Problem 116 - Chemistry The Central Science In Si Units

Question

Sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) is produced by plants as follows: \(12 \mathrm{CO}_{2}(g)+11 \mathrm{H}_{2} \mathrm{O}(I) \longrightarrow \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}+12 \mathrm{O}_{2}(g)\) $$ \Delta H=5645 \mathrm{~kJ} $$ About \(4.8 \mathrm{~g}\) of sucrose is produced per day per square meter of the earth's surface. The energy for this endothermic reaction is supplied by the sunlight. About \(0.1 \%\) of the sunlight that reaches the earth is used to produce sucrose. Calculate the total energy the sun supplies for each square meter of surface area. Give your answer in kilowatts per square meter \(\left(\mathrm{kW} / \mathrm{m}^{2}\right.\) where \(\left.1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s}\right)\)

Step-by-Step Solution

Answer

The sun supplies approximately 0.915 kW/m² of energy per day.

Step 1: Calculate moles of sucrose produced per day

First, determine the molar mass of sucrose, C\(_{12}\)H\(_{22}\)O\(_{11}\):\[\text{Molar mass of sucrose} = (12 \times 12.01) + (22 \times 1.01) + (11 \times 16.00) = 342.34 \text{ g/mol}\]Next, calculate the moles of sucrose produced per square meter per day using:\[\text{Moles of sucrose} = \frac{4.8 \text{ g}}{342.34 \text{ g/mol}} \approx 0.0140 \text{ mol}\]

Step 2: Calculate the energy required for sucrose production

Given that the enthalpy change \(\Delta H = 5645 \text{ kJ/mol}\), the energy required to produce the calculated moles of sucrose is:\[\text{Energy required} = 0.0140 \text{ mol} \times 5645 \text{ kJ/mol} \approx 79.03 \text{ kJ}\]

Step 3: Calculate total sunlight energy used for sucrose production

Since only 0.1\% of sunlight is used for sucrose production, the energy supplied by the sun is:\[\text{Total sunlight energy} = \frac{79.03 \text{ kJ}}{0.001} = 79030 \text{ kJ}\]

Step 4: Convert energy to kilowatts per square meter

First, convert the energy from kilojoules to watts as follows since 1 kilowatthour (kWh) = 3600 kilojoules (kJ):\[79030 \text{ kJ} = \frac{79030}{3600} \text{ kWh} \approx 21.95 \text{ kWh}\]Since this is per day per square meter, convert it to kilowatts per square meter: \[21.95 \text{ kWh/day} = \frac{21.95 \text{ kWh}}{24 \text{ h}} \approx 0.915 \text{ kW/m}^{2}\]

Key Concepts

Sucrose production

Sucrose is a form of sugar that plants produce and store as an energy reserve. This production occurs through photosynthesis, where plants convert carbon dioxide and water into sucrose and oxygen using the energy they absorb from sunlight. The chemical reaction for sucrose production can be written as:

• The carbon dioxide (4CO extsubscript{2}) and water (4H extsubscript{2}O) react to form sucrose, 4C extsubscript{12}H extsubscript{22}O extsubscript{11}, and oxygen (4O extsubscript{2}).
• This reaction releases oxygen as a by-product, which is essential for the survival of most living organisms on Earth.

During this process, not all the light energy from the sun is used efficiently by plants. Only around 0.1% of the sunlight that hits the Earth contributes to sucrose production. This is a crucial aspect of photosynthesis because it indicates that a vast amount of solar energy is needed to produce only a small amount of sucrose. Despite this, the conversion of solar energy to chemical energy in the form of sucrose is vital for sustaining the food chain and maintaining oxygen levels in the atmosphere.

Endothermic reaction

An endothermic reaction is a type of chemical reaction that requires an input of energy to proceed. Unlike exothermic reactions, which release energy, endothermic reactions absorb energy from their surroundings. The process of photosynthesis, particularly sucrose production, is an excellent example of an endothermic reaction.
The enthalpy change 0H for sucrose production is positive (5645 kJ/mol), indicating that energy is absorbed from the sun during the reaction. This energy absorption is what makes photosynthesis an endothermic process. Plants use solar energy to drive the reaction forward, converting simple molecules like carbon dioxide and water into complex sugars like sucrose.
It's crucial to note that endothermic reactions, like that of sucrose production, contribute significantly to the broader energy cycles in nature. By drawing energy from the sun, plants can sustain themselves and create a form of energy storage that is accessible to other living organisms on the planet.

Solar energy

Solar energy is the magnificent power derived from the sun, and it's at the heart of photosynthesis. Plants capture just a tiny fraction of this abundant energy to carry out the endothermic reaction required for sucrose production.

• Only about 0.1% of the sunlight that reaches the Earth's surface is used in the production of sucrose, highlighting the inefficiency but also the sheer abundance of solar energy available.
• This small percentage contributes greatly to the Earth's energy balance and ecosystems.

To understand its impact, let's consider the total sunlight energy used. Though only a fraction is used for photosynthesis, the overall amount is considerable due to the vast surface area available globally. The calculation from the energy conversion shows that approximately 0.915 kW/m7 of sunlight energy is harnessed for this purpose daily. Solar energy provides the necessary power for the biosphere, emphasizing its central role in life on Earth by enabling processes like photosynthesis, which maintains the oxygen and carbon dioxide cycles. Solar energy, therefore, not only facilitates plant growth but also supports life across various trophic levels by influencing food availability.