Problem 9 - University Physics with Modern Physics

Question

A gas in a cylinder expands from a volume of 0.110 \(\mathrm{m}^{3}\) to 0.320 \(\mathrm{m}^{3}\) . Heat flows into the gas just rapidly enough to keep the pressure constant at \(1.80 \times 10^{5}\) Pa during the expansion. The total heat added is \(1.15 \times 10^{5} \mathrm{J}\) . (a) Find the work done by the gas. (b) Find the change in internal energy of the gas. it matter whether the gas is ideal? Why or why not?

Step-by-Step Solution

Answer

(a) The work done by the gas is \(3.78 \times 10^{4}\) J. (b) The change in internal energy is \(7.72 \times 10^{4}\) J. The gas type does not affect these calculations.

Step 1: Understand the Scenario

The gas in a cylinder expands under constant pressure. You're given the initial and final volumes, the constant pressure, and the total heat added to the system. You need to find the work done by the gas during expansion and the change in internal energy of the gas.

Step 2: Calculate Work Done by the Gas

Work done by the gas during expansion at constant pressure is calculated using the formula \( W = P \Delta V \), where \( P \) is the pressure and \( \Delta V \) is the change in volume.\[ \Delta V = V_{final} - V_{initial} = 0.320 \ \mathrm{m}^{3} - 0.110 \ \mathrm{m}^{3} = 0.210 \ \mathrm{m}^{3} \]Plug the values into the formula:\[ W = 1.80 \times 10^{5} \times 0.210 = 3.78 \times 10^{4} \ \mathrm{J} \]

Step 3: Use the First Law of Thermodynamics

The first law of thermodynamics states \( \Delta U = Q - W \), where \( \Delta U \) is the change in internal energy, \( Q \) is the heat added, and \( W \) is the work done by the system. We calculated \( W \) in the previous step, and \( Q \) is given.\[ \Delta U = 1.15 \times 10^{5} - 3.78 \times 10^{4} \]\[ \Delta U = 7.72 \times 10^{4} \ \mathrm{J} \]

Step 4: Consider if Gas Type Matters

For this problem, whether the gas is ideal does not matter. The calculations are purely thermodynamic and do not depend on the specific properties of the gas beyond its behavior of expanding at constant pressure.

Key Concepts

First Law of Thermodynamics

The First Law of Thermodynamics is an essential principle in physics that provides a clear understanding of how energy interactions occur within a thermodynamic system. This law is essentially about energy conservation.
It states that the change in internal energy of a system, denoted as \( \Delta U \), is equal to the heat added to the system, \( Q \), minus the work done by the system, \( W \). This relationship is mathematically expressed as:
\[ \Delta U = Q - W \]

• \( \Delta U \): Change in internal energy
• \( Q \): Heat added
• \( W \): Work done by the system

This equation helps us analyze energy changes during processes like heating, cooling, and work done by gases. If energy enters a system as heat, it can either be stored within the system as internal energy or leave as work done by the system. Understanding this basic idea helps us solve problems in thermodynamics with greater ease.

Constant Pressure Process

In a constant pressure process, the pressure of the system remains unchanged throughout the entire process. This is also known as an isobaric process. Such processes are common in real-world scenarios and are essential for studying the behavior of gases in dynamic environments.
During this type of process, as seen in the exercise, the gas expands while maintaining a fixed pressure. We use the formula for work done at constant pressure:\[ W = P \Delta V \]
where:

• \( P \): Constant pressure
• \( \Delta V \): Change in volume, \( V_{final} - V_{initial} \)

This formula allows us to calculate the work done by the gas as it expands against the constant pressure in the cylinder. The work required for expansion depends not only on the pressure but also significantly on the change in volume.

Internal Energy Change

Internal energy change, \( \Delta U \), is a core concept in thermodynamics and relates to the energy stored within a system. It signifies how much energy is held inside the system due to factors like temperature and molecular interaction.
When analyzing any thermodynamic process, it's crucial to determine how much the internal energy changes. Using the first law of thermodynamics, \( \Delta U = Q - W \), enables us to compute it.
In this exercise, after the system undergoes an expansion at constant pressure, the calculation of \( \Delta U \) involves:

• Identifying \( Q \) (heat added, \( 1.15 \times 10^{5} \) J)
• Determining \( W \) (work done by the gas, \( 3.78 \times 10^{4} \) J)
• Applying them in the equation to find \( \Delta U \)

These calculations show how much energy has shifted within the gas system as part of the internal energy change, helping us to better understand energy flow and transformation in thermodynamic processes.