Problem 5 - Biochemistry: Concepts and Connections

Question

The flux of solar energy reaching the Earth's surface is approximately \(7 \mathrm{~J} / \mathrm{cm}^{2}\). Assume that \(a l l\) of this energy is used by a green leaf \(\left(10 \mathrm{~cm}^{2}\right.\) in area), with maximal efficiency of \(35 \%\). How many moles of hexose could the leaf theoretically generate in an hour? Use \(600 \mathrm{~nm}\) for an average wavelength.

Step-by-Step Solution

Answer

The leaf can theoretically generate approximately 0.031 moles of hexose in an hour.

Step 1: Calculate Energy Input per Hour

First, calculate the total solar energy reaching the leaf in one hour. The flux of solar energy is given as \(7 \text{ J/cm}^2\). The area of the leaf is \(10 \text{ cm}^2\). Multiply these numbers to find the energy input per second and then multiply by the number of seconds in an hour (3600 seconds): \[7 \text{ J/cm}^2 \times 10 \text{ cm}^2 \times 3600 \text{ s} = 252,000 \text{ J}\]

Step 2: Calculate Usable Energy

Next, calculate the usable energy, assuming a conversion efficiency of \(35\%\). Multiply the total energy by the efficiency:\[252,000 \text{ J} \times 0.35 = 88,200 \text{ J}\]

Step 3: Calculate Energy Required per Mole of Hexose

The energy required to synthesize one mole of hexose from photosynthesis can be calculated. Assuming the average wavelength of light is \(600 \text{ nm}\), convert this wavelength to energy using Planck's equation: \[E = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \text{ J s} \times 3 \times 10^8 \text{ m/s}}{600 \times 10^{-9} \text{ m}} = 3.31 \times 10^{-19} \text{ J/photon} \]The energy needed to produce one mole (approximately 180 g) of hexose is about \(2870 \text{ kJ/mol}\), which is equivalent to \(2870000 \text{ J/mol}\).

Step 4: Calculate Moles of Hexose Produced

Finally, calculate the number of moles of hexose that could be produced from the usable energy. Divide the usable energy by the energy needed per mole:\[\frac{88,200 \text{ J}}{2870000 \text{ J/mol}} \approx 0.0307 \text{ moles}\]

Step 5: Conclusion: Amount of Hexose

Based on the calculations, the leaf can theoretically generate approximately \(0.031\) moles of hexose in an hour with the given parameters.

Key Concepts

Solar Energy Conversion

Photosynthesis is a fascinating process by which plants convert solar energy into chemical energy, stored within glucose and other carbohydrates. The sun provides a tremendous amount of energy to our planet, with a solar flux reaching the Earth's surface of about 7 J/cm². This energy, when captured by a leaf, serves as the driving force for photosynthesis.

A key aspect of solar energy conversion in plants involves chlorophyll, a pigment found in the chloroplasts of green leaves. Chlorophyll absorbs sunlight, specifically harnessing photons of light energy which are then transformed through a series of biochemical reactions. As the energy from these photons excites electrons, it initiates a chain of events leading to the synthesis of carbohydrates.

• Photon absorption is crucial for energy transfer.
• Excited electrons facilitate chemical reactions.
• These reactions build up complex sugars like glucose.

This whole process not only supports the plant but is also fundamental to the sustenance of life on Earth, providing oxygen and forming the base of the food chain.

Efficiency in Energy Conversion

In the context of photosynthesis, efficiency refers to how effectively a plant can convert solar energy into chemical energy. Realistically, plants do not use all available solar energy; however, they have a maximum theoretical efficiency under ideal conditions.

In our scenario, a green leaf has an efficiency of about 35%. This means that out of the total solar energy absorbed—calculated earlier as 252,000 J—the leaf can effectively harness only a portion of it to create glucose. After applying the efficiency rate, the usable energy becomes 88,200 J. This concept helps us understand how plants manage their energy resources and optimize them for growth and development.

It's important to note that several factors can affect this efficiency, including:

• Leaf orientation and surface area.
• Environmental conditions such as light intensity and temperature.
• Intrinsic metabolic pathways within the plant.

Understanding energy conversion efficiency not only sheds light on plant biology but also guides agricultural practices and renewable energy technologies.

Hexose Production

Hexose sugars, such as glucose, are a primary product of photosynthesis and play a crucial role in plant metabolism. During this process, plants take carbon dioxide from the atmosphere and use the energy stored in ATP and NADPH to synthesize these sugars, providing essential energy and building materials for the plant.

In our calculation, we've determined that for every 88,200 J of usable energy, approximately 0.031 moles of hexose can be produced. The synthesis of hexose involves specific energy costs, measured in kilojoules per mole; in this case, about 2870 kJ/mol, equivalent to 2870000 J/mol.

This process involves multiple steps and enzymes that catalyze the conversion of intermediate molecules into hexose. These sugars can then be used for:

• Immediate energy needs within cells.
• Biosynthesis of starch or cellulose for structural support.
• Storage for later energy access.

Understanding hexose production gives insight into how plants meet their energy needs and sustain their growth and survival. It's a brilliant demonstration of biochemical efficiency, ensuring that every bit of captured solar energy is put to good use.