Question

Given a fixed quantity of a gas at constant temperature, calculate the new pressure the gas would exert if the volume were changed as shown in the following table. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Initial } \\ \text { Pressure } \end{array} & \begin{array}{c} \text { Initial } \\ \text { Volume } \end{array} & \begin{array}{c} \text { Final } \\ \text { Volume } \end{array} & \begin{array}{c} \text { Final } \\ \text { Pressure } \end{array} \\ \hline 845 \text { torr } & 155 \mathrm{~mL} & 1.55 \mathrm{~L} & ? \\ \hline 5.30 \mathrm{~atm} & 2.85 \mathrm{~L} & 4.50 \mathrm{~L} & ? \\ \hline 755 \text { torr } & 2.00 \mathrm{~L} & 5500 \mathrm{~mL} & ? \\ \hline \end{array} $$

Short answer

The final pressures for the three scenarios are approximately 8435 torr, 3.37 atm, and 2736 torr respectively.

Step-by-step solution

Interpret the data in the table

Each row in the table represents a different scenario. In each row, the first column gives the initial pressure of the gas, the second column gives the initial volume, and the third column gives the final volume. The question mark in the last column indicates that the final pressure is unknown and needs to be calculated.

Apply Boyle's Law

Boyle's Law states that the product of the pressure and volume of a gas is constant as long as the temperature of the gas stays the same. This can be written in the equation P1 * V1 = P2 * V2 where P1 and V1 represent the initial pressure and volume, while P2 and V2 represent the final pressure and volume. The final pressure P2 can be calculated using the rearranged formula P2 = P1 * V1 / V2.

Perform the calculations for each scenario

Substitute the initial pressures, volumes and final volumes from each row of the table into the equation P2 = P1 * V1 / V2. Remember that the units of volume must be the same when performing the calculation. Therefore, convert all volumes to the same unit before substituting them into the equation. After performing the calculations, round the final pressure to an appropriate number of significant figures.

Key concepts

Gas Law Calculations

Gas law calculations are essential in understanding how different gas parameters relate to each other. Boyle's Law, specifically, helps us predict how pressure and volume change in relation to one another for a given mass of gas at a constant temperature. It's one of the cornerstone principles of chemistry and physics, allowing scientists and engineers to manipulate gases for various applications.

To perform gas law calculations, especially when dealing with Boyle's Law, it's crucial to familiarize oneself with the equation \( P_1 V_1 = P_2 V_2 \), where \( P \) represents the pressure and \( V \) indicates the volume of the gas. This equation shows that if you have a gas at a known pressure and volume, and you change the volume, you'll be able to calculate the new pressure exerted by the gas (or vice versa), provided the temperature remains constant.

Key Steps for Gas Law Calculations:

• Make sure all units are consistent before performing calculations.
• Use Boyle's Law equation to set the product of initial pressure and initial volume equal to the product of final pressure and final volume.
• Solve for the unknown, ensuring that you respect significant figures based on the measured values.

Performing these calculations accurately is crucial for lab work, industrial applications, and even in understanding natural phenomena like breathing and weather patterns.

Pressure-Volume Relationship

The pressure-volume relationship, often referred to as Boyle's Law in the context of ideal gases, describes how the pressure of a gas tends to decrease as the volume of the container increases, provided the temperature of the gas does not change. This inverse relationship is a fundamental aspect of understanding gas behavior and is used extensively in chemistry problem solving.

Boyle's Law can be visualized as a syringe being compressed: as the volume inside the syringe decreases due to the plunger being pushed in, the pressure inside increases—assuming we can't change the temperature, which is a common assumption in these problems.

To articulate this concept mathematically, Boyle's Law is expressed as \( P_1 V_1 = P_2 V_2 \). In this equation:\

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

When solving problems, it's vital to handle the units properly; typically, pressures are given in atmospheres (atm) or torr, and volumes are in liters (L) or milliliters (mL). Converting to consistent units allows for direct substitution into the Boyle's Law equation to find the missing variable.

Chemistry Problem Solving

Chemistry problem solving often involves understanding and applying laws and principles to find a solution. Tackling problems using Boyle's Law involves systematic steps that begin with basic interpretation of data and end with calculation and analysis.

Here's what a typical problem-solving process looks like:

• Interpreting Data: Identify the knowns and unknowns from the information provided. In gas law problems, this involves recognizing initial and final pressures and volumes.
• Applying the Law: Understand which law applies to the scenario. In cases dealing with pressure and volume at constant temperature, Boyle's Law is relevant.
• Performing Calculations: Use algebra to manipulate the equations. Always keep an eye on unit consistency and significant figures, as precision matters in scientific calculations.
• Checking Work: After finding the answers, double-check calculations and reasoning. Ensure that the results follow the principles of the law. For instance, if volume increases, pressure should inversely decrease.

By working through problems methodically, students learn critical thinking and analytical skills. These problem-solving capabilities extend beyond chemistry, offering essential tools for various scientific and engineering disciplines.