Finding work for a gas in a piston cylinder

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Discussion Overview

The discussion revolves around a homework problem involving a gas in a piston-cylinder assembly, specifically focusing on the relationship between pressure and volume defined by the equation pV²=Constant. Participants are tasked with determining the final volume and the work done during the process, exploring the mathematical integration involved in the solution.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant presents an initial attempt at solving for the final volume and work, stating V₂=0.03333 m³ and W=-19.97 kJ, but expresses uncertainty about the correctness of their solution.
  • Another participant points out that the integration and determination of the constant C are incorrect, suggesting that C should be defined as C=p₁V₁²=p₂V₂².
  • A later reply questions the integration process, affirming that the integral of 1/x² is indeed -1/x, but indicates that the limits of integration were substituted incorrectly.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correctness of the initial solution, as there are disagreements regarding the integration process and the determination of the constant C.

Contextual Notes

The discussion highlights potential limitations in the integration steps and assumptions about the relationship between pressure and volume, but these remain unresolved.

jdawg
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Homework Statement


A gas in a piston-cylinder assembly undergoes a process for which the relationship between pressure and volume is pV2=Constant. The initial pressure is 1 bar, the initial volume is 0.1 m3, and the final pressure is 9 bar. Determine the final volume and the work for the process.

Homework Equations

The Attempt at a Solution


V2=sqrt((p1)(V12)/(p2))
V2= 0.03333 m3

W=∫ pdV
W=∫ C/(V2) dV
=C[-1/V]
=C[-1/(V2-V1)]
=[(p2V22)-(p1V12)][-1/(V2-V1)]

=[((900000Pa)(0.0333m3)2)-((100000Pa)(0.1m3)2))][-1/(0.0333-0.1)]

I think the answer is supposed to be W= -19.97KJ, I'm not sure what I'm doing wrong.
 
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jdawg said:

Homework Statement


A gas in a piston-cylinder assembly undergoes a process for which the relationship between pressure and volume is pV2=Constant. The initial pressure is 1 bar, the initial volume is 0.1 m3, and the final pressure is 9 bar. Determine the final volume and the work for the process.

Homework Equations

The Attempt at a Solution


V2=sqrt((p1)(V12)/(p2))
V2= 0.03333 m3

W=∫ pdV
W=∫ C/(V2) dV
=C[-1/V]
=C[-1/(V2-V1)]
=[(p2V22)-(p1V12)][-1/(V2-V1)]

=[((900000Pa)(0.0333m3)2)-((100000Pa)(0.1m3)2))][-1/(0.0333-0.1)]

I think the answer is supposed to be W= -19.97KJ, I'm not sure what I'm doing wrong.
Your integration is incorrect, and your determination of the constant C is incorrect. ##C=p_1V_1^2=p_2V_2^2##
 
Oh ok, the constant part makes sense. How is the integration wrong though?
I thought ∫1/x^2 dx = -1/x ?
 
jdawg said:
Oh ok, the constant part makes sense. How is the integration wrong though?
I thought ∫1/x^2 dx = -1/x ?
Yes, that's correct, but you substituted the integration limits incorrectly.

Chet
 
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Thanks for your help!
 

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