Question

Sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) is produced by plants as follows: $$ \begin{aligned} 12 \mathrm{CO}_{2}(g)+11 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}+12 \mathrm{O}_{2}(g) \\ \Delta H=5645 \mathrm{~kJ} \end{aligned} $$ 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 \(\left.1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s}\right).\)

Short answer

The total energy the sun supplies daily for each square meter of surface area is approximately \(0.9146 \mathrm{~kW/m}^2\).

Step-by-step solution

Calculate the energy required to produce sucrose per day per square meter

Since we know the reaction of sucrose production, we can use stoichiometry to calculate the energy requirement. First, we must find the amount of energy required to produce 4.8 grams of sucrose per day. We are given the value of \(\Delta H = 5645 kJ\) for the entire reaction. The molar mass of sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) is: \(12(12.01) + 22(1.01) + 11(16.00) = 144.12 + 22.22 + 176 = 342.34 \mathrm{~g/mol}\) Now we find the moles of sucrose produced daily per square meter: \(\frac{4.8 \mathrm{~g}}{342.34 \mathrm{~g/mol}} = 0.014 \mathrm{~mol}\) Next, we calculate the energy required to produce 0.014 mol of sucrose per day per square meter using the given value of \(\Delta H\): Energy = \(\Delta H \times \mathrm{moles}\) Energy = \(5645 \mathrm{~kJ} \times 0.014 \mathrm{~mol} = 79.03 \mathrm{~kJ}\)

Determine the total energy supplied by the sun per day per square meter

We are given that only 0.1% of the sunlight that reaches the earth is used to produce sucrose. Therefore, we can find the total energy supplied by the sun daily per square meter by dividing the energy used to produce sucrose (79.03 kJ) by the efficiency (0.1%): Total solar energy = \(\frac{Energy \, for \, sucrose \, production}{Percentage \, efficiency}\) Total solar energy = \(\frac{79.03 \mathrm{~kJ}}{0.001} = 79030 \mathrm{~kJ}\)

Convert the energy to kilowatts per square meter

We have calculated the total solar energy supplied per day per square meter in kJ. Now, we need to convert this to kW/m². First, we should convert kJ to J: \(79030 \mathrm{~kJ} = 79030 \times 1000 = 79,030,000 \mathrm{~J}\) Since 1 second has 86400 seconds, now we can convert the energy to watts per square meter: Energy rate = \(\frac{Energy}{Time}\) Energy rate = \(\frac{79,030,000 \mathrm{~J}}{86400 \mathrm{~s}} = 914.583 \mathrm{~W/m}^2\) Finally, we can convert the energy rate to kilowatts per square meter: Total solar energy = \(914.583 \mathrm{~W/m}^2 \times \frac{1\, kW}{1000 \, W} = 0.9146 \mathrm{~kW/m}^2\) The total energy the sun supplies daily for each square meter of surface area is approximately \(0.9146 \mathrm{~kW/m}^2\).

Key concepts

Endothermic Reactions

Endothermic reactions are processes where heat is absorbed from the surroundings to facilitate a chemical change. In these reactions, the energy required to break the bonds of the reactants is greater than the energy released in forming the bonds of the products. Hence, they need an external energy source to proceed.
Photosynthesis, where plants convert carbon dioxide and water into glucose and oxygen using sunlight, is a classic example of an endothermic reaction. The energy from the sun provides the necessary fuel to power the complex chemical reaction. In the case of sucrose production, the reaction absorbs energy, reflected by the positive enthalpy change (ΔH = 5645 kJ).
This energy absorption makes endothermic reactions crucial for life, allowing organisms to store energy in chemical bonds, which can later be released when needed. It's the opposite of exothermic reactions, which release heat into the environment.

Stoichiometry

Stoichiometry is the field of chemistry that looks at the quantities of reactants and products in a chemical reaction. It allows us to calculate the number of molecules involved, ensuring that chemical equations are balanced correctly.
In the sucrose production equation, stoichiometry helps us determine how much energy is required to form a specific amount of sucrose. Given that the molar mass of sucrose is 342.34 g/mol, stoichiometry allows us to find out how many moles of sucrose are formed per square meter daily.
By knowing the ΔH value, which represents the change in enthalpy, we can calculate the energy needed for producing a defined mass of sucrose. This approach ensures that we accurately account for all atoms and energy exchanges, which is fundamental for predicting the outcomes of any chemical reaction.

Solar Energy Conversion

Solar energy conversion is the process by which plants harness the power of sunlight to fuel chemical reactions, primarily photosynthesis. This mechanism is essential for converting light energy into chemical energy stored as glucose or, in some cases, sucrose.
Plants only use a small fraction of the sunlight they receive; in the case of our sucrose production exercise, only 0.1% of the available sunlight is used. However, this minor percentage is sufficient for maintaining life on Earth through the creation of organic molecules.
The conversion efficiency may seem minimal, but the amount of energy provided by the sun is colossal. By calculating the total solar energy supplied, we see that each square meter of the Earth's surface receives enough energy daily to sustain plant growth and other ecological processes. This conversion is a vital part of the global energy cycle, underlining the role of photosynthesis in supporting life.