Question

At the end of 2009 , global population was about 6.8 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: \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(s)+6 \mathrm{O}_{2}(g) \longrightarrow 6 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(l)\) \(\Delta H^{\mathrm{o}}=-2803 \mathrm{~kJ}\)

Short answer

Approximately 1.00 x 10^12 kg of glucose would be needed to provide 1500 Calories of nourishment per day to the global population of 6.8 billion people for one year.

Step-by-step solution

Convert Calories to energy in kilojoules (kJ)

Given the energy requirement per person per day is 1500 Cal, we will convert Calories to kilojoules (kJ) since we have energy value in kJ for metabolizing glucose. Use the conversion factor 1 Cal = 4.184 kJ Energy per person per day in kJ = (1500 Cal)(4.184 kJ/Cal) = 6276 kJ

Calculate glucose needed (in moles) per person per day

We know that glucose's metabolism yields \(\Delta H^{\mathrm{o}} = -2803 \mathrm{~kJ}\), which is energy released. As we are looking for the glucose required, we must use the absolute value of this value. Energy released per mole of glucose = 2803 kJ Now we can calculate the moles of glucose required per person per day: Moles of glucose per person per day = Energy per person per day / Energy released per mole of glucose = 6276 kJ / 2803 kJ/mol ≈ 2.24 mol

Calculate mass of glucose needed per person per day

In this step, we will find out the mass of glucose required per person per day. Molar mass of glucose: (6×12.01(C) + 12×1.01(H) + 6×16.00(O)) g/mol = 180.18 g/mol Mass of glucose per person per day = Moles of glucose per person per day × Molar mass of glucose = 2.24 mol × 180.18 g/mol ≈ 403.2 g

Calculate total mass of glucose needed for global population for one year

Now, we know the mass of glucose required per person per day, we will multiply this by the global population (6.8 billion) and the number of days in a year (365): Total mass of glucose = Mass of glucose per person per day × Global population × Days in a year Total mass of glucose = 403.2 g × 6.8 × 10^9 people × 365 days Total mass of glucose = 1,001,573,760,000 kg Therefore, approximately 1.00 x 10^12 kg of glucose would be needed to provide 1500 Calories of nourishment per day to the global population of 6.8 billion people for one year.

Key concepts

Thermochemical Equations

Thermochemical equations are an essential part of understanding the energetics of chemical reactions. These equations not only indicate the reactants and products, but also the amount of heat (energy) absorbed or released during the reaction. For example, the provided exercise involves the combustion of glucose, which is represented by the equation

\( \mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6}(s) + 6\mathrm{O}_{2}(g) \longrightarrow 6\mathrm{CO}_{2}(g) + 6\mathrm{H}_{2}\mathrm{O}(l) \)
Followed by the energy change \( \Delta H^{\mathrm{o}} = -2803 \mathrm{~kJ} \),
which indicates that 2803 kJ of energy is released when one mole of glucose is metabolized. Heat is mentioned with a negative sign because it is exothermic (heat is being released). When solving related problems, we use the absolute value of this heat change to calculate the amount of substance needed to release or absorb a specific amount of energy.

Mole Concept

The mole concept is fundamental in chemistry, as it is the bridge between the microscopic world of atoms and molecules and the macroscopic world we can measure. A mole is defined as the amount of substance that contains as many entities (atoms, ions, molecules, etc.) as there are atoms in 12 grams of carbon-12. This number is known as Avogadro's number, approximately \( 6.022 \times 10^{23} \).

In the exercise, we calculate the moles of glucose needed per person per day to estimate the mass of glucose required for the world's population. By equating the energy needed by a person to the energy released by a mole of glucose, the mole concept enables us to convert between energy and mass.

Molar Mass Calculation

Molar mass is the mass of one mole of a substance, normally expressed in grams per mole (g/mol). It's a vital constant that converts moles to grams. To calculate molar mass, we sum the product of the number of atoms of each element in a compound and their respective atomic masses. For instance, glucose \( \mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6} \) has a molecular formula that includes carbon (C), hydrogen (H), and oxygen (O) atoms. As completed in the exercise solution, the molar mass of glucose is calculated as
\( (6 \times 12.01\mathrm{(C)} + 12 \times 1.01\mathrm{(H)} + 6 \times 16.00\mathrm{(O)}) \mathrm{g/mol} = 180.18 \mathrm{g/mol} \).
This step is crucial because it translates the calculated moles of glucose required for energy into a physical quantity of glucose measured in grams.

Energy Conversion

Energy conversion in chemistry involves changing one form of energy to another. In the context of our exercise, we're dealing with the conversion of caloric energy to kilojoules and then relating it to chemical energy in the form of glucose. The first step of the problem requires converting the daily caloric energy requirement from Calories (nutritional calories, where 1 Cal is equal to 1000 cal) to kilojoules using the conversion factor \( 1 \mathrm{Cal} = 4.184 \mathrm{kJ} \).

The concept of energy conservation is pivotal in chemistry, as it ensures that the energy being calculated corresponds correctly between different units of measure. Understanding energy conversion allows students to bridge the gap between nutrition (often discussed in Calories) and the chemical energy (frequently measured in joules or kilojoules) involved in metabolic processes.