Question

The Sun supplies about 1.0 kilowatt of energy for each square meter of surface area \(\left(1.0 \mathrm{~kW} / \mathrm{m}^{2},\right.\) where a watt \(\left.=1 \mathrm{~J} / \mathrm{s}\right)\) Plants produce the equivalent of about \(0.20 \mathrm{~g}\) of sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) per hour per square meter. Assuming that the sucrose is produced as follows, calculate the percentage of sunlight used to produce sucrose. $$ \begin{aligned} 12 \mathrm{CO}_{2}(g)+11 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}+12 \mathrm{O}_{2}(g) \\ \Delta H=5645 \mathrm{~kJ} \end{aligned} $$

Short answer

The percentage of sunlight used by plants to produce sucrose is approximately 0.0917%.

Step-by-step solution

Calculate the energy of sunlight per hour per square meter

We are given that the Sun supplies 1.0 kW per square meter, which is equivalent to 1000 joules per second per square meter. To calculate the energy per hour per square meter, we will multiply this energy rate by the number of seconds in an hour. Energy = Power × Time Energy = 1.0 kW × 1 hour There are 3600 seconds in an hour, so the energy is: Energy = 1000 J/s × 3600 s = 3,600,000 J

Calculate the energy required to produce sucrose per hour per square meter.

We are given that plants produce 0.20 g of sucrose per hour per square meter and that the enthalpy of the reaction is 5645 kJ. First, we'll convert this enthalpy in Joules: Enthalpy change (ΔH) = 5645 kJ × 1000 J/kJ = 5,645,000 J Next, We will use molar masses to find the energy required to produce 0.20 g of sucrose per hour per square meter. A sucrose molecule has the formula C12H22O11. Therefore, its molar mass is 12×12 + 22 + 11×16 = 342 g/mol. To find the moles of sucrose in 0.20 g, we'll use the formula: Moles of sucrose = Mass / Molar mass Moles of sucrose = 0.20 g / 342 g/mol = 0.0005848 mol Now, we can calculate the energy required for the production of this amount of sucrose by multiplying the moles of sucrose produced by the enthalpy change: Energy required = Moles of sucrose × ΔH Energy required = 0.0005848 mol × 5,645,000 J/mol = 3300.394 J

Calculate the percentage of sunlight used to produce sucrose.

We have found the energy of sunlight per hour per square meter (3,600,000 J) and the energy required to produce sucrose per hour per square meter (3300.394 J). Now, we will calculate the percentage of sunlight used in sucrose production: Percentage = (Energy required / Energy of sunlight) × 100 Percentage = (3300.394 J / 3,600,000 J) × 100 Percentage ≈ 0.0917% The percentage of sunlight used by plants to produce sucrose is approximately 0.0917%.

Key concepts

Sucrose Production in Plants

Plants are remarkable factories that produce sugars like sucrose from basic elements such as carbon dioxide and water using sunlight as their energy source. This process of sugar production is a vital part of photosynthesis, where light energy is converted into chemical energy stored in the bonds of glucose and sucrose molecules.

When we explore the intricacies of sucrose production, we find a series of complex enzymatic reactions. Sucrose, a disaccharide consisting of glucose and fructose units, becomes a form of energy that can be transported to different parts of the plant and used or stored. In just one hour, a square meter of a plant's surface can generate approximately 0.20 grams of this essential sugar. Despite the complexity, a simple equation represents the overall transformation:
$$12 \text{CO}_2(g) + 11 \text{H}_2\text{O}(l) \rightarrow \text{C}_{12}\text{H}_{22}\text{O}_{11} + 12 \text{O}_2(g)$$

This formula hides the detailed steps and massive orchestration of plant cells but points out the general ingredients and products of the photosynthesis recipe. The basis of life on Earth hinges on this process, showcasing the plant's role in the broader ecosystem as not just passive organisms but as active energy converters.

Photosynthetic Energy Utilization

In the grand scheme of energy flow within an ecosystem, plants have a special job: they utilize the Sun’s raw energy and convert it into a form that can be consumed and used by other living organisms. Photosynthetic energy utilization reflects how efficient plants are in turning sunlight into chemical energy. The Sun beams down about 1 kilowatt of energy per square meter, a generous amount when you consider that a single plant leaf is a powerhouse capturing this energy.

However, this process isn't perfectly efficient. As seen in the exercise, plants convert only a small fraction of this sunlight energy into the chemical energy of sucrose. With an output of 0.20 grams of sucrose per square meter per hour, if we translate this to energy terms, the photosynthetic energy conversion efficiency is approximately 0.0917%. This number is quite low, not because plants are wasteful, but because they face many challenges, including light absorption capacity, photorespiration, and various energy losses during the conversion processes. The small percentage captured illustrates the selectivity and intricacy of photosynthesis, emphasizing the delicate balance nature maintains to sustain life.

Enthalpy Change in Chemical Reactions

The enthalpy change, denoted by ΔH, in a chemical reaction is an expression of the heat absorbed or released as reactants transform into products. This concept is crucial in understanding the energetics of reactions, such as the synthesis of sucrose in plants. When we look at the textbook exercise, it states that the ΔH for the formation of sucrose is 5645 kJ.

Enthalpy changes can be positive for endothermic reactions, where energy is absorbed from the environment. Conversely, for exothermic reactions, which release energy, enthalpy changes are negative. In the case of sucrose production, the reaction has a large positive ΔH, which tells us that to create sucrose bonds, plants must put in energy – energy that they derive from sunlight during photosynthesis.

The energy required to produce 0.20 grams of sucrose, which we calculated as 3300.394 J per square meter per hour, is only a fraction of the total solar energy available. This disparity between available energy and utilized energy encapsulates the concept of enthalpy change in the real-world scenario of plant metabolism and showcases the complex interplay between biological systems and thermodynamic laws.