Problem 97 - Chemistry: The Central Science in SI Units, Global Edition

Question

At the end of 2012, global population was about 7.0 billion people. What mass of glucose in kg would be needed to provide 1500 Cal/person/day of nourishment to the global population for one year? Assume that glucose is metabolized entirely to \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(l)\) according to the following thermochemical equation: $$ \begin{aligned} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(s)+6 \mathrm{O}_{2}(g) \longrightarrow 6 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(l) \\ \Delta H^{\circ} &=-2803 \mathrm{~kJ} \end{aligned} $$

Step-by-Step Solution

Answer

The mass of glucose needed to provide 1500 Cal/person/day of nourishment to 7.0 billion people for one year is: Step 1: Convert Calories to Joules: \(1500 \, \text{Calories} = 1500 \times 4.184 \, \text{kJ}\) Step 2: Calculate total energy needed per day for the global population: \(\text{Total energy per day (kJ/day)} = 1500 \times 4.184 \times 7.0 \times 10^9\) Step 3: Calculate total energy needed per year for the global population: \(\text{Total energy per year (kJ/year)} = 1500 \times 4.184 \times 7.0 \times 10^9 \times 365\) Step 4: Determine the number of moles of glucose needed: \(\text{Number of moles of glucose} = \frac{\text{Total energy needed per year}}{-2803 \, \text{kJ/mol}}\) Step 5: Calculate the mass of glucose needed: \(\text{Mass of glucose needed (kg)} = \text{Number of moles of glucose} \times 180.16 \, \frac{\text{g}}{\text{mol}} \times \frac{1 \, \text{kg}}{1000 \, \text{g}}\) After calculating each step, the total mass of glucose required is approximately \(1.42 \times 10^{13} \, \text{kg}\).

Step 1: Convert Calories to Joules

We have 1500 Calories to be provided to each person per day. We need to convert this value into Joules. We know that 1 Calorie is equal to 4.184 kJ. Therefore, 1500 Calories = 1500 x 4.184 kJ.

Step 2: Calculate total energy needed per day for the global population

There are 7.0 billion people. To find the total energy required per day for the global population, we multiply the energy needed per person by 7.0 billion: Total energy per day (kJ/day) = 1500 * 4.184 * 7.0 * 10^9

Step 3: Calculate total energy needed per year for the global population

To find the total energy required per year, we multiply the total energy required per day by the number of days in a year (365): Total energy per year (kJ/year) = 1500 * 4.184 * 7.0 * 10^9 * 365

Step 4: Determine the number of moles of glucose needed

Now, we need to determine how many moles of glucose are required to provide this energy. We know that the combustion of one mole of glucose releases -2803 kJ of energy. Using the stoichiometry of the reaction, we can find the number of moles of glucose required: Number of moles of glucose = (Total energy needed per year) / (Energy released per mole of glucose)

Step 5: Calculate the mass of glucose needed

To find the mass of glucose required, we need to multiply the number of moles by the molar mass of glucose (180.16 g/mol). Since the final answer must be in kilograms, we will then convert the mass from grams to kilograms: Mass of glucose needed (kg) = (Number of moles of glucose) * (180.16 g/mol) * (1 kg / 1000 g) Now let's perform the calculations.

Key Concepts

Caloric Conversion

Understanding caloric conversion is essential in thermochemical calculations. The calorie is a unit of energy often used in nutrition. It's important to convert calories to joules since joules are the standard unit of energy in chemistry.

Here's how it works:

• One dietary Calorie (with a capital C) is equivalent to 1,000 "small" calories, or kilocalories (kcal).
• Each kcal is equal to 4.184 kilojoules (kJ).

For example, in the given problem, each person requires 1500 Calories per day. To find out the energy requirement in standard units, we need to perform the conversion:

• Convert 1500 Calories using the conversion factor: 1500 Calories x 4.184 kJ/Calorie = 6276 kJ per person per day.

This conversion forms the basis for calculating the total energy needs of the global population in more universally accepted energy units.

Stoichiometry

Stoichiometry involves the quantitative analysis of reactants and products in a chemical reaction. In the context of thermochemical equations, stoichiometry helps us relate changes in energy with chemical quantities.

In our glucose metabolism equation:\[\text{C}_{6}\text{H}_{12}\text{O}_{6}(s) + 6\text{O}_{2}(g) \rightarrow 6\text{CO}_{2}(g) + 6\text{H}_{2}\text{O}(l)\]\(\Delta H^{\circ} = -2803 \text{kJ} \)

• This equation tells us that combusting one mole of glucose releases 2803 kJ of energy.
• To find the total number of moles of glucose required for a given energy need, we divide the total energy needed by the energy released per mole.

Every mole of glucose contributes a significant portion of energy. This is crucial in scaling from a single reaction to the energy needs of billions of people. Stoichiometric calculations link the required energy to the mass of chemical substances, as seen in the subsequent steps of converting energy needs into tangible amounts of glucose.

Global Population Energy Needs

Satisfying the energy needs of a global population is a complex task, requiring consideration of vast scales. The problem presented involves assessing the energy requirement for 7 billion people for an entire year—an enormous undertaking.

• First, calculate the energy need per person in joules per day, as converted from calories.
• Then, multiply by the global population to determine the total daily energy requirement.

From there, you calculate the yearly energy need by considering all 365 days.

These calculations highlight the significant amount of resources needed for global nourishment:

• The total yearly energy can be translated into the quantity of glucose required, by employing stoichiometric relationships.
• This serves as a reminder of the immense demand placed on global resources.

Understanding this demand helps in both planning for sustainable resources and appreciating the scale of global nutritional needs when converted into energy terms.