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Chapter 15: Problem 45

An ideal gas expands while the pressure is kept constant. During this process, does heat flow into the gas or out of the gas? Justify your answer.

Short Answer

Expert verified
Heat flows into the gas.

Step by step solution

01

Understanding the Concept

When an ideal gas expands under constant pressure, it means that the volume of the gas increases while the pressure remains the same. According to the first law of thermodynamics, the change in internal energy is equal to the heat added to the system minus the work done by the system.
02

Applying the First Law of Thermodynamics

The first law of thermodynamics is given by the equation: \[\Delta U = Q - W\]where \(\Delta U\) is the change in internal energy, \(Q\) is the heat added to the system, and \(W\) is the work done by the system. In a process at constant pressure, the work done by the gas is \(P \Delta V\), where \(P\) is pressure and \(\Delta V\) is the change in volume.
03

Analyzing the Work Done

Since the gas is expanding, \(\Delta V > 0\), which implies that the work done \(W = P \Delta V > 0\). This means the system is doing work on the surroundings as the gas expands.
04

Considering the Heat Flow

To maintain a constant pressure while the gas expands, energy must be added to the system to compensate for the work done (\(W > 0\)). This means that the heat \(Q\) must be positive, indicating that heat flows into the gas.
05

Conclusion

Therefore, during the expansion of an ideal gas at constant pressure, heat flows into the gas to accomplish the work done during the expansion.

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Key Concepts

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

ideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that interact only through elastic collisions. This simplification allows us to model the behavior of real gases under certain conditions.
Key characteristics of an ideal gas include:
  • No intermolecular forces: The gas particles do not attract or repel each other.
  • Elastic collisions: When gas particles collide, they do so without losing energy.
  • Point particles: The size of the particles is negligible compared to the distances between them.
These assumptions are captured by the Ideal Gas Law equation: \[ PV = nRT \]where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin.
The Ideal Gas Law provides a simple way to understand the relationship between pressure, volume, and temperature within a closed system.
first law of thermodynamics
The first law of thermodynamics is a form of the law of conservation of energy, adapted for thermodynamic systems. It states that the total energy of an isolated system is constant.
The mathematical expression of the first law of thermodynamics is:\[ \Delta U = Q - W \]where:
  • \(\Delta U\) is the change in internal energy of the system.
  • \(Q\) is the heat exchanged with the surroundings (positive if added to the system).
  • \(W\) is the work done by the system on the surroundings (positive if done by the system).
This law allows us to understand how energy flows within a system, and it is especially useful for dealing with heat and work interactions in processes such as expansion or compression of gases.
constant pressure process
A constant pressure process is a thermodynamic process during which the pressure remains fixed while other variables like volume and temperature may change. It is also known as an isobaric process.
In such processes, the work done by or on the system can be calculated by:\[ W = P \Delta V \]where \(P\) is the constant pressure, and \(\Delta V\) is the change in volume.
These processes are common in engineering applications, such as in piston engines where gases expand or compress at constant pressure, resulting in work being performed. Understanding this process is crucial for predicting how systems respond to various thermodynamic changes.
heat flow in gases
Heat flow in gases occurs when energy is transferred between a system and its surrounding due to a temperature difference. This can happen by various mechanisms such as conduction, convection, or radiation.
In the context of expanding gases, particularly under constant pressure, heat flow is guided by the need to balance the energy equation as explained by the first law of thermodynamics:\[ Q = \Delta U + W \]where positive \(Q\) indicates that heat is entering the system.
  • When an ideal gas expands, its volume increases, and the gas does work on its surroundings.
  • For the expansion to continue at constant pressure, energy in the form of heat must flow into the system to offset the work done by gas.
This process underlines the fundamental principle that heat input is essential to maintain constant pressure as volume changes.

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Most popular questions from this chapter

Calculate the volume of \(1.00 \mathrm{~mol}\) of liquid water at a temperature of \(20^{\circ} \mathrm{C}\) (at which its density is \(998 \mathrm{~kg} / \mathrm{m}^{3}\) ), and compare this volume with the volume occupied by \(1.00 \mathrm{~mol}\) of steam at \(200^{\circ} \mathrm{C}\). Assume the steam is at atmospheric pressure and can be treated as an ideal gas.

Helium gas with a volume of \(2.60 \mathrm{~L}\) under a pressure of 1.30 atm and at a temperature of \(41.0^{\circ} \mathrm{C}\) is warmed until both the pressure and volume of the gas are doubled. (a) What is the final temperature? (b) How many grams of helium are there? The molar mass of helium is \(4.00 \mathrm{~g} / \mathrm{mol}\).

A cylinder contains \(0.250 \mathrm{~mol}\) of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) gas at a temperature of \(27.0^{\circ} \mathrm{C}\). The cylinder is provided with a frictionless piston, which maintains a constant pressure of 1.00 atm on the gas. The cylinder is placed on a hot plate and \(920 \mathrm{~J}\) of heat flows into the gas, thereby raising its temperature to \(127.0^{\circ} \mathrm{C}\). Assume that the \(\mathrm{CO}_{2}\) may be treated as an ideal gas. (a) Draw a \(p V\) diagram of this process. (b) How much work is done by the gas in the process? (c) On what is this work done? (d) What is the change in internal energy of the gas? (e) How much work would have been done if the pressure had been 0.50 atm?

One type of gas mixture used in anesthesiology is a \(50 \% / 50 \%\) mixture (by volume) of nitrous oxide \(\left(\mathrm{N}_{2} \mathrm{O}\right)\) and oxygen \(\left(\mathrm{O}_{2}\right),\) which can be premixed and kept in a cylinder for later use. Because these two gases don't react chemically at or below 2000 psi, at typical room temperatures they form a homogeneous single- gas phase, which can be considered an ideal gas. If the temperature drops below \(-6^{\circ} \mathrm{C},\) however, \(\mathrm{N}_{2} \mathrm{O}\) may begin to condense out of the gas phase. Then any gas removed from the cylinder will initially be nearly pure \(\mathrm{O}_{2}\). As the cylinder empties, the proportion of \(\mathrm{O}_{2}\) will decrease until the gas coming from the cylinder is nearly pure \(\mathrm{N}_{2} \mathrm{O}\). In a hospital, pure oxygen may be delivered at 50 psi (gauge pressure) and then mixed with \(\mathrm{N}_{2} \mathrm{O}\). What volume of oxygen at \(20^{\circ} \mathrm{C}\) and 50 psi (gauge pressure) should be mixed with \(1.7 \mathrm{~kg}\) of \(\mathrm{N}_{2} \mathrm{O}\) to get a \(50 \% / 50 \%\) mixture by volume at \(20^{\circ} \mathrm{C} ?\) A. \(0.21 \mathrm{~m}^{3}\) B. \(0.27 \mathrm{~m}^{3}\) C. \(1.9 \mathrm{~m}^{3}\) D. \(100 \mathrm{~m}^{3}\)

An ideal gas at 4.00 atm and \(350 \mathrm{~K}\) is permitted to expand adiabatically to 1.50 times its initial volume. Find the final pressure and temperature if the gas is (a) monatomic with \(C_{p} / C_{V}=\frac{5}{3},\) (b) diatomic with \(C_{p} / C_{V}=\frac{7}{5}\).
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