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Chapter 6: Problem 64

A cylinder/piston contains \(1 \mathrm{~kg}\) propane gas at \(100 \mathrm{kPa}, 300 \mathrm{~K}\). The gas is compressed reversibly to a pressure of \(800 \mathrm{kPa}\). Calculate the work required if the process is adiabatic.

Short Answer

Expert verified
Work required is calculated using adiabatic relations.

Step by step solution

01

Understand the Adiabatic Process

In an adiabatic process, there is no heat transfer into or out of the system. The first law of thermodynamics for a closed system becomes \[ \Delta U = W_{on} \]where \( \Delta U \) is the change in internal energy and \( W_{on} \) is the work done on the system. For an ideal gas, use the relation between initial and final states in an adiabatic process:\[ PV^{\gamma} = \text{constant} \]where \( \gamma = \frac{C_p}{C_v} \), the ratio of specific heats.
02

Determine Specific Heat Ratio and Initial Conditions

For propane, the specific heat ratio \( \gamma \) is approximately 1.13. The initial conditions provided are Pressure \( P_1 = 100 \) kPa and Temperature \( T_1 = 300 \) K. We need to find the initial volume \( V_1 \) using the ideal gas law:\[ PV = nRT \]where \( n \) is the number of moles, \( R = 8.314 \frac{J}{mol \cdot K} \), and the molar mass of propane \( = 44.1 \frac{g}{mol} \).

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Key Concepts

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

Thermodynamics
Thermodynamics is the study of energy, its transformations, and its relationship with matter. It explains how thermal energy is converted to and from other forms of energy and how it affects matter. In thermodynamics, an important concept is a system which can be defined as the part of the universe being studied. Everything outside the system is known as the surroundings.
  • Adiabatic processes occur when a system exchanges no heat with its surroundings. This means that all changes in the system are due to work done on or by the system.
  • The laws of thermodynamics, including the first and second laws, govern how energy moves and changes form.
  • An adiabatic process can be reversible or irreversible, meaning it could reach the same end state by reversing the process or not.
Understanding thermodynamic principles provides the foundation for analyzing energy changes in physical and chemical processes, which is crucial for engineering applications and scientific studies.
Ideal Gas Law
The Ideal Gas Law is a fundamental equation in thermodynamics that describes how gases behave under various conditions. The formula is given by the equation:\[ PV = nRT \]where:
  • \( P \) is the pressure of the gas.
  • \( V \) is the volume of the gas.
  • \( n \) is the number of moles of gas.
  • \( R \) is the universal gas constant (approximately 8.314 \( J/mol \cdot K \)).
  • \( T \) is the temperature in Kelvin.
The Ideal Gas Law assumes no interactions between the gas molecules except collisions, and it works best under low-pressure and high-temperature conditions.
Adiabatic processes can be well-explained using the Ideal Gas Law by understanding how any change in one of the properties (like pressure or volume) must affect the others if the gas remains ideal.
Specific Heat Ratio
The specific heat ratio, denoted as \( \gamma \) (gamma), is the ratio of the specific heat at constant pressure \( (C_p) \) to the specific heat at constant volume \( (C_v) \). It is crucial in describing adiabatic processes. Mathematically, it's expressed as:\[ \gamma = \frac{C_p}{C_v} \]
  • This ratio impacts how gases expand and compress adiabatically.
  • A higher \( \gamma \) indicates that a gas requires more work for a given adiabatic process, which typically results in larger temperature changes.
  • For propane, a common specific heat ratio value is approximately 1.13. This is used to calculate changes within adiabatic conditions, helping determine initial volumes and work done.
An understanding of the specific heat ratio allows engineers to predict the energy requirements and efficiency in systems like heat engines.
First Law of Thermodynamics
The First Law of Thermodynamics is often stated as the conservation of energy principle. It asserts that energy can neither be created nor destroyed, only transformed from one form to another. In equation form for a closed system, the first law is:\[ \Delta U = Q - W \]where:
  • \( \Delta U \) is the change in internal energy of the system.
  • \( Q \) is the heat added to the system.
  • \( W \) is the work done by the system.
In adiabatic processes, \( Q = 0 \), simplifying the equation to \( \Delta U = -W \). This means that changes in internal energy are entirely due to the work done on or by the system.
The First Law of Thermodynamics is fundamental in understanding how energy transformations are calculated in processes like compression and expansion of gases, such as those in a piston-cylinder assembly.

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Most popular questions from this chapter

A rigid, insulated vessel contains superheated vapor steam at \(3 \mathrm{MPa}, 400^{\circ} \mathrm{C}\). A valve on the vessel is opened, allowing steam to escape, as shown in Fig. P6.35. The overall process is irreversible, but the steam remaining inside the vessel goes through a reversible adiabatic expansion. Determine the fraction of steam that has escaped when the final state inside is saturated vapor.

Nitrogen at \(200^{\circ} \mathrm{C}, 300 \mathrm{kPa}\) is in a piston/cylinder, volume \(5 \mathrm{~L}\), with the piston locked with a pin. The forces on the piston require a pressure inside of \(200 \mathrm{kPa}\) to balance it without the pin. The pin is removed, and the piston quickly comes to its equilibrium position without any heat transfer. Find the final \(P, T\) and the entropy generation due to this partly unrestrained expansion.

A cylinder/piston contains \(4 \mathrm{ft}^{3}\) air at 16 lbf/in. \({ }^{2}\), \(77 \mathrm{~F}\). The air is compressed in a reversible polytropic process to a final state of \(120 \mathrm{lbf} / \mathrm{in} .^{2}, 400\) F. Assume the heat transfer is with the ambient at \(77 \mathrm{~F}\) and determine the polytropic exponent \(n\) and the final volume of the air. Find the work done by the air, the heat transfer, and the total entropy generation for the process.

A cylinder fitted with a movable piston contains water at \(3 \mathrm{MPa}, 50 \%\) quality, at which point the volume is \(20 \mathrm{~L}\). The water now expands to \(1.2 \mathrm{MPa}\) as a result of receiving \(600 \mathrm{~kJ}\) of heat from a large source at \(300^{\circ} \mathrm{C}\). It is claimed that the water does \(124 \mathrm{~kJ}\) of work during this process. Is this possible?

A \(750 \mathrm{~W}\) electric heater is on for 15 minutes and delivers a heat transfer from its \(800 \mathrm{~K}\) surface to \(25^{\circ} \mathrm{C}\) air. How much entropy is generated in the heater element? How much entropy, if any, is generated outside the heater element?
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