Problem 101 - Chemistry: The Central Science in SI Units, Global Edition

Question

Carbon dioxide, which is recognized as the major contributor to global warming as a "greenhouse gas," is formed when fossil fuels are combusted, as in electrical power plants fueled by coal, oil, or natural gas. One potential way to reduce the amount of \(\mathrm{CO}_{2}\) added to the atmosphere is to store it as a compressed gas in underground formations. Consider a 1000-megawatt coal-fired power plant that produces about \(6 \times 10^{6}\) tons of \(\mathrm{CO}_{2}\) per year. (a) Assuming ideal-gas behavior, \(101.3 \mathrm{kPa}\), and \(27^{\circ} \mathrm{C}\), calculate the volume of \(\mathrm{CO}_{2}\) produced by this power plant. (b) If the \(\mathrm{CO}_{2}\) is stored underground as a liquid at \(10^{\circ} \mathrm{C}\) and \(12.16 \mathrm{MPa}\) and a density of \(1.2 \mathrm{~g} / \mathrm{cm}^{3},\) what volume does it possess? (c) If it is stored underground as a gas at \(30^{\circ} \mathrm{C}\) and \(7.09 \mathrm{MPa}\), what volume does it occupy?

Step-by-Step Solution

Answer

(a) The volume of CO2 produced under the given conditions is approximately \(\frac{(6 \times 10^{12}/44) \times 8.314 \times 300.15}{101.3 \times 10^3} \: \mathrm{m}^{3}\). (b) The volume of CO2 stored underground as a liquid at the given conditions is \(\frac{6 \times 10^{12}}{1.2 \: \mathrm{g}/\mathrm{cm}^{3}} \: \mathrm{cm}^{3}\). (c) The volume of CO2 stored underground as a gas at the given conditions is \(\frac{6 \times 10^{12}}{2.45 \: \mathrm{g}/\mathrm{cm}^{3}} \: \mathrm{cm}^{3}\).

Step 1: Convert the mass of CO2 from tons to grams

: Given, the power plant produces \(6 \times 10^6\) tons of CO2 per year. We need to convert it into grams: \(1 \: \text{ton} = 1000 \: \text{kg}\) and \(1 \: \text{kg} = 1000 \: \text{g}\). So, the mass of CO2 in grams is: \(6 \times 10^6 \times 1000 \times 1000 = 6 \times 10^{12} \: \text{g}\).

Step 2: Calculate the number of moles of CO2

: The molecular weight of CO2 is approximately \(44 \: \text{g/mol}\). We will use this to calculate the number of moles of CO2 produced: Number of moles = \( \frac{6 \times 10^{12}}{44} \: \text{moles}\).

Step 3: Calculate the volume of CO2 at given conditions for part (a)

: For part (a), we are given the conditions as \(101.3 \: \text{kPa}\) pressure and \(27^{\circ} \mathrm{C}\). We will first convert the temperature to Kelvin: \(T = 27 + 273.15 = 300.15 \: \mathrm{K}\). Now, we will use the ideal gas law formula: \(PV = nRT\), where P is the pressure, V is the volume of the gas, n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin. We can rearrange the formula to solve for V: \(V = \frac{nRT}{P}\). Substituting the values for temperature, pressure, and the universal gas constant (R = 8.314 J/mol·K) into the equation, we can calculate the volume of CO2 produced under part (a) conditions: \(V = \frac{(6 \times 10^{12}/44) \times 8.314 \times 300.15}{101.3 \times 10^3} \: \mathrm{m}^{3}\).

Step 4: Calculate the volume for parts (b) and (c) using given densities

: In parts (b) and (c), the density of CO2 is given, and we are asked to find the volume that it occupies. To find the volume, we can use the formula: Volume = \(\frac{\text{mass}}{\text{density}}\). For part (b), the mass of CO2 is \(6 \times 10^{12}\) g, and its density is \(1.2 \: \mathrm{g}/ \mathrm{cm}^{3}\). Volume = \(\frac{6 \times 10^{12}}{1.2 \: \mathrm{g}/\mathrm{cm}^{3}} \: \mathrm{cm}^{3}\). For part (c), we are given a density of \(2.45 \: \text{g/cm}^{3}\). Volume = \(\frac{6 \times 10^{12}}{2.45 \: \mathrm{g}/\mathrm{cm}^{3}} \: \mathrm{cm}^{3}\). Note that the final volumes should be expressed in appropriate units (e.g., \(\mathrm{m}^{3}\) or \(\mathrm{cm}^{3}\)) and converted if necessary.

Key Concepts

Carbon Dioxide Storage

Carbon dioxide (CO₂) is a significant greenhouse gas contributing to climate change. One method to mitigate its impact is through carbon capture and storage (CCS). This involves capturing CO₂ emissions from sources like power plants and storing it underground.
Deep geological formations, such as depleted oil and gas fields or deep saline aquifers, can serve as storage sites.

• **Transportation:** CO₂ can be transported via pipelines to storage sites.
• **Injection:** CO₂ is injected into underground formations at high pressure.
• **Sequestration:** Over time, CO₂ may react with the reservoir rock to form stable carbonate minerals.

By storing CO₂, we reduce its atmospheric concentration and help mitigate global warming.

Greenhouse Gases

Greenhouse gases, such as carbon dioxide, methane, and nitrous oxide, trap heat in the Earth's atmosphere. This phenomenon is known as the greenhouse effect, which leads to global warming and climate change. CO₂ is emitted through various human activities, including:

• Combustion of fossil fuels for energy and transportation.
• Industrial processes and agricultural practices.
• Deforestation and land-use changes.

Reducing CO₂ emissions by enhancing energy efficiency, transitioning to renewable sources, and adopting CCS technologies are crucial in combating climate change.

Moles and Molar Mass

Understanding moles and molar mass is crucial for chemical calculations, especially when using the ideal gas law. The mole is a unit that measures the amount of a substance. Molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol).
For CO₂, with a molar mass of 44 g/mol, it’s essential to convert the mass of CO₂ produced into moles:

• **Mass to Moles:** \(\text{Number of moles} = \frac{\text{mass of CO₂}}{\text{molar mass}}\).
• This conversion helps in calculating the volume of gas using the ideal gas law equation.

Understand these concepts to comprehend chemical reactions and quantitate gaseous products.

Gas Volume Calculations

Gas volume calculations often utilize the ideal gas law, which states that \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature in Kelvin. This formula helps calculate the volume of gases under different conditions. Key steps include:

• **Convert Temperature:** Always convert Celsius to Kelvin by adding 273.15.
• **Rearrange for Volume:** \( V = \frac{nRT}{P} \) to find the gas volume.
• **Consider Density for Non-Ideal Conditions:** In cases of high pressure and low temperatures, density can also be used: \( \text{Volume} = \frac{\text{mass}}{\text{density}} \).

Understanding gas volume calculation is essential for accurately predicting how gases will behave under varying environmental conditions.