Problem 18 - University Physics with Modern Physics

Question

During an isothermal compression of an ideal gas, 335 \(\mathrm{J}\) of heat must be removed from the gas to maintain constant temperature. How much work is done by the gas during the process?

Step-by-Step Solution

Answer

The work done by the gas is -335 J.

Step 1: Understand the Process

In an isothermal compression process, the temperature (T) of the gas remains constant. For an ideal gas, this means the internal energy change (\( \Delta U \)) is zero. So, according to the first law of thermodynamics, \( \Delta U = Q - W = 0 \), where \( Q \) is the heat exchanged and \( W \) is the work done by the gas.

Step 2: Apply the First Law of Thermodynamics

From the first law, since \( \Delta U = 0 \), we have \( Q = W \). Therefore, the work done by the gas equals the heat removed. Given that 335 \( \mathrm{J} \) of heat is removed, thus \( W = Q = 335 \mathrm{J} \) , but since the work is done on the gas during compression, \( W = -335 \mathrm{J} \).

Step 3: Conclude the Calculation

Since we are calculating the work \( W \) done by the gas and it has a negative value because it is compression, the work done by the gas is \(-335 \mathrm{J} \). This negative sign indicates that work is done on the gas, not by the gas.

Key Concepts

Ideal Gas

An ideal gas is a hypothetical concept that simplifies the study of gases by assuming certain conditions. These conditions include:

• The gas consists of a large number of tiny particles (molecules) that are in constant random motion.
• The particles are so small that their volume is negligible compared to the total volume of the gas.
• There are no interactions between the particles except during elastic collisions.

These assumptions make it easier to apply mathematical models to predict the behavior of gases. One of the key equations used for ideal gases 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 equation helps us understand how changes in volume, pressure, and temperature relate when dealing with an ideal gas, especially during processes like isothermal compression.

First Law of Thermodynamics

The first law of thermodynamics is a foundational principle in physics that deals with the conservation of energy. It can be stated as:\[ \Delta U = Q - W \]Here:

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

In the context of an isothermal process, where the temperature of an ideal gas remains constant, the change in internal energy \( \Delta U \) is zero. This simplifies the equation to \( Q = W \), meaning the heat exchanged is equal to the work done. In the case of isothermal compression, if heat is removed, it indicates work is done on the gas. If 335 J of heat is removed, the work done is \( -335 \) J, illustrating energy balance.

Work and Heat Transfer

Understanding work and heat transfer is crucial when analyzing thermodynamic processes: - **Work** is the energy transferred when a force is applied over a distance. For a gas, work is associated with volume change during processes. In the case of compression, work is done on the gas, often resulting in a decrease in volume. - **Heat Transfer** occurs when thermal energy is exchanged between a system and its surroundings. Heat can be absorbed or released depending on whether the system is heated or cooled. For isothermal compression of an ideal gas:

• Heat is removed to maintain a constant temperature.
• The work done on the gas results is energy being transferred out of the gas.

This illustrates the interdependence of work and heat transfer, demonstrating how energy changes form but is conserved according to the first law of thermodynamics.