric coefficients - the numbers in front of molecules - which dictate the mole ratio of reactants and products.

• If 12 molecules of glucose react, we multiply the stoichiometric ratio by 12: resulting in 72 molecules of \( \mathrm{CO}_{2} \) being produced, because for every 1 glucose, 6 \( \mathrm{CO}_{2} \) are formed.
• Similarly, if you have 30 molecules of \( \mathrm{O}_{2} \), you would solve for glucose by dividing the molecules of \( \mathrm{O}_{2} \) by 6 (the amount needed per glucose), yielding 5 glucose molecules processed.

Stoichiometry is a powerful tool that helps chemists conserve resources, predict yields, and understand material transformations in everything from metabolic pathways to industrial scale processes.