A rigid vessel containing a 3:1 mol ratio of carbon dioxide and water vapor is held at 200°C where it has a total pressure of 2.00 atm. If the vessel is cooled to 10°C so that all of the water vapor condenses, what is the pressure of carbon dioxide? Neglect the volume of the liquid water that forms on cooling.

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Identify the initial conditions: The vessel contains a 3:1 mol ratio of CO₂ to H₂O vapor at 200°C with a total pressure of 2.00 atm.

Use the ideal gas law to express the initial partial pressures: Let P_CO2 and P_H2O be the partial pressures of CO₂ and H₂O, respectively. Since the mol ratio is 3:1, P_CO2 = 3x and P_H2O = x, where 3x + x = 2.00 atm.

Solve for x to find the initial partial pressures: 4x = 2.00 atm, so x = 0.50 atm. Therefore, P_CO2 = 1.50 atm and P_H2O = 0.50 atm.

Consider the cooling process: When the vessel is cooled to 10°C, all the water vapor condenses, meaning P_H2O becomes 0 atm.

Determine the final pressure of CO₂: Since the volume of the vessel is constant and the temperature changes, use the ideal gas law to find the new pressure of CO₂ at 10°C, considering only the CO₂ gas remains.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for understanding how gases behave under varying conditions of temperature and pressure, particularly in rigid vessels where volume remains constant.

Partial Pressure

Partial pressure is the pressure exerted by a single component of a gas mixture. According to Dalton's Law, the total pressure of a gas mixture is the sum of the partial pressures of its individual components. This concept is crucial for determining the pressure of carbon dioxide after the water vapor condenses.

Phase Changes and Vapor Pressure

Phase changes, such as the condensation of water vapor, occur when a substance transitions between solid, liquid, and gas states. The vapor pressure of a liquid at a given temperature indicates the pressure exerted by its vapor in equilibrium with its liquid. Understanding this helps in calculating the remaining pressure of carbon dioxide after the water vapor has condensed.

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