Problem 86 - Chemistry and Chemical Reactivity

Question

living organisms derive energy from the oxidation of food, typified by glucose. $$\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(\mathrm{aq})+6 \mathrm{O}_{2}(\mathrm{g}) \longrightarrow 6 \mathrm{CO}_{2}(\mathrm{g})+6 \mathrm{H}_{2} \mathrm{O}(\ell)$$ Electrons in this redox process are transferred from glucose to oxygen in a series of at least 25 steps. It is instructive to calculate the total daily current flow in a typical organism and the rate of energy expenditure (power). (See T.P. Chirpich: Journal of Chemical Education, Vol. 52 , p. \(99,1975 .)\) (a) The molar enthalpy of combustion of glucose is \(-2800 \mathrm{kJ} .\) If you are on a typical daily diet of \(2400 \mathrm{Cal}\) (kilocalories), what amount of glucose (in moles) must be consumed in a day if glucose is the only source of energy? What amount of \(\mathrm{O}_{2}\) must be consumed in the oxidation process? (b) How many moles of electrons must be supplied to reduce the amount of \(\mathrm{O}_{2}\) calculated in part (a)? (c) Based on the answer in part (b), calculate the current flowing, per second, in your body from the combustion of glucose. (d) If the average standard potential in the electron transport chain is \(1.0 \mathrm{V},\) what is the rate of energy expenditure in watts?

Step-by-Step Solution

Answer

a) 3.586 moles of glucose, 21.516 moles of O2; b) 86.064 moles of electrons; c) 96.12 A; d) 96.12 W.

Step 1: Convert Calories to Kilojoules

First, note that 1 Calorie (kcal) is equivalent to 4.184 kJ. Given the daily diet is 2400 Cal, convert this energy into kilojoules:\[ 2400 \text{ Cal} \times 4.184 \text{ kJ/Cal} = 10041.6 \text{ kJ} \]

Step 2: Calculate Moles of Glucose Consumed

Using the molar enthalpy of combustion of glucose, which is -2800 kJ/mol, determine the amount of glucose in moles needed to produce 10041.6 kJ.\[ \frac{10041.6 \text{ kJ}}{2800 \text{ kJ/mol}} = 3.586 \text{ moles} \]

Step 3: Determine Amount of O2 Needed

From the balanced chemical equation, 1 mole of glucose requires 6 moles of oxygen. Therefore, calculate the moles of \( \mathrm{O}_2 \):\[ 3.586 \text{ moles of glucose} \times 6 \text{ moles of } \mathrm{O}_2/\text{mol of glucose} = 21.516 \text{ moles of } \mathrm{O}_2 \]

Step 4: Calculate Moles of Electrons Needed

In the reaction, each molecule of \( \mathrm{O}_2 \) consumes 4 electrons. Thus, calculate the total electrons required for 21.516 moles of \( \mathrm{O}_2 \):\[ 21.516 \text{ moles of } \mathrm{O}_2 \times 4 \text{ moles of electrons/mole of } \mathrm{O}_2 = 86.064 \text{ moles of electrons} \]

Step 5: Calculate Current Flow

Using the fact that 1 mole of electrons corresponds to a charge of 96500 Coulombs (Faraday's constant), find the total charge transferred and divide by the number of seconds in a day (86400 s) to get current:\[ 86.064 \text{ moles of electrons} \times 96500 \text{ C/mol} = 8304180 \text{ C} \]\[ \frac{8304180 \text{ C}}{86400 \text{ s}} = 96.12 \text{ A} \]

Step 6: Calculate Power in Watts

Using the current calculated and the average standard potential of 1.0 V, calculate power (P = IV):\[ P = 96.12 \text{ A} \times 1.0 \text{ V} = 96.12 \text{ W} \]

Key Concepts

Redox Reactions

Redox reactions are integral to the combustion of glucose. In these reactions, electrons are transferred from one substance to another. In the case of glucose oxidation, glucose (\(\text{C}_6\text{H}_{12}\text{O}_6\)) donates electrons to oxygen, transforming into carbon dioxide (\(\text{CO}_2\)) and water (\(\text{H}_2\text{O}\)). This electron transfer is crucial for breaking down glucose to release energy.

In this process, glucose is oxidized as it loses electrons, and oxygen is reduced as it gains electrons. Think of oxidation as losing electrons and reduction as gaining them. It's like an exchange that provides the energy required for life processes.

• Glucose loses electrons – it is oxidized.
• Oxygen gains electrons – it is reduced.

This redox reaction is the foundation for energy production in cells, driving many other biological processes.

Electron Transport Chain

The electron transport chain (ETC) is a series of protein complexes in the mitochondria. During glucose metabolism, electrons are transferred through these complexes. Think of the ETC as a relay race where electrons are handed off from one complex to the next.

The movement of electrons in the ETC creates a flow of protons across the mitochondrial membrane. This generates a proton gradient, much like a battery charging up. The energy from this gradient is used to produce ATP, the energy currency of cells.

• Electrons move through proteins in the mitochondria.
• A proton gradient is formed, driving ATP synthesis.
• ATP provides energy for cellular activities.

The ETC is essential for energy conversion, transforming the energy from glucose into a usable form for the body.

Molar Enthalpy of Combustion

The molar enthalpy of combustion of glucose is a measure of energy released when one mole of glucose is burned completely. In the context of metabolism, this energy is harnessed for life functions. Here, it is given as \(-2800\, \text{kJ/mol}\).

Understanding this value helps us calculate how much glucose is needed for our daily energy requirements. If your diet provides 2400 kilocalories, you can convert this to kilojoules to find out how much glucose you need.

• Convert calories to kilojoules to relate diet energy to glucose.
• Calculate moles of glucose with the formula: \(\frac{\text{total kJ}}{2800 \text{kJ/mol}}\).

This calculation shows how energy from food is quantified and managed in metabolic processes.

Energy Conversion in Metabolism

Metabolism transforms food into energy. Let's break down how this happens with glucose. When you eat, glucose is metabolized through several steps, eventually leading to energy production.

First, glucose undergoes glycolysis. Then, in the presence of oxygen, it enters the citric acid cycle, and finally, the electron transport chain. Through these steps, glucose is converted into carbon dioxide, water, and ATP.

• Glucose is broken down for maximum energy release.
• Processes like glycolysis, citric acid cycle, and ETC play roles.
• The end result is ATP, which fuels body functions.

This coordinated sequence ensures efficient energy conversion from dietary glucose, impacting everything from cellular repair to muscle contraction.