Question

A sample of argon measuring \(3.00 \mathrm{~L}\) at \(27.0^{\circ} \mathrm{C}\) and 765 torr was bubbled through and collected over water. The pressure of the new system is 778 torr at \(15.0^{\circ} \mathrm{C}\). What is the volume occupied by the gas? The vapor pressure of water at \(15.0^{\circ} \mathrm{C}\) is 13 torr.

Short answer

The volume occupied by the gas in the new system is 2.88 L.

Step-by-step solution

Identify Key Information and Formulas

We have the initial conditions of the gas (Volume = 3.00L, Temperature = 27.0C = 300.15K, Pressure = 765torr). It is then placed in a new scenario with a new total pressure (778 torr). We need to find out the volume in the new scenario. We also have the vapor pressure of water at the new scenario's temperature (13 torr). Key Formulas: Ideal Gas Law: \(P_iV_i = P_fV_f\), Dalton's Law: \(P_{total} = P_{gas} + P_{water}\)

Determine the Pressure of the Argon Gas in the System

Use Dalton's Law to determine the pressure of the gas in the new system, independent of the pressure of water. Rearrange the formula to \(P_{gas} = P_{total} - P_{water}\). Plugging in our known values, 778 torr (total pressure) and 13 torr (water vapor pressure), we find that \(P_{gas} = 765 torr\). The pressure of the gas remains constant.

Find the Volume of the Gas in the New System

Since the pressure of the gas remains constant, and the temperature changes, we can use a form of the Ideal Gas Law that states the volume and temperature of a gas are directly proportional (when pressure is constant). So, we derive: \(V_f = V_i \cdot T_f/T_i\). Convert the final temperature from degrees celsius to Kelvin: 15 C = 288.15 K. Plugging in our known values, we have \(V_f = 3 L \cdot 288.15 K / 300.15 K\). Calculation gives \(V_f = 2.88 L\).

Key concepts

Dalton's Law

Dalton's Law is a principle that helps us understand how gases behave when they mix. According to this law, the total pressure exerted by a mixture of gases is the sum of the pressures exerted by each gas individually. In simple terms, if you have multiple gases in a container, each gas contributes to the total pressure as if it were alone in the container.

For our exercise, we used Dalton’s Law to separate the pressure contributions from argon gas and water vapor. The total pressure in the system was found to be 778 torr, but this included the vapor pressure of water, which is 13 torr at 15°C. By subtracting the water vapor pressure from the total pressure, we identified the pressure exerted purely by argon gas as 765 torr. Here’s the formula used:

• \( P_{gas} = P_{total} - P_{water} \)

Understanding Dalton's Law allows us to analyze situations where multiple gases coexist, and specifically for our problem, it lets us calculate the actual gas pressure we need for further calculations.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid at a given temperature. It is an essential concept because it influences the total pressure in a system where liquid and vapor are present, such as when gas is collected over water.

In the exercise scenario, the argon gas is collected over water, meaning water vapor is also present, contributing to the total pressure. At 15°C, the vapor pressure of water is 13 torr. This is crucial to know because it helps us use Dalton's Law accurately to find the actual pressure of argon gas by subtracting the vapor pressure from the total system pressure. This knowledge of vapor pressure is important for accurate gas volume calculations because it ensures that we are considering only the intended gas’s pressure in our system.

• Recognizing the vapor pressure value helps in deducting it from total pressure.
• It depicts how temperature variations can change vapor pressure values.

Gas Volume Calculation

Calculating the volume of a gas under changing conditions is a central task when dealing with gases. The Ideal Gas Law's principles support these calculations by linking pressure, volume, and temperature. In scenarios where pressure and temperature change while the gas's mass remains constant, the law helps predict the new state of the gas.

In this exercise, after identifying that the argon gas pressure stays at 765 torr, we needed to calculate its new volume when the temperature decreases from 27°C to 15°C. By using the Ideal Gas Law for constant pressure and finalizing with the formula for volume and temperature relationship:

• \( V_f = V_i \cdot \frac{T_f}{T_i} \)

where \( V_i \) is the initial volume, \( T_f \) is the final temperature, and \( T_i \) is the initial temperature—a simple division provides the new volume of argon. In this calculation, converting temperatures to Kelvin (which are absolute values) ensures accuracy, resulting in a new volume of 2.88 L. This conversion and formula application are key steps in correctly applying the Ideal Gas Law.