Problem 1 - University Physics with Modern Physics

Question

Two moles of an ideal gas are heated at constant pressure from \(T=27^{\circ} \mathrm{C}\) to \(T=107^{\circ} \mathrm{C}\) (a) Draw a \(p V\) -diagram for this process. (b) Calculate the work done by the gas.

Step-by-Step Solution

Answer

The work done by the gas is 1326.24 Joules.

Step 1: Understand the Problem

This exercise deals with an ideal gas that is heated at constant pressure. We need to analyze the changes it undergoes and calculate the work done during these changes given the initial and final temperatures of the gas.

Step 2: Set Up the Initial and Final States

Convert the given temperatures from Celsius to Kelvin for accurate calculations involving gases. The initial temperature \(T_i\) is \(27^{\circ}C = 300K\), and the final temperature \(T_f\) is \(107^{\circ}C = 380K\).

Step 3: Plot the PV Diagram

In a \(pV\) diagram for a constant pressure process, the line is horizontal as pressure remains constant. Label the initial point (state 1) with \(V_1\) and the final point (state 2) with \(V_2\) on this horizontal line, where \(V_2 > V_1\) due to the increase in temperature.

Step 4: Recall the Ideal Gas Law

The ideal gas law equation is \(pV = nRT\), where \(p\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature in Kelvin. Since pressure \(p\) is constant, the volume \(V\) is proportional to temperature \(T\).

Step 5: Calculate Work Done by the Gas

The work done by the gas at constant pressure is given by \(W = p\Delta V\). However, we can also express this as \(W = nR(T_f - T_i)\) since \(\Delta V = \frac{nR(T_f - T_i)}{p}\). Substitute \(n = 2\ \mathrm{moles}, R = 8.314\ \mathrm{J/(mol\cdot K)}, T_f = 380\ \mathrm{K}, T_i = 300\ \mathrm{K}\) to calculate \(W = 2 \times 8.314 \times (380 - 300)\).

Step 6: Perform the Calculation

Calculate the value: \[W = 2 \times 8.314 \times 80 = 1326.24\ \mathrm{J}\]. Thus, the work done by the gas is 1326.24 Joules.

Key Concepts

Constant Pressure Process

When dealing with a constant pressure process, imagine you are inflating a balloon. The external pressure of the balloon remains the same as you add air. Similarly, in our exercise, the gas undergoes heating at a constant pressure. This is also known as an isobaric process. During this, although pressure remains unchanged, other variables like volume and temperature do change.

Some key points about a constant pressure process include:

• Pressure remains the same throughout the process.
• As the gas is heated, its volume will increase if it is not confined.
• This is important in understanding how gases behave under varied conditions while pressure stays the same.

Since pressure is constant, according to Boyle's Law, any increase in temperature will lead to an increase in volume. Thus, the relationship between volume and temperature becomes direct.

Work Done by Gas

Work done by gas can be visualized as the energy transferred from the gas as it expands. In practical terms, think about pushing a piston in a car engine, where the gas pushes the piston out as it expands. The work done is essentially the amount of energy needed to move the piston—this is similar to what happens during the gas expansion.

For gases, specifically at constant pressure, the work done (W) by the gas can be calculated using:\[W = p \Delta V\]Or using the ideal gas law, reformed:\[W = nR(T_f - T_i)\]Where:

• W is work done by the gas.
• p is the constant pressure.
• \Delta V is the change in volume.
• n is the number of moles of the gas.
• R is the universal gas constant (8.314 J/mol·K).
• T_f and T_i are the final and initial temperatures in Kelvin, respectively.

This mathematical formulation helps in solving problems where thermal energy transfers result in mechanical work.

PV Diagram

The PV diagram, or pressure-volume diagram, is a graphical representation of the changes in pressure and volume in a system. It is a crucial tool in thermodynamics for visualizing how a gas undergoes different processes. In our specific exercise dealing with constant pressure, the PV diagram simplifies the visual understanding.

In a constant pressure process, the PV diagram will display a horizontal straight line. Here’s why this happens:

• The horizontal axis represents volume (V), and the vertical axis represents pressure (P).
• Since the pressure is constant, the line is horizontal instead of slanted or vertical.
• The initial state of the gas starts at the left (lower volume), and as the gas heats, it expands, moving right (to a higher volume) along the line.

A PV diagram not only shows the initial and final states but also helps in calculating work done, as the area under the line is indicative of the work exerted by or on the system.