Problem 125 - Chemistry The Molecular Science

Question

An ideal gas was contained in a glass vessel of unknown volume with a pressure of \(0.960 \mathrm{~atm} .\) Some of the gas was withdrawn from the vessel and used to fill a \(25.0-\mathrm{mL}\) glass bulb to a pressure of \(1.00 \mathrm{~atm}\). The pressure of the gas remaining in the vessel of unknown volume was 0.882 atm. All the measurements were done at the same temperature. Determine the volume of the vessel.

Step-by-Step Solution

Answer

The volume of the vessel is approximately 300.0 mL.

Step 1: Understand the Problem

We have an ideal gas initially in a vessel at 0.960 atm. Some of this gas is removed to fill a 25.0 mL bulb at 1.00 atm. The remaining gas in the vessel is at 0.882 atm. We need to find the initial volume of the vessel.

Step 2: Apply Ideal Gas Law

Since we are dealing with the ideal gas under constant temperature, we can use the ideal gas law PV = nRT. However, since R and T are constant and canceled out (because temperature is constant and not given), we can apply Boyle's law: \( P_1 V_1 = P_2 V_2 \).

Step 3: Calculate Moles Used for the Bulb

First, calculate the moles of gas used in the bulb using \( P_2 V_2 = nRT \). As R, n, and T are constants, we only consider \( P_2 \) and \( V_2 \). Thus, for the bulb, \( n = \frac{P_{bulb} \times V_{bulb}}{RT} \).

Step 4: Formulate Equation for Initial Vessel

Let \( V_{vessel} \) be the unknown volume of the vessel. Initially, \( P_{initial} V_{vessel} = n' RT\). After removing some gas, \( P_{remaining} (V_{vessel} - V_{bulb}) = n'' RT \).

Step 5: Use Boyle's Law for Change in Conditions

We use Boyle's Law on the vessel before and after some gas is withdrawn. Initially, \( 0.960 \, atm \times V_{vessel} = 0.882 \, atm \times V_{remaining} \), where \( V_{remaining} = V_{vessel} - V_{bulb}'s \) contribution.

Step 6: Express Remaining Volume in Terms of Initial Volume

From the rearranged Boyle's law, \( V_{remaining} = \frac{0.960}{0.882} \times V_{vessel} \). Also, \( V_{remaining} = V_{vessel} - 25.0 \, mL \).

Step 7: Solve for V_{vessel}

Now, equate the two expressions for \( V_{remaining} \): \( \frac{0.960}{0.882} \times V_{vessel} = V_{vessel} - 25.0 \, mL \). Solve this equation for \( V_{vessel} \).

Step 8: Rearrange and Calculate Solution

Simplify and solve the equation: \( V_{vessel} ( \frac{0.960}{0.882} - 1) = 25.0 \, mL \). Evaluate to find \( V_{vessel} \approx 300.0 \, mL \).

Key Concepts

Boyle's Law

Boyle's Law is a fundamental principle involving gas behavior under varying pressure and volume conditions. It describes how the pressure of a gas tends to decrease as the volume increases, provided the temperature remains constant. This principle can be mathematically expressed as:\[ P_1 V_1 = P_2 V_2 \]where:

• \( P_1 \) and \( V_1 \) are the initial pressure and volume,
• \( P_2 \) and \( V_2 \) are the pressure and volume after a change.

Boyle's Law is used when analyzing situations where a gas is transferred between containers or its volume changes due to external pressures. This legal relationship is vital to solve problems in chemistry and physics, especially when working with ideal gases.To apply Boyle’s law effectively, always keep in mind that the temperature must remain constant, as its validity hinges on this assumption. This makes it particularly useful in controlled environments where the temperature can be regulated or deemed unchanging.

Gas Pressure

Gas pressure is a measure of the force exerted by gas molecules as they collide with the walls of their container. This pressure is a result of molecular motion within the gas, governed by temperature and volume conditions. In the context of our exercise, gas molecules in the initial vessel exert a pressure of 0.960 atm, which changes as some gas is transferred to another bulb. Inside the bulb, the pressure reaches 1.00 atm. The remaining gas in the vessel then exerts a pressure of 0.882 atm. The unit of atmospheric pressure, atm, is commonly used when discussing pressures in chemistry because it is roughly equivalent to the average pressure at sea level. Understanding how pressure changes relate to volume adjustments is critical in gas law calculations, as these parameters are often intertwined when describing gas behaviors. In problem-solving scenarios, knowing the correct way to measure and interpret pressure changes is crucial—such as safely recognizing how these pressures are affected by removing gas from a system. Always consider how pressures affect subsequent calculations and how they interconnect with Boyle's Law.

Volume Measurement

In gas law problems, accurately understanding and measuring volume is crucial. Volume measurement refers to the determination of the space occupied by the gas. In our example, initially, the ideal gas occupies the entire unknown volume of a vessel at specific pressure. When some gas is transferred to a 25.0 mL bulb, this volume is subtracted from the original space, leading to a new pressure within the initial container. Volume is often measured in milliliters (mL) or liters (L), with 1 L equaling 1000 mL. Consistency in volume units is essential, especially when performing calculations involving the Ideal Gas Law or Boyle's Law. For effective problem-solving, remember:

• Always note the standard or derived volume units used in calculations.
• Make conversions between units if needed to ensure calculations are accurate.
• Understand how volume relates to other parameters like temperature and pressure, especially when using equations like Boyle's Law.

Understanding these concepts will help ensure precision in experimental setups and theoretical calculations involving gases.