Problem 109 - Introductory Chemistry Atoms First

Question

(a) If the temperature of a gas is doubled while the pressure is kept constant, the volume of the gas (b) If the pressure of a gas is halved while the temperature is kept constant, the volume of the gas

Step-by-Step Solution

Answer

(a) When the temperature of a gas is doubled while the pressure is kept constant, the volume of the gas is doubled (\(V2 = 2 * V1\)). (b) When the pressure of a gas is halved while the temperature is kept constant, the volume of the gas is doubled (\(V2 = 2 * V1\)).

Step 1: Write down the ideal gas law equation

The ideal gas law states that (P * V) / T = constant.

Step 2: Find the initial and final states

In this case, the initial state has temperature T1, pressure P1, and volume V1. The final state has temperature 2 * T1 (since the temperature is doubled), pressure P1 (since the pressure is constant), and volume V2.

Step 3: Apply the ideal gas law for the initial and final states

For the initial state: (P1 * V1) / T1 = C For the final state: (P1 * V2) / (2 * T1) = C

Step 4: Compare the initial and final states

Since the constant C is the same for both equations, we can set them equal to each other: (P1 * V1) / T1 = (P1 * V2) / (2 * T1)

Step 5: Solve for the final volume V2

By cross-multiplying and simplifying, we get: V2 = 2 * V1 So, the volume of the gas when the temperature is doubled and the pressure is kept constant is double the initial volume. #Scenario 2: Pressure halved while the temperature is kept constant#

Step 1: Write down the ideal gas law equation

The ideal gas law states that (P * V) / T = constant.

Step 2: Find the initial and final states

In this case, the initial state has temperature T1, pressure P1, and volume V1. The final state has temperature T1 (since the temperature is constant), pressure P1/2 (since the pressure is halved), and volume V2.

Step 3: Apply the ideal gas law for the initial and final states

For the initial state: (P1 * V1) / T1 = C For the final state: ((P1/2) * V2) / T1 = C

Step 4: Compare the initial and final states

Since the constant C is the same for both equations, we can set them equal to each other: (P1 * V1) / T1 = ((P1/2) * V2) / T1

Step 5: Solve for the final volume V2

By cross-multiplying and simplifying, we get: V2 = 2 * V1 So, the volume of the gas when the pressure is halved and the temperature is kept constant is double the initial volume.

Key Concepts

temperature and volume relationship

When exploring the relationship between temperature and volume in gases, we need to consider Charles's Law. It is a part of the ideal gas law and states that, at constant pressure, the volume of a gas is directly proportional to its temperature. This means that as the temperature of a gas increases, its volume will also increase if the pressure is constant.
For example:

• Doubling the temperature of an ideal gas will double the volume, assuming the pressure stays the same.
• Conversely, if the temperature is halved, the volume is also halved.

The relationship can be mathematically represented as \( V_1/T_1 = V_2/T_2 \). This formula allows us to calculate how the volume changes with temperature under constant pressure conditions. This understanding is crucial for working with gases in real-world settings, such as inflating a balloon or operating a combustion engine.

pressure and volume relationship

Another key relationship in gas behavior is explained by Boyle's Law. This principle highlights that the volume of a gas is inversely proportional to its pressure when temperature remains constant. Practically, this means if you increase the pressure exerted on a gas, its volume will decrease and vice versa.
Consider these scenarios:

• When pressure is doubled, the volume of the gas is reduced by half.
• If the pressure is halved, the volume will double.

The mathematical representation of Boyle's Law is \( P_1V_1 = P_2V_2 \). By using this equation, you can anticipate how changes in pressure will affect the volume, considering the temperature doesn't change. This relationship is vital for understanding phenomena like how a bicycle pump works or why a soda can bursts when over-pressurized.

gas laws in chemistry

The behavior of gases is primarily governed by three classical gas laws: Boyle’s Law, Charles’s Law, and Avogadro's Law, collectively integrated into the Ideal Gas Law. The ideal gas law formulates how pressure, volume, temperature, and the number of molecules relate to each other in a typical gas setting. It is expressed with the equation \( PV = nRT \), where:

• \( P \) stands for pressure,
• \( V \) is volume,
• \( n \) signifies the number of moles of gas,
• \( R \) is the ideal gas constant,
• \( T \) symbolizes temperature.

Understanding gas laws helps students and professionals predict and manipulate how gases will react under various conditions. These laws are pivotal in numerous chemistry and physics applications, such as predicting molecular behavior in different environments, designing pressurized vessels, and calculating the energy potential in fuels.