Problem 31 - College Physics

Question

(a) If the temperature of an ideal gas increases and its volume decreases, will the pressure of the gas (1) increase, (2) remain the same, or (3) decrease? Why? (b) The Kelvin temperature of an ideal gas is doubled and its volume is halved. How is the pressure affected?

Step-by-Step Solution

Answer

(a) Increase; (b) Pressure quadruples.

Step 1: Understanding the Ideal Gas Law

The ideal gas law is given by \( PV = nRT \), where \( P \) is the pressure of the gas, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. Our task is to understand how changes in temperature \( T \) and volume \( V \) affect the pressure \( P \).

Step 2: Analyzing Part (a)

In part (a), the temperature \( T \) of the gas increases, and its volume \( V \) decreases. According to the ideal gas law, if \( T \) increases and \( V \) decreases, the pressure \( P \) must increase to keep the product \( PV = nRT \) constant with constant \( n \) and \( R \).

Step 3: Understanding Part (b) Conditions

In part (b), the Kelvin temperature \( T \) is doubled, and the volume \( V \) is halved. So, the changes can be expressed as \( T' = 2T \) and \( V' = V/2 \). We need to find the effect on pressure.

Step 4: Calculating Pressure Change for Part (b)

By substituting the changed variables into the ideal gas equation: the new pressure \( P' \) is given by \( P'V' = nRT' \). Hence, \( P' \cdot \frac{V}{2} = nR \,\cdot 2T \). Simplifying, we get \( P' \times V = 2 \times 2 \times nRT \) which implies that \( P' = 4P \). Thus, the pressure increases by a factor of 4.

Key Concepts

Pressure Changes

Pressure is a measure of the force that gas particles exert on the walls of their container. According to the ideal gas law expressed as \( PV = nRT \), the pressure \( P \) is directly related to the temperature \( T \) and inversely related to the volume \( V \) when the number of moles \( n \) and the ideal gas constant \( R \) remain unchanged.
If the volume of a gas decreases while the temperature increases, these changes affect how often and how forcefully the gas particles collide with the container walls, leading to changes in pressure.

• An increase in temperature generally means particles move faster and collide more energetically with the walls. This tends to increase pressure.
• A decrease in volume causes particles to collide more frequently with the walls due to the reduced space, also leading to increased pressure.

Thus, an increase in temperature combined with a decrease in volume generally results in increased pressure.

Temperature and Volume

Temperature and volume are two key factors affecting the behavior of a gas under the ideal gas law. They help determine the pressure when changes occur within a closed system of gas.
According to Charles's Law, volume is directly proportional to the Kelvin temperature when all other factors are constant. This means that as the temperature of a gas increases, its volume should also expand if pressure remains constant.
However, thinking back to the ideal gas law, if both temperature and pressure increase, but volume decreases, the interplay becomes more dynamic:

• If temperature doubles while volume is halved, without changing other variables, there will be a significant pressure change. For instance, if the temperature doubles and the volume halves, the pressure will quadruple.

This relationship highlights how interconnected these factors are within the framework of gas behavior.

Kelvin Temperature Effect

The Kelvin scale is crucial when using the ideal gas law because it directly ties the temperature to the kinetic energy of gas particles. The Kelvin scale starts at absolute zero, the point where particles theoretically stop moving.
Increases in Kelvin temperature translate to increased kinetic energy in gas particles, which has several effects:

• An increase in Kelvin temperature leads to faster particle movement, increasing both the frequency and force of collisions within the volume of gas.
• When temperature increases under constant volume conditions, pressure will increase, since particles collide more often and more forcefully with the container walls.

Understanding the Kelvin temperature effect is essential for accurately predicting how pressure will change when adjustments in temperature are made, such as doubling the temperature as seen in the original exercise.