One mole of methane in a piston-cylinder system undergoes an adiabatic compression from and initial state (P1 = 0.5 bar, V1 = 0.05 m3/mol) to a final state (P2 = 10 bar, V2 = 0.003 m3/mol). How much work is done on the system?
One mole of methane in a piston-cylinder system undergoes an adiabatic compression from and initial state (P1 = 0.5 bar, V1 = 0.05 m3/mol) to a final state (P2 = 10 bar, V2 = 0.003 m3/mol). How much work is done on the system?
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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![**Adiabatic Compression of Methane in a Piston-Cylinder System**
One mole of methane in a piston-cylinder system undergoes an adiabatic compression from an initial state (\(P_1 = 0.5 \text{ bar}, V_1 = 0.05 \text{ m}^3/\text{mol}\)) to a final state (\(P_2 = 10 \text{ bar}, V_2 = 0.003 \text{ m}^3/\text{mol}\)). How much work is done on the system?
In this scenario, the process is adiabatic, meaning no heat is exchanged with the surroundings. The work done on the system during this adiabatic process can be calculated using thermodynamic principles.
Variables:
- \(P_1\) = Initial Pressure = 0.5 bar
- \(V_1\) = Initial Volume = 0.05 m³/mol
- \(P_2\) = Final Pressure = 10 bar
- \(V_2\) = Final Volume = 0.003 m³/mol
To solve for the work done on the system, you would typically use the adiabatic process equations relating \(P\), \(V\), and work (\(W\)). The relationship between pressure and volume in an adiabatic process for an ideal gas is given by:
\[P_1 V_1^\gamma = P_2 V_2^\gamma\]
Where:
- \(\gamma\) is the adiabatic index or ratio of specific heats (Cp/Cv).
For methane (approximated as an ideal gas), the adiabatic index \(\gamma\) is typically around 1.3.
The work done (\(W\)) can be found using the expression for work in an adiabatic process:
\[W = \frac{P_2 V_2 - P_1 V_1}{1 - \gamma}\]
**Note:** Ensure to use consistent units when performing calculations (e.g., converting bars to Pascals for pressure, cubic meters for volume).
This problem demonstrates how thermodynamic principles are applied in practical scenarios involving gas compression in piston-cylinder systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44dce5c9-bcec-4e4b-a98e-07662a0f3f15%2F7e5f80f7-2fd9-4f98-8956-b65e79aedc38%2Fhjiog3a.png&w=3840&q=75)
Transcribed Image Text:**Adiabatic Compression of Methane in a Piston-Cylinder System**
One mole of methane in a piston-cylinder system undergoes an adiabatic compression from an initial state (\(P_1 = 0.5 \text{ bar}, V_1 = 0.05 \text{ m}^3/\text{mol}\)) to a final state (\(P_2 = 10 \text{ bar}, V_2 = 0.003 \text{ m}^3/\text{mol}\)). How much work is done on the system?
In this scenario, the process is adiabatic, meaning no heat is exchanged with the surroundings. The work done on the system during this adiabatic process can be calculated using thermodynamic principles.
Variables:
- \(P_1\) = Initial Pressure = 0.5 bar
- \(V_1\) = Initial Volume = 0.05 m³/mol
- \(P_2\) = Final Pressure = 10 bar
- \(V_2\) = Final Volume = 0.003 m³/mol
To solve for the work done on the system, you would typically use the adiabatic process equations relating \(P\), \(V\), and work (\(W\)). The relationship between pressure and volume in an adiabatic process for an ideal gas is given by:
\[P_1 V_1^\gamma = P_2 V_2^\gamma\]
Where:
- \(\gamma\) is the adiabatic index or ratio of specific heats (Cp/Cv).
For methane (approximated as an ideal gas), the adiabatic index \(\gamma\) is typically around 1.3.
The work done (\(W\)) can be found using the expression for work in an adiabatic process:
\[W = \frac{P_2 V_2 - P_1 V_1}{1 - \gamma}\]
**Note:** Ensure to use consistent units when performing calculations (e.g., converting bars to Pascals for pressure, cubic meters for volume).
This problem demonstrates how thermodynamic principles are applied in practical scenarios involving gas compression in piston-cylinder systems.
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