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Chapter 15: Problem 8

A gas, while expanding under isobaric conditions, does \(480 \mathrm{~J}\) of work. The pressure of the gas is \(1.6 \times 10^{5} \mathrm{~Pa}\), and its initial volume is \(1.5 \times 10^{-3} \mathrm{~m}^{3}\). What is the final volume of the gas?

Short Answer

Expert verified
The final volume is \(4.5 \times 10^{-3} \; \text{m}^3.\)

Step by step solution

01

Understanding the Problem

We need to find the final volume of a gas that expands isobarically, meaning at constant pressure, while performing 480 J of work. The initial volume and pressure are given.
02

Applying the Isobaric Work Formula

For an isobaric process, the work done by the gas is given by the formula \( W = P \Delta V \), where \( W \) is the work done, \( P \) is the pressure, and \( \Delta V = V_f - V_i \) is the change in volume. Plug in the known values: \( W = 480 \; \text{J} \), \( P = 1.6 \times 10^5 \; \text{Pa} \), and \( V_i = 1.5 \times 10^{-3} \; \text{m}^3 \).
03

Rearranging the Formula

Rearrange the formula to solve for \( \Delta V \): \[ \Delta V = \frac{W}{P} = \frac{480}{1.6 \times 10^5} \; \text{m}^3. \]
04

Calculating the Change in Volume

Calculate \( \Delta V \): \[ \Delta V = \frac{480}{1.6 \times 10^5} \approx 3.0 \times 10^{-3} \; \text{m}^3. \]
05

Finding the Final Volume

Add the initial volume to the volume change to find the final volume: \[ V_f = V_i + \Delta V = 1.5 \times 10^{-3} + 3.0 \times 10^{-3} = 4.5 \times 10^{-3} \; \text{m}^3. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Work Done by Gas
When dealing with a gas that is expanding or compressing, it's important to understand how work is done. The work done by a gas in an isobaric process, which means the process happens at constant pressure, can be calculated using the formula:
  • \( W = P \Delta V \)
Here \( W \) is the work done by the gas, \( P \) is the pressure, and \( \Delta V \) is the change in volume of the gas.
This formula is straightforward. It takes the pressure (a constant factor in an isobaric process) and multiplies it by how much the volume changes.
For our exercise, the gas performed work by expanding, meaning it used energy to increase its volume. The energy spent in this process is exactly 480 J.
Understanding this concept helps in analyzing how much energy a gas utilizes during its expansion or compression.
Initial Volume
The initial volume of a gas is the starting point in its expansion or compression process. For our problem, the initial volume \( V_i \) was given as \( 1.5 \times 10^{-3} \, \text{m}^3 \).
This measurement reflects how much space the gas occupied at the beginning of the observation.
Knowing the initial volume is crucial because it serves as a baseline to determine how much the volume of the gas changes during the procedure.
In our situation, this initial condition helps us solve for the final volume after the gas has done its work and expanded.
Final Volume
The final volume of a gas represents the new volume it occupies after some change has occurred. In an isobaric process like ours, the formula to find the final volume \( V_f \) is:
  • \( V_f = V_i + \Delta V \)
where \( \Delta V \) is the change in volume.
After calculating \( \Delta V = 3.0 \times 10^{-3} \, \text{m}^3 \), we add this to the initial volume:
  • \( V_f = 1.5 \times 10^{-3} + 3.0 \times 10^{-3} = 4.5 \times 10^{-3} \, \text{m}^3 \)
The final volume tells us how much space the gas occupies after it has expanded.
It is essential for understanding the conditions of the gas after performing work in an isobaric process.
Pressure of Gas
Pressure is a measure of how much force the gas particles exert on the walls of their container. In our scenario, the gas pressure is given as a constant \( 1.6 \times 10^{5} \, \text{Pa} \) (Pascals), as the process is isobaric.
Isobaric conditions ensure that this pressure remains the same throughout the expansion.
Constant pressure means the force exerted per unit area doesn鈥檛 change, which makes the calculation of work done straightforward.
It's critical to keep this in mind, as variations in pressure could significantly alter the work done and subsequently, the final volume of the gas.

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Most popular questions from this chapter

Multiple-Concept Example 6 deals with the concepts that are important in this problem. In doing \(16600 \mathrm{~J}\) of work, an engine rejects \(9700 \mathrm{~J}\) of heat. What is the efficiency of the engine?

Concept Questions Two Carnot air conditioners, \(\mathrm{A}\) and \(\mathrm{B},\) are removing heat from different rooms. The outside temperature is the same for both, but the room temperatures are different. The room serviced by unit \(\mathrm{A}\) is kept colder than the room serviced by unit B. The heat removed from both rooms is the same, (a) Which unit requires the greater amount of work? (b) Which unit deposits the greater amount of heat outside? Explain. Problem The outside temperature is \(309.0 \mathrm{~K}\). The room serviced by unit \(\mathrm{A}\) is kept at a temperature of \(294.0 \mathrm{~K},\) while the room serviced by unit \(\mathrm{B}\) is kept at \(301.0 \mathrm{~K}\). The heat removed from either room is \(4330 \mathrm{~J} .\) For both units, find the magnitude of the work required and the magnitude of the heat deposited outside. Verify that your answers are consistent with your answers to the Concept Questions.

Refer to Interactive Solution 15.87 at for help in solving this problem. A diesel engine does not use spark plugs to ignite the fuel and air in the cylinders. Instead, the temperature required to ignite the fuel occurs because the pistons compress the air in the cylinders. Suppose air at an initial temperature of \(21^{\circ} \mathrm{C}\) is compressed adiabatically to a temperature of \(688^{\circ} \mathrm{C}\). Assume the air to be an ideal gas for which \(\gamma=\frac{7}{5} .\) Find the compression ratio, which is the ratio of the initial volume to the final volume.

Two Camot engines, \(A\) and \(B\), utilize the same hot reservoir, but engine \(\mathrm{A}\) is less efficient than engine \(\mathrm{B}\). (a) Which engine produces more work for a given heat input? (b) Which engine has the lower cold-reservoir temperature? Give your reasoning. Carnot engine A has an efficiency of \(0.60\), and Carnot engine \(\mathrm{B}\) has an efficiency of \(0.80\). Both engines utilize the same hot reservoir, which has a temperature of \(650 \mathrm{~K}\) and delivers \(1200 \mathrm{~J}\) of heat to each engine. Find the magnitude of the work produced by each engine and the temperatures of the cold reservoirs that they use. Check to see that your answers are consistent with your answers to the Concept Questions.

Suppose the gasoline in a car engine burns at \(631^{\circ} \mathrm{C},\) while the exhaust temperature (the temperature of the cold reservoir) is \(139^{\circ} \mathrm{C}\) and the outdoor temperature is \(27^{\circ} \mathrm{C}\) Assume that the engine can be treated as a Carnot engine (a gross oversimplification). In an attempt to increase mileage performance, an inventor builds a second engine that functions between the exhaust and outdoor temperatures and uses the exhaust heat to produce additional work. Assume that the inventor's engine can also be treated as a Carnot engine. Determine the ratio of the total work produced by both engines to that produced by the first engine alone.
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