Question

As a result of photosynthesis, an acre of forest \(\left(1 \text { acre }=4047 \text { square meters) can take up } 1000 . \text { kg of } \mathrm{CO}_{2}\right.\) Assuming air is \(0.0314 \% \mathrm{CO}_{2}\) by volume, what volume of air is required to provide \(350 .\) kg of \(\mathrm{CO}_{2}\) ? Assume \(T=310 \mathrm{K}\) and \(P=1.00 \mathrm{atm}\).

Short answer

To provide 350 kg of CO2, approximately 6,430,149,360 L of air is required under the given temperature (310 K) and pressure (1.00 atm).

Step-by-step solution

Convert mass of CO2 to moles

First, we need to find out how many moles of CO2 are there in 350 kg. We can do this by using the molar mass of CO2 which is (12.01 g/mol for carbon and 16.00 g/mol for oxygen, and there are two oxygen atoms in CO2). Molar mass of CO2 = 12.01 + 2(16.00) = 44.01 g/mol Now, we can convert the mass of CO2 (350 kg) to moles: moles of CO2 = mass / molar mass moles of CO2 = (350,000 g) / (44.01 g/mol) = 7950.01 moles

Calculate the volume of CO2 required

Next, we need to find the volume of pure CO2 required for these 7950.01 moles. Using the ideal gas law, PV=nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant (0.0821 L atm/mol K), and T is the temperature. We are given T = 310 K and P = 1.00 atm: V (CO2) = (nRT) / P V (CO2) = (7950.01 moles * 0.0821 L atm/mol K * 310 K) / 1.00 atm V (CO2) = 2,018,593.19 L

Calculate the volume of air required

Now, we need to find the volume of air required to provide this volume of CO2. We know that air contains 0.0314% CO2 by volume. So, we can set up a proportion: (Volume of CO2) / (Volume of air) = 0.0314 / 100 Volume of air = (Volume of CO2) / (0.0314 / 100) Volume of air = 2,018,593.19 L / (0.0314 / 100) Volume of air = 6,430,149,360 L Thus, approximately 6,430,149,360 L of air is required to provide 350 kg of CO2 under the given temperature and pressure.

Key concepts

Understanding the Mole Concept

The mole concept is a fundamental principle in chemistry that relates the macroscopic world to the microscopic world of atoms and molecules. It's the bridge that allows chemists to count individual atoms by weighing them. One mole is defined as the amount of substance that contains as many entities (atoms, molecules, ions, or other particles) as there are atoms in 12 grams of carbon-12. The number of entities in a mole is called Avogadro's number, approximately equal to \(6.022 \times 10^{23}\).

To grasp its application, let's revert to the exercise where we needed to find out how many moles of \(CO_2\) correspond to 350 kg. To do this task, we first related the given mass to the molar mass of \(CO_2\). Understanding the mole concept ensures you can handle such calculations in stoichiometry, the study of the quantitative relationships of reactants and products in a chemical reaction, such as photosynthesis in this case. Learning the mole concept helps predict how much reactants are required or products formed in a chemical process.

Applying the Ideal Gas Law

The ideal gas law is an equation of state for a hypothetical ideal gas. It is a good approximation of the behavior of many gases under a variety of conditions, although it has limitations. The law is usually stated as \(PV = nRT\), where P is the pressure of the gas, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the absolute temperature.

In the context of our exercise, the ideal gas law is used to find the volume of \(CO_2\) that 7950.01 moles would occupy at a temperature of 310 K and a pressure of 1.00 atm. By rearranging the law to solve for V and inputting the known values, we obtained the volume of \(CO_2\). This step is crucial because knowing the amount of gas in terms of volume is more practical for understanding how plants convert atmospheric \(CO_2\) into oxygen through photosynthesis.

Determining Molar Mass

Molar mass is the mass of one mole of a substance (atom, molecule, ion, etc.), typically expressed in grams per mole (g/mol). It is the molecular weight of a substance but scaled from atomic mass units to grams. The molar mass is used to convert grams of a substance to moles, which is a key step in stoichiometric calculations.

In our exercise about photosynthesis, the molar mass of carbon dioxide (\(CO_2\)) was calculated based on the molar masses of carbon and oxygen. It proved essential to convert 350 kg of \(CO_2\) to moles. Accurate calculation of molar mass is vital in chemistry because it allows chemists to measure specific amounts of substances for reactions, ensuring that the ratios of reactants match the balanced chemical equation.