Problem 199 - Introductory Chemistry Atoms First

Question

Explain what effect compressing a gas has on the density of the gas.

Step-by-Step Solution

Answer

Compressing a gas leads to an increase in its density because the pressure inside the gas increases while the temperature remains constant. This relationship can be observed through the equation \(\rho = \frac{MP}{RT}\), where \(\rho\) is the density, M is the molecular weight, P is the pressure, R is the gas constant, and T is the temperature. Since density is directly proportional to pressure and inversely proportional to temperature, an increase in pressure due to compression and constant temperature results in increased density.

Step 1: Understand the Ideal Gas Law

The ideal gas law is an equation of state for a hypothetical ideal gas. It relates the pressure (P), volume (V), and temperature (T) of an ideal gas to the number of molecules (n) and the gas constant (R). The ideal gas law is given by the equation: \[PV = nRT\] Where, P = Pressure of the gas V = Volume of the gas n = Number of molecules of the gas R = Gas constant (8.314 J/(mol·K)) T = Temperature of the gas in Kelvin

Step 2: Define Density and Relate it to Gas Laws

Density (\(\rho\)) is defined as mass per unit volume: \[\rho = \frac{m}{V}\] Now, we know that the number of molecules (n) in a gas is related to its mass (m) by the molecular weight (M) of the gas: \[n = \frac{m}{M}\] We can rewrite the ideal gas law equation using density: \[P = \frac{nRT}{V} = \frac{mRT}{MV}\] Now, we replace the mass-to-volume ratio with density and get the equation relating the density of a gas to its pressure and temperature: \[\rho = \frac{MP}{RT}\]

Step 3: Discuss the Effect of Compression on Gas Density

Compressing a gas means decreasing its volume while keeping the temperature constant. When we decrease the volume of the gas, the pressure inside the gas increases in order to maintain the ideal gas law relationship (since temperature is constant). According to the density equation we derived above, the density of a gas (\(\rho\)) is directly proportional to the pressure (P) and inversely proportional to the temperature (T). Thus, when the pressure increases due to compression and temperature remains constant, the density of the gas also increases. In conclusion, compressing a gas increases its density because the pressure inside the gas increases while the temperature remains constant.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in understanding the properties of gases. It provides a relationship between pressure, volume, temperature, and the number of molecules in a gas. The law is represented by the formula:\[ PV = nRT \]In this equation:

• P: Pressure of the gas
• V: Volume of the gas
• n: Number of moles
• R: Universal gas constant (8.314 J/(mol·K))
• T: Temperature in Kelvin

Understanding this equation helps explain how changes in pressure, volume, and temperature can affect a gas. When compressing a gas, knowing these relationships will help predict changes in the gas's behavior.

Gas Density

Gas density refers to the mass of gas per unit volume. It can be calculated using the formula:\[ \rho = \frac{m}{V} \]where:

• m: Mass of the gas
• V: Volume of the gas

By using the Ideal Gas Law, we can express density in terms of pressure and temperature:\[ \rho = \frac{MP}{RT} \]This equation shows that density is directly proportional to pressure and inversely proportional to temperature, making it a pivotal aspect in understanding how compressing a gas affects its density.

Pressure-Volume Relationship

The pressure-volume relationship is an essential principle of gas behavior. According to Boyle’s Law, at constant temperature, the volume of a given amount of gas is inversely proportional to its pressure:\[ PV = \text{constant} \]This means that if the volume of a gas decreases, its pressure increases. When a gas is compressed:

• Volume decreases
• Pressure increases

This relationship helps in understanding the changes that happen within a gas when compressed, especially in constant temperature scenarios. It directly links to why the density increases, as greater pressure on a reduced volume results in a higher density.

Molecular Weight

Molecular weight, or molar mass, is the mass of a given molecule and is a necessary factor in calculating the number of moles, as denoted in the ideal gas law. It is represented in grams per mole (g/mol). Understanding molecular weight links directly to the number of gas molecules and overall mass:

• With known molecular weight: \( n = \frac{m}{M} \)
• Where m is mass and M is molecular weight.

The importance of molecular weight in these equations illustrates its role in determining gas properties, along with how it influences factors like density when the gas undergoes changes such as compression.