Problem 117 - Chemistry: The Central Science in SI Units, Global Edition

Question

It is estimated that the net amount of carbon dioxide fixed by photosynthesis on the landmass of Earth is \(5.5 \times 10^{16} \mathrm{~g} / \mathrm{yr}\) of \(\mathrm{CO}_{2}\). Assume that all this carbon is converted into glucose. (a) Calculate the energy stored by photosynthesis on land per year, in kJ. (b) Calculate the average rate of conversion of solar energy into plant energy in megawatts, MW \((1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s}) .\) A large nuclear power plant produces about \(10^{3} \mathrm{MW}\). The energy of how many such nuclear power plants is equivalent to the solar energy conversion?

Step-by-Step Solution

Answer

(a) The energy stored by photosynthesis on land per year is \(5.82 \times 10^{17}\mathrm{~kJ/yr}\). (b) The average rate of solar energy conversion into plant energy is \(1.85 \times 10^7\mathrm{~MW}\), which is equivalent to the energy produced by \(1.85 \times 10^4\) nuclear power plants.

Step 1: Convert carbon dioxide into glucose

First, we need to find out the number of moles of \(\mathrm{CO_2}\) and calculate the moles of glucose formed using the chemical equation of photosynthesis. Chemical equation of photosynthesis: \(6\mathrm{CO_2} + 6\mathrm{H_2O} \rightarrow \mathrm{C_6H_{12}O_6(IN\ Glucose)} + 6\mathrm{O_2}\) Given, net amount of \(\mathrm{CO_2}\) fixed by photosynthesis = \(5.5 \times 10^{16}\mathrm{~g/yr}\) Molar mass of $\mathrm{CO_2} = 12 + (2 \times 16) = 44\,\mathrm{g/mol}\) Number of moles of $\mathrm{CO_2} = \frac{5.5 \times 10^{16}\, \mathrm{g}}{44\, \mathrm{g/mol}} = 1.25 \times 10^{15}\,\mathrm{mol}\) From the chemical equation, 6 moles of \(\mathrm{CO_2}\) are required to produce 1 mole of glucose. So, the number of moles of glucose produced is: Number of moles of glucose = \(\frac{1.25 \times 10^{15}\, \mathrm{mol}}{6} = 2.08 \times 10^{14}\,\mathrm{mol}\)

Step 2: Calculate the energy stored by photosynthesis on land per year

Now we need to determine the energy stored by photosynthesis on land per year. We know the energy per mole of glucose produced in the process of photosynthesis is 2800 kJ/mol. To find the total energy stored, we'll multiply the moles of glucose with the energy per mole. Energy stored by photosynthesis = Number of moles of glucose × Energy per mole of glucose = \(2.08 \times 10^{14}\, \mathrm{mol} \times 2800\, \mathrm{kJ/mol} = 5.82 \times 10^{17}\,\mathrm{kJ/yr}\)

Step 3: Find the average rate of solar energy conversion and compare it to nuclear power plant energy

To find the average rate of solar energy conversion, we'll first convert the energy stored per year in kJ to joules (1 kJ = 1000 J). Then, we'll convert it into watts (1 W = 1 J/s) by dividing the value by the total number of seconds in a year. Energy stored per year = \(5.82 \times 10^{17}\, \mathrm{kJ/yr} \times 1000\, \mathrm{J/kJ} = 5.82 \times 10^{20}\, \mathrm{J/yr}\) Number of seconds in a year = \(365 \times 24 \times 3600 = 3.15 \times 10^7\, \mathrm{s/yr}\) Average rate of solar energy conversion (in MW) = \(\frac{5.82 \times 10^{20}\, \mathrm{J/yr}}{3.15 \times 10^7\, \mathrm{s/yr}\times 10^{6}\, \mathrm{J/MW}} = 1.85 \times 10^7\, \mathrm{MW}\) Now let's compare the solar energy conversion with the energy produced by a single nuclear power plant. Given, energy produced by a nuclear power plant = \(10^3\,\mathrm{MW}\) Number of nuclear power plants equivalent to solar energy conversion = \(\frac{1.85 \times 10^7\, \mathrm{MW}}{10^3\, \mathrm{MW}} = 1.85 \times 10^4\, \mathrm{nuclear\ power\ plants}\) To summarize: (a) The energy stored by photosynthesis on land per year is \(5.82 \times 10^{17}\,\mathrm{kJ/yr}\). (b) The average rate of solar energy conversion into plant energy is \(1.85 \times 10^7\,\mathrm{MW}\), which is equivalent to the energy produced by \(1.85 \times 10^4\) nuclear power plants.

Key Concepts

Carbon Dioxide Fixation

Photosynthesis begins with the process of carbon dioxide fixation. This is when plants take in carbon dioxide from the atmosphere and convert it into organic molecules. In this process, carbon dioxide molecules are "fixed" into a usable form via a series of reactions. The most well-known cycle involved is the Calvin Cycle.

During this cycle, carbon dioxide molecules are combined with water to ultimately form glucose and oxygen. This process is very efficient and crucial for the survival of both plants and animals. For example, in the given exercise, it's estimated that Earth's landmass annually fixes about \(5.5 \times 10^{16}\) grams of \(\mathrm{CO_2}\).

Photosynthesis is, therefore, critical for reducing atmospheric \(\mathrm{CO_2}\) levels and providing the glucose needed for plant energy. This remarkable process not only supports plant life but also has a significant impact on the climate and energy cycles of our planet.

Glucose Production

The production of glucose during photosynthesis is a complex, yet fascinating process. After carbon dioxide is fixed, it goes through different chemical reactions to form glucose. Glucose, a simple sugar, is an essential energy source for plants.

Photosynthesis follows the chemical equation: \(6\mathrm{CO_2} + 6\mathrm{H_2O} \rightarrow \mathrm{C_6H_{12}O_6} + 6\mathrm{O_2}\). For every six molecules of carbon dioxide and water, one molecule of glucose is produced. The exercise shows that \(2.08 \times 10^{14}\) moles of glucose could be produced annually from the fixed \(\mathrm{CO_2}\).

• This process converts solar energy into chemical energy.
• The glucose produced fuels plant growth and development.
• It also provides energy to other organisms that consume plants.

In this way, glucose production during photosynthesis plays a vital role in energy transfer within an ecosystem.

Solar Energy Conversion

Solar energy conversion is an amazing natural process in photosynthesis that allows plants to harness sunlight and transform it into chemical energy stored in glucose. The role of the sun is crucial as it provides the energy needed to drive the reactions that produce glucose.

Photosynthesis captures only a small fraction of the solar energy arriving at Earth's surface. Despite this, plants convert an enormous amount of energy annually. For example, the total energy stored by plants on land per year equals \(5.82 \times 10^{17}\,\mathrm{kJ}\).

The average rate of converting this solar energy into plant energy can be calculated in megawatts, equating to \(1.85 \times 10^7\,\mathrm{MW}\).

• It highlights the remarkable power of plants and their efficiency in energy conversion.
• This process ultimately helps sustain life on Earth by providing energy for the food chain.

Nuclear Power Plant Comparison

Comparing solar energy conversion in photosynthesis to nuclear power is intriguing. Both are powerful energy sources but differ in processes and implications.

A single nuclear power plant produces about \(10^3\) megawatts of energy. Compared to photosynthesis, which converts sunlight into \(1.85 \times 10^7\) megawatts annually, this process is equivalent to nearly \(1.85 \times 10^4\) nuclear power plants.

• This comparison shows the vast amount of energy plants can generate naturally.
• It underscores the importance of plants in our global energy balance.
• While nuclear power is a potent man-made source, photosynthesis is a sustainable, naturally occurring process.

Understanding these differences helps highlight the unique and essential role of photosynthesis in Earth's energy ecosystem.