Problem 9 - General Chemistry: Principles and Modern Applications

Question

A sample of \(\mathrm{O}_{2}(\mathrm{g})\) has a volume of \(26.7 \mathrm{L}\) at 762 Torr. What is the new volume if, with the temperature and amount of gas held constant, the pressure is (a) lowered to 385 Torr; (b) increased to 3.68 atm?

Step-by-Step Solution

Answer

The new volumes of \(O_{2}\) gas under different pressures are: (a) 53.1L at 385 Torr; (b) 7.26L at 3.68 atm.

Step 1: Conversion of pressure units

Since the units of pressure in the given exercise are not the same, we need to convert them to the same units for calculations. We have 1 atm = 760 Torr. Hence, 762 Torr = \( \frac{762}{760} = 1.003 \) atm. Now the pressure values are: initial pressure (P1) = 1.003 atm, pressure for part (a) (P2) = \( \frac{385}{760} = 0.506 \) atm, and pressure for part (b) (P3) = 3.68 atm.

Step 2: Use Boyle's Law to find the new volume

Boyle's Law is given by \(P1(V1) = P2(V2)\), where P1 and V1 are initial pressure and volume respectively, P2 is the new pressure and V2 is the new volume to be found. For part (a), we have \(1.003(V1) = 0.506(V2)\). Substituting \(V1 = 26.7L\) gives \(V2 = \frac{1.003 * 26.7}{0.506} = 53.1L\). For part (b), we have \(1.003 * 26.7 = 3.68 * V3\). Solving for V3 gives \(V3 = \frac{1.003 * 26.7}{3.68} = 7.26L\)

Step 3: Interpret the result

The results of the calculations indicate that when the pressure decreases, the volume of the gas sample increases, while when the pressure increases, the volume decreases. This is consistent with the principle of Boyle's Law.

Key Concepts

Gas Laws

Understanding the fundamental gas laws is crucial in chemistry. These laws describe how gases behave under various conditions, specifically concerning pressure, volume, and temperature changes. One of the most important gas laws is Boyle’s Law, which states that the pressure of a gas is inversely proportional to its volume when temperature and the number of gas particles remain constant.
This principle is essential for predicting how a gas will react to changes in its environment. When working with gases, it's crucial to determine which gas law applies and how to perform calculations using these laws. Boyle's Law, for instance, is a powerful tool in determining the volume of a gas when its pressure is altered under constant temperature conditions.

Pressure-Volume Relationship

The pressure-volume relationship described by Boyle’s Law is one of the cornerstones of gas behavior. According to the law \[ P1 \times V1 = P2 \times V2 \] where \(P1\) and \(V1\) are the initial pressure and volume, and \(P2\) and \(V2\) are the pressure and volume after a change has occurred. This relationship shows that:

• As pressure increases, volume decreases.
• As pressure decreases, volume increases.

This inverse relationship means that if you double the pressure on a gas, its volume will halve, assuming temperature and the amount of gas remain constant. This principle can be visualized by imagining gas molecules packed closer together as pressure increases, thus occupying less space. Applying Boyle's Law allows you to predict these changes accurately.

Unit Conversion

Unit conversion is a necessary step when working with gas laws because pressure might be expressed in different units. It's important to ensure that pressures are in the same units before performing calculations. Common units for pressure include atmospheres (atm), millimeters of mercury (mmHg or Torr), and Pascals (Pa). In our problem:

• 1 atm is equivalent to 760 Torr.
• Pressure values must be converted to the same unit. For example, 762 Torr was converted to approximately 1.003 atm and 385 Torr to 0.506 atm.

Accurate unit conversion ensures the integrity of the calculations, allowing you to solve problems correctly. Mastery of unit conversion is essential for any student working in scientific fields.

Principles of Chemistry

In chemistry, understanding the principles that govern reactions and behaviors is crucial for predicting outcomes. The gas laws, such as Boyle's Law, are fundamental principles that allow us to comprehend how gases behave.

• These principles are not just theoretical; they are applied in practical scenarios, such as the compression of gases in cylinders or when using air pumps.
• Boyle's Law teaches us that gases are compressible and that manipulating their pressure will influence their volume.

Understanding the principles of chemistry provides a framework for solving complex problems, predicting the results of chemical reactions, and designing experiments or processes safely and effectively. These concepts build a solid foundation for exploring more advanced topics.