Problem 30 - University Physics with Modern Physics

Question

In an adiabatic process for an ideal gas, the pressure decreases. In this process does the internal energy of the gas increase or decrease? Explain your reasoning.

Step-by-Step Solution

Answer

The internal energy of the gas decreases in this adiabatic process.

Step 1: Understanding Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. This means that any change in internal energy of the gas is solely due to work done on or by the system.

Step 2: Relating Pressure Change to Volume Change

In an adiabatic process, the equation \( PV^\gamma = ext{constant} \) (known as Poisson's equation) is applicable, where \( P \) is the pressure, \( V \) is the volume, and \( \gamma \) (gamma) is the heat capacity ratio \( C_p/C_v \). A decrease in pressure usually implies that the volume of the gas is increasing, as they are inversely related in an adiabatic expansion.

Step 3: Using the First Law of Thermodynamics

According to the first law of thermodynamics, \( \Delta U = Q - W \), where \( \Delta U \) is the change in internal energy, \( Q \) is the heat added to the system (which is zero in an adiabatic process), and \( W \) is the work done by the system. Since there is no heat input, \( \Delta U = -W \). Therefore, the internal energy decreases if work is done by the system.

Step 4: Concluding Effect on Internal Energy

When the gas expands during the adiabatic process, it does work on the surroundings, which means it uses up its internal energy. Therefore, if the pressure decreases and the volume increases, the internal energy of the gas decreases.

Key Concepts

Ideal Gas Laws

The Ideal Gas Laws describe the behavior of an ideal gas in terms of pressure, volume, and temperature. The equation commonly used is the Ideal Gas Law: \[ PV = nRT \]where:

• \( P \) is the pressure
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant
• \( T \) is the temperature in Kelvin

This law helps us understand how changes in one of the variables can affect the others, assuming the amount of gas remains constant. In an adiabatic process, however, we use a modified relationship, as discussed later. Understanding these laws is crucial as they apply to many fundamental processes in thermodynamics.
The Ideal Gas Laws are particularly useful for calculating how a gas will respond to changes in its environment, such as those seen in pressure or volume changes, which are key variables in adiabatic processes.

Internal Energy

Internal energy refers to the total energy contained within a system, primarily stemming from the motion and interactions of molecules within the gas. It is a fundamental concept in thermodynamics. Changes in internal energy are represented by:\[ \Delta U = Q - W \]where:

• \( \Delta U \) is the change in internal energy
• \( Q \) is the heat added to the system
• \( W \) is the work done by the system

In an adiabatic process, this becomes much simpler, as there is no heat exchange \( Q = 0 \). Thus, any change in internal energy is directly related to the work done by or on the system. When a gas expands and performs work on its surroundings, it adopts energy from its own reserves, leading to a decrease in its internal energy. This relationship is vital when analyzing energy changes in processes like those involving ideal gases.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, and energy. The first law, known as the Law of Energy Conservation, is particularly important in understanding adiabatic processes. It states:\[ \Delta U = Q - W \]This law asserts that the internal energy change in a system is equal to the heat added minus the work done by the system. In an adiabatic process where no heat is exchanged, the system’s energy changes solely through the work done.
During an adiabatic expansion, the system does work on the surroundings, diminishing its internal energy. Consequently, understanding thermodynamics is crucial for predicting how energy transformations occur, such as those in an adiabatic process.
Thermodynamics guides us by providing the rules that dictate how energy is conserved and transformed, making it indispensable in studying processes that affect an ideal gas.

Poisson's Equation

In the context of adiabatic processes for ideal gases, Poisson's equation plays a key role. It is given by:\[ PV^\gamma = \text{constant} \]where:

• \( P \) is the pressure
• \( V \) is the volume
• \( \gamma = \frac{C_p}{C_v} \) is the heat capacity ratio, representing specific heat at constant pressure over specific heat at constant volume.

This equation explains the relationship between pressure and volume during an adiabatic process. When the gas expands, pressure decreases due to the increase in volume, as they are inversely proportional.
Understanding Poisson's equation is crucial in predicting how pressure and volume will behave during an adiabatic change. It illustrates how these variables are intricately linked in processes where no heat is exchanged with the environment.