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Ch 19: The First Law of Thermodynamics
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
Chapter 19, Problem 12c

A gas in a cylinder is held at a constant pressure of 1.80×1051.80\(\times\)10^5 Pa and is cooled and compressed from 1.701.70 m3 to 1.201.20 m3. The internal energy of the gas decreases by 1.40×1051.40\(\times\)10^5 J. Does it matter whether the gas is ideal? Why or why not?

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To determine whether it matters if the gas is ideal, we need to consider the properties and assumptions of an ideal gas. An ideal gas follows the ideal gas law, which is expressed as: 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 the context of this problem, the gas is undergoing a process where its volume changes while the pressure remains constant. This is an isobaric process. For an ideal gas, the change in internal energy depends only on the change in temperature, not on the volume or pressure directly.
The first law of thermodynamics states that the change in internal energy ΔU is equal to the heat added to the system Q minus the work done by the system W: ΔU=Q-W. For an ideal gas, the work done during an isobaric process is given by W=PΔV, where ΔV is the change in volume.
In this problem, the internal energy decreases, which implies that the work done by the gas is greater than the heat added to it. For an ideal gas, this relationship is straightforward because the internal energy change is directly related to temperature change.
Therefore, whether the gas is ideal or not can affect the interpretation of how internal energy changes with temperature. However, the problem provides the change in internal energy directly, so the ideal gas assumption primarily affects how we understand the relationship between temperature, pressure, and volume changes. In this specific problem, the ideal gas assumption simplifies the analysis but does not change the given data about internal energy change.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics, expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. It describes the behavior of an ideal gas, assuming no interactions between molecules and that the volume of the molecules is negligible. Understanding whether a gas behaves ideally helps determine if this law can be applied to predict changes in state variables.
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First Law of Thermodynamics

The First Law of Thermodynamics, also known as the law of energy conservation, states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system on its surroundings. Mathematically, it is expressed as ΔU = Q - W. This principle is crucial for analyzing energy changes in processes involving gases, such as cooling and compression.
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Real vs. Ideal Gases

Real gases deviate from ideal behavior due to intermolecular forces and the finite volume of gas molecules, especially at high pressures and low temperatures. Whether a gas is ideal affects calculations of state changes, as real gases require more complex equations of state, like the Van der Waals equation, to account for these deviations. Understanding the nature of the gas helps determine the accuracy of predictions using the Ideal Gas Law.
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Related Practice
Textbook Question

Five moles of an ideal monatomic gas with an initial temperature of 127127°C expand and, in the process, absorb 15001500 J of heat and do 21002100 J of work. What is the final temperature of the gas?

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Textbook Question

A gas in a cylinder expands from a volume of 0.1100.110 m3 to 0.3200.320 m3 . Heat flows into the gas just rapidly enough to keep the pressure constant at 1.65×1051.65\(\times\)10^5 Pa during the expansion. The total heat added is 1.15×1051.15\(\times\)10^5 J. Find the work done by the gas.

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Textbook Question

The pVpV-diagram in Fig. E19.1319.13 shows a process abcabc involving 0.4500.450 mol of an ideal gas. How much heat had to be added during the process to increase the internal energy of the gas by 15,00015,000 J?

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Textbook Question

When water is boiled at a pressure of 2.002.00 atm, the heat of vaporization is 2.20×1062.20\(\times\)10^6 J/kg and the boiling point is 120120°C. At this pressure, 1.001.00 kg of water has a volume of 1.00×1031.00\(\times\)10^{-3} m3, and 1.001.00 kg of steam has a volume of 0.8240.824 m3. Compute the increase in internal energy of the water.

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Textbook Question

The process abcabc shown in the pVpV-diagram in Fig. E19.1119.11 involves 0.01750.0175 mol of an ideal gas. What was the lowest temperature the gas reached in this process? Where did it occur?

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Textbook Question

When water is boiled at a pressure of 2.002.00 atm, the heat of vaporization is 2.20×1062.20\(\times\)10^6 J/kg and the boiling point is 120120°C. At this pressure, 1.001.00 kg of water has a volume of 1.00×1031.00\(\times\)10^{-3} m3, and 1.001.00 kg of steam has a volume of 0.8240.824 m3. Compute the work done when 1.001.00 kg of steam is formed at this temperature.

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