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 coalfired power plant that produces about \(6 \times 10^{6}\) tons of \(\mathrm{CO}_{2}\) per year. (a) Assuming ideal-gas behavior, \(1.00 \mathrm{~atm}\), 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 \(120 \mathrm{~atm}\) 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 \(36^{\circ} \mathrm{C}\) and \(90 \mathrm{~atm},\) what volume does it occupy?

Short answer

In short, to calculate the volume of CO₂ produced by the power plant and its volume when stored as a liquid or a gas, follow these steps: 1. Convert the total mass of CO₂ to moles using the molar mass of CO₂ (44.01 g/mol). 2. Use the Ideal Gas Law (PV=nRT) to calculate the volume of CO₂ produced at 27°C and 1 atm. 3. Convert mass of CO₂ to volume when stored as a liquid using the given density (1.2 g/cm³). 4. Use the Ideal Gas Law again to calculate the volume of CO₂ when stored as a gas at 36°C and 90 atm. Computing these volumes will provide the answers for all three storage conditions.

Step-by-step solution

1. Convert the total mass of \(\mathrm{CO}_{2}\) to moles.

To do this, we need the molar mass of \(\mathrm{CO}_{2}\). The molar mass of carbon is 12.01 g/mol and the molar mass of oxygen is 16.00 g/mol. So, the molar mass of \(\mathrm{CO}_{2}\) is: \[M_{CO2} = 12.01 + 2 \times 16.00 = 44.01 \, \mathrm{g/mol}\] The mass of \(\mathrm{CO}_{2}\) produced is \(6 \times 10^{6}\) tons or \(6 \times 10^{6} \times 10^3\) kg or \(6 \times 10^{9}\) g. Now, we can compute the number of moles: \[n = \frac{m}{M_{CO2}} = \frac{6 \times 10^9 \, \mathrm{g}}{44.01 \, \mathrm{g/mol}}\]

2. Calculate the volume of \(\mathrm{CO}_{2}\) produced.

To compute the volume of the \(\mathrm{CO}_{2}\) produced, we will use the Ideal Gas Law: \[V = \frac{nRT}{P}\] where \(R = 0.08206 \frac{\mathrm{L} \cdot \mathrm{atm}}{\mathrm{mol} \cdot \mathrm{K}}\), \(T = 27^{\circ} \mathrm{C} + 273.15 = 300.15 \, \mathrm{K}\), and \(P = 1.00 \, \mathrm{atm}\). Substitute these values and compute the volume V.

3. Calculate the volume if the \(\mathrm{CO}_{2}\) is stored underground as a liquid.

To do this, we'll use the given density of \(\rho = 1.2 \, \mathrm{g/cm}^3\). First, convert the total mass of \(\mathrm{CO}_{2}\) produced to volume: \[V_\text{liquid} = \frac{\text{mass of }CO_2}{\rho} = \frac{6 \times 10^9 \, \mathrm{g}}{1.2 \, \mathrm{g/cm}^3}\] Convert the volume to more convenient units if necessary.

4. Calculate the volume if the \(\mathrm{CO}_{2}\) is stored underground as a gas.

To do this, we'll use the given temperature and pressure: \(T = 36^{\circ}\mathrm{C} + 273.15 = 309.15\,\mathrm{K}\) and \(P = 90\,\mathrm{atm}\). Use the Ideal Gas Law and the calculated number of moles n to compute the volume: \[V_\text{gas} = \frac{nRT}{P}\] Substitute the values and compute the volume. Now you have calculated the volumes for each storage condition.

Key concepts

Carbon Dioxide and Global Warming

The greenhouse effect is a natural process that warms the Earth’s surface and is critical for life as we know it. However, human activities, notably the combustion of fossil fuels, have increased concentrations of greenhouse gases, such as carbon dioxide (CO₂), in the atmosphere. This elevation in CO₂ levels enhances the greenhouse effect, leading to global warming, with serious ecological and socio-economic consequences.

When we burn coal, oil, or natural gas for energy, it releases an enormous amount of CO₂, which traps heat in the atmosphere. Over the past century, increasing CO₂ emissions have significantly contributed to the warming of the planet, melting ice caps, rising sea levels, and extreme weather changes. Understanding how to calculate the volume of CO₂ produced by activities such as operating a power plant is crucial for developing strategies to mitigate global warming, such as carbon capture and sequestration, which we will explore further in this article.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure, volume, temperature, and the number of moles of an ideal gas. It is usually expressed as PV=nRT, where P is the pressure, V is the volume, T is the temperature in Kelvin, n is the number of moles of the gas, and R is the Ideal Gas Constant, which equals 0.08206 L·atm/(mol·K).

This law assumes several idealizations, such as no interactions between the gas molecules and that the molecules occupy no volume. Although real gases do not perfectly obey the Ideal Gas Law, it provides a good approximation under many conditions, especially at high temperatures and low pressures. Using this law allows chemists and engineers to predict the behavior of gases, including how much space a specified amount of gas will occupy under certain temperature and pressure conditions, as demonstrated in the solved textbook exercise.

Molar Mass

Molar mass is a physical property defined as the mass of a given substance (chemical element or chemical compound) divided by the amount of substance. It is expressed in units of grams per mole (g/mol). It is an important concept in chemistry because it provides a link between the mass of a material and the number of molecules or atoms it contains.

For example, the molar mass of carbon dioxide (CO₂) can be calculated by adding the molar masses of one carbon atom (12.01 g/mol) and two oxygen atoms (16.00 g/mol each), resulting in a molar mass of 44.01 g/mol for CO₂. In the exercise provided, understanding the molar mass of CO₂ is essential for converting a large mass of CO₂ emissions into moles, which can then be used in conjunction with the Ideal Gas Law to find the volume of gas produced or required storage space.

Gas Storage and Sequestration

Gas storage and sequestration refer to the methods and techniques used to control and reduce the amount of greenhouse gases, particularly CO₂, released into the atmosphere. One such method involves capturing CO₂ at the source, such as a power plant, and storing it in various forms to prevent it from contributing to the greenhouse effect.

There are two common ways to store CO₂, as elucidated by the steps in the textbook exercise: either as a compressed liquid in underground geological formations or as a supercritical or gas phase under high-pressure conditions. These techniques, known as carbon capture and storage (CCS), are considered essential to mitigate global warming. Determining the volume of CO₂ that can be stored in different states is fundamental for evaluating the feasibility and capacity of storage sites. For example, CO₂ stored as a liquid occupies less volume but requires careful management to maintain the necessary pressure and temperature conditions.