Problem 24 - University Physics with Modern Physics

Question

An ideal gas expands while the pressure is kept constant. During this process, does heat flow into the gas or out of the gas? Justify your answer.

Step-by-Step Solution

Answer

Heat flows into the gas because work is done by the gas during expansion.

Step 1: Understanding the Process

This process is an isobaric expansion, meaning that the pressure of the gas remains constant while the volume increases.

Step 2: Applying the First Law of Thermodynamics

According to the first law of thermodynamics, the change in internal energy \( \Delta U \) of a system is given by \( \Delta U = Q - W \), where \( Q \) is the heat added to the system and \( W \) is the work done by the system.

Step 3: Analyze the Work Done by the Gas

For a gas expanding at constant pressure, the work done by the gas is given by \( W = P \Delta V \), where \( P \) is the pressure and \( \Delta V \) is the change in volume. Since the volume increases, \( \Delta V > 0 \), thus \( W \) is positive.

Step 4: Determine the Direction of Heat Flow

To keep the internal energy change \( \Delta U \) balanced, heat \( Q \) must be added to the system to compensate for the positive work done by the gas. Therefore, \( Q > 0 \), meaning heat flows into the gas.

Key Concepts

Isobaric Process

An isobaric process is a thermodynamic process where the pressure remains constant while other parameters, such as volume or temperature, may change. In the context of an ideal gas, an isobaric process often involves a gas expanding or compressing. When we say that the pressure is constant, it means that every expansion or contraction of the gas occurs without any change in pressure.
Such a process is typically seen in systems where there is enough time to adjust to pressure changes, such as a piston slowly moving in an engine.Some key aspects of an isobaric process include:

• Pressure, denoted as \( P \), is constant throughout the process.
• The volume change, \( \Delta V \), is directly related to the work done.
• The first law of thermodynamics can be applied to understand energy flow in the system.

Understanding these aspects helps us predict how heat and work interplay in such processes. The constant pressure condition simplifies many calculations and allows for clearer analysis of other changing variables.

First Law of Thermodynamics

The first law of thermodynamics is a statement of energy conservation. It tells us that energy cannot be created or destroyed, only transformed from one form to another. In the context of thermodynamics, it is expressed as:\[ \Delta U = Q - W \]Here, \( \Delta U \) represents the change in internal energy of a system. \( Q \) stands for the heat added to the system, and \( W \) is the work done by the system.
This law is fundamental in analyzing thermodynamic processes. It provides a mathematical framework to predict how energy changes form:

• If heat is added, \( Q > 0 \), it increases the internal energy or does work on the surroundings.
• If work is done by the system, \( W > 0 \), it either reduces internal energy or requires heat input.
• The direction of heat transfer and work done dictates changes in internal energy.

The first law of thermodynamics aids in understanding energy flow in processes like heating, cooling, and phase changes.

Heat Transfer

Heat transfer in thermodynamic processes is a crucial aspect as it influences how systems reach equilibrium. When discussing heat transfer, we consider whether heat moves into or out of a system.
There are three main methods of heat transfer:

• Conduction: Heat transfer through direct contact.
• Convection: Heat transfer due to fluid movement.
• Radiation: Heat transfer via electromagnetic waves.

In an isobaric process where a gas expands, the concern is whether heat needs to be added to balance the work done by the gas. According to the first law of thermodynamics, if a gas does work by expanding at a constant pressure, heat must flow into it. This maintains the internal energy balance by compensating for the energy used to perform work on the surroundings. Thus, heat transfer during an isobaric process becomes crucial to maintain the desired state of the gas.

Ideal Gas Behavior

Ideal gas behavior is a simplified model to describe how gases behave under varying conditions of temperature, pressure, and volume. Ideal gases follow the equation:\[ PV = nRT \]Where \( P \) is pressure, \( V \) is volume, \( n \) is the amount of gas in moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin.
This equation assumes there are no interactions between the gas particles except during elastic collisions and that the volume of the gas particles is negligible compared to the space they occupy. Although no real gases perfectly follow this model, the ideal gas law provides a close approximation for many gases under typical conditions.In an isobaric process:

• The pressure \( P \) is held constant, while changes in volume \( V \) directly affect temperature \( T \).
• When the volume increases (as in expansion), the temperature tends to rise if the amount of gas remains the same.
• This behavior is often used to understand and predict the changes in energy states when a gas undergoes thermal processes.

Understanding ideal gas behavior is fundamental in solving problems related to expansion, compression, and other thermodynamic processes.