Problem 16 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

You expend about \(100 \mathrm{kJ}\) a day keeping your heart beating. What is the minimum mass of glucose you must oxidize per day in order to produce this much energy? (Section 14.5 ) $$\begin{array}{r}\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(\mathrm{s})+6 \mathrm{O}_{2}(\mathrm{g}) \rightarrow 6 \mathrm{CO}_{2}(\mathrm{g})+6 \mathrm{H}_{2} \mathrm{O} / \\\\\Delta G^{0}=-2872 \mathrm{kJ} \mathrm{mol}^{-1}\end{array}$$

Step-by-Step Solution

Answer

You need at least 6.27 grams of glucose to produce 100 kJ of energy.

Step 1: Understanding the Reaction

The given reaction shows the oxidation of glucose: \( \text{C}_6\text{H}_{12}\text{O}_6 + 6\text{O}_2 \rightarrow 6\text{CO}_2 + 6\text{H}_2\text{O} \). This process releases energy, as reflected by the negative \( \Delta G^0 = -2872 \, \text{kJ/mol} \). This means that each mole of glucose releases 2872 kJ of energy.

Step 2: Calculate Moles of Glucose Needed

We need to find how many moles of glucose are required to produce 100 kJ of energy. Since each mole of glucose releases 2872 kJ, set up the equation: \[ \text{Number of moles} = \frac{100 \, \text{kJ}}{2872 \, \text{kJ/mol}} \] Calculate the moles: \[ \frac{100}{2872} \approx 0.0348 \, \text{mol} \]

Step 3: Calculate Mass of Glucose Needed

We will now convert the moles of glucose to mass. The molar mass of glucose \( \text{C}_6\text{H}_{12}\text{O}_6 \) can be calculated as follows: \[ 6(12.01) + 12(1.008) + 6(16.00) = 180.16 \, \text{g/mol} \]To find the mass, multiply the number of moles by the molar mass: \[ \text{Mass} = 0.0348 \, \text{mol} \times 180.16 \, \text{g/mol} \approx 6.27 \, \text{g} \]

Key Concepts

Energy Production in Metabolism

In the grand scheme of biology, metabolism is crucial for providing the energy that our cells need to operate efficiently. This energy is often derived from the breakdown of glucose, which is a type of carbohydrate. When glucose is oxidized, it releases energy that is harnessed by the body for various functions.

The chemical reaction provided, \( \text{C}_6\text{H}_{12}\text{O}_6 + 6\text{O}_2 \rightarrow 6\text{CO}_2 + 6\text{H}_2\text{O} \), describes this metabolic process. During this reaction, the glucose molecule combines with oxygen, leading to the production of carbon dioxide, water, and energy.

• The energy produced is indicated by the negative \( \Delta G^0 \) value, which is \(-2872 \, \text{kJ/mol}\).
• This tells us that for every mole of glucose oxidized, 2872 kJ of energy is released.

The negative sign of \( \Delta G^0 \) denotes that this is an exergonic reaction, meaning it proceeds with a net release of energy. This is why glucose is such an effective fuel for our bodies, efficiently powering everything from cellular repair to heartbeats.

Molar Mass Calculation

Calculating the molar mass of a substance like glucose is essential for understanding how much of it we need for certain reactions. Molar mass can be thought of as the weight of one mole of a substance, and it’s expressed in grams per mole (g/mol). For glucose, \( \text{C}_6\text{H}_{12}\text{O}_6 \), we identify the molar masses of its individual atoms:

• Carbon (C) has an atomic mass of 12.01 g/mol.
• Hydrogen (H) has an atomic mass of 1.008 g/mol.
• Oxygen (O) has an atomic mass of 16.00 g/mol.

To calculate glucose's molar mass, sum up the individual atomic masses based on their counts in the formula:
\[ 6(12.01) + 12(1.008) + 6(16.00) = 180.16 \, \text{g/mol} \]
This tells us that one mole of glucose weighs 180.16 grams. Molar mass is a foundational concept in chemistry, as it aids in converting moles to grams, a necessary step in many calculations of chemical reactions.

Moles and Mass Conversion

Moles and mass conversion is a fundamental aspect of chemistry that involves switching back and forth between chemical quantities in moles and their corresponding mass. Once we know how much energy is needed and the energy produced per mole, we convert that requirement into moles of reacting substances.

Starting with the need to produce 100 kJ of energy, we calculate the number of moles of glucose required using the relation:
\[ \text{Number of moles} = \frac{\text{Energy required in kJ}}{\text{Energy per mole in kJ/mol}} \]
Substituting in the values, \( \frac{100}{2872} \approx 0.0348 \, \text{mol} \). Thus, approximately 0.0348 moles of glucose are needed to produce 100 kJ of energy.

• This figure can then be converted to mass using the molar mass of glucose calculated previously.
• Using the formula: \( \text{Mass} = \text{Number of moles} \times \text{Molar mass} \).

For glucose:
\[ 0.0348 \, \text{mol} \times 180.16 \, \text{g/mol} \approx 6.27 \, \text{g} \]
So, to keep your heart beating while expending around 100 kJ a day, you would need to oxidize at least 6.27 grams of glucose. This conversion is especially vital in chemistry and biology for linking the microscopic world of atoms and molecules to the macroscopic world of laboratory measurements.