Polytropic process vs perfect gas eq

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Discussion Overview

The discussion revolves around the relationship between the polytropic process and the perfect gas equation, specifically examining how both can be valid for the same gas under different conditions. Participants explore the implications of the polytropic law and the ideal gas law, particularly in scenarios where temperature remains constant or changes during the process.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant notes that the polytropic law states P1V1^n = P2V2^n, while the perfect gas equation is expressed as PV = mRT, leading to the question of how both can be true simultaneously when n ≠ 1.
  • Another participant explains that if T1 = T2, the process described by the polytropic expression corresponds to n = 1, indicating an isothermal process where PV remains constant.
  • A participant expresses confusion regarding the use of n in different contexts, questioning how n can be both 1 and a variable in the derivation of related equations, suggesting a potential inconsistency in the application of these equations.
  • It is mentioned that for polytropic processes with n ≠ 1, the temperature of the gas changes, but the ideal gas law still applies, maintaining the relationship P1V1/T1 = P2V2/T2.

Areas of Agreement / Disagreement

Participants express differing views on the application of the polytropic law and the ideal gas law, particularly regarding the value of n and its implications for temperature changes. The discussion remains unresolved, with no consensus reached on the relationship between the equations.

Contextual Notes

Participants reference specific pages from a textbook to support their arguments, indicating that the discussion may depend on interpretations of the material presented in those texts. There are unresolved questions about the consistency of using n in different equations.

imsmooth
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The polytropic law states:

(1) P1V1n = P2V2n

The perfect gas equation states:

PV = mRT --> P1V1/T1 = P2/V2/T2

If T1 = T2 then
(2) P1V1 = P2V2

So, how can equation 1 and 2 both be true for the same gas? If the gas follows a polytropic process, where n ≠ 1, how can 2 be correct when there is no temperature change?
 
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Merry Christmas imsmooth,

The polytropic law describes a process that a gas would follow from state 1 to state 2 during a compression or expansion. The value of n can be anything from 0 to infiniti for a set process.

For your question if T1=T2, then this process is descibed by the polytropic expression with the value of n = 1. This means that during the process of compression or expansion PV = a constant = mRT ( since m, R, T are all fixed values for only this process where n = 1 ). Where the temperature does not change, the process is called an isothermal process and the state of the gas follows a constant temperature profile called isotherms.

For any other value of n, there are other descriptions of the process, during which for an ideal gas, the equation PV=mRT will hold true, and as P or V are altered so will the value of T alter.

This has a brief summary:
http://web2.clarkson.edu/projects/fluidflow/kam/courses/2004/es340/chap3-ext.pdf

all the best.
 
I appreciate the answer, but I know this.

On page 116 of Rayner Joel's Engineering Thermodynamics, both equations are used to derive another set of equations. One equation is setting n = 1; the other is just leaving it as n. This does not make sense as n should be the same for both equations for deriving the third.

Even using your reference on page 8, PV = mRT. mR is a constant. Thus, P1V1 = T = P2V2. Here, n = 1. This is rearranged to have V1 = mRT/P2 and subsituted into PV^n

How can n = 1 for PV = nRT, but it is just n for PV^n?
 
imsmooth said:
I appreciate the answer, but I know this.

On page 116 of Rayner Joel's Engineering Thermodynamics, both equations are used to derive another set of equations. One equation is setting n = 1; the other is just leaving it as n. This does not make sense as n should be the same for both equations for deriving the third.

Even using your reference on page 8, PV = mRT. mR is a constant. Thus, P1V1 = T = P2V2. Here, n = 1. This is rearranged to have V1 = mRT/P2 and subsituted into PV^n

How can n = 1 for PV = nRT, but it is just n for PV^n?
For polytropic processes with n≠1, the temperature of the gas changes during the process. This does not mean that the ideal gas law doesn't also apply to these processes. In such cases, P1V1/T1 = P2V2/T2.
 
That makes sense. Thanks.
 

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