What is the Correct Value of the Index n in Gas Compression?

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Discussion Overview

The discussion revolves around calculating the value of the index n in the context of gas compression, specifically using the equation P1V1^n = P2V2^n. Participants explore various mathematical approaches and corrections related to the transposition and input of values in the equation.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant attempts to calculate n using the equation and arrives at n = -2.38, expressing uncertainty about the correctness of their approach.
  • Another participant identifies mistakes in the transposition and input of values, suggesting that the relationship should be P1/P2 = (V2/V1)^n instead.
  • A later reply indicates that the exponent of volume should not be cubed, leading to a recalculation of n = 0.79.
  • Further corrections highlight that the transposition of the logarithmic equation was incorrect, leading to a new expression for n = logP1/P2 / logV2/V1, resulting in n = 1.25 after substituting values.
  • One participant suggests that if digit significance is a concern, n could be approximated as 1.

Areas of Agreement / Disagreement

Participants express differing views on the correct approach to the problem, with multiple competing calculations for the value of n. The discussion remains unresolved as participants propose different values based on their interpretations of the equation and corrections made.

Contextual Notes

There are unresolved issues regarding the assumptions made in the calculations, particularly concerning the treatment of volume and the logarithmic relationships. The discussion reflects a progression of thought with corrections but does not reach a consensus on the final value of n.

Tiberious
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Homework Statement


[/B]
In a closed system a gas is compressed from a pressure of 3 bar absolute to a pressure of 5 bar absolute, and its volume changes from 0.9m^3 to 0.6m^3 according to the law P1V1^n = P2V2^n. Calculate:

The value of the index n.

Homework Equations



P1V1^n = P2V2^n

The Attempt at a Solution



Given:

P1V1^n = P2V2^n

Transposing:

P1/P2 = (V1/V2)^n

Taking logs of both sides:

logP1/P2 = n*logV1/V2

Further transposition:

n = logV1/V2 / logP1/P2

Inputting our given values:

n = log0.9^3/0.6^3 / log3*10^6/5*10^6

n = -2.38

I know I'm doing something wrong. Any assistance is appreciated.
 
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Mistake #1:
Tiberious said:
Transposing:

P1/P2 = (V1/V2)^n

Mistake #2:
Tiberious said:
Inputting our given values:

n = log0.9^3/0.6^3 / log3*10^6/5*10^6

I won't spell out the errors to give you a chance to find them.
 
Transposing:

P1/P2 = (V2/V1)^n

Value's in reverse = 2.38.

Still, that does not seem right.
 
Good work addressing Mistake #1. You should make plans to address Mistake #2.

To be clear, Mistake #1 propagated into the expression identified as Mistake #2. But Mistake #2 has an issue independent of Mistake #1.
 
Last edited:
Would Mistake #2 be the volume ^3.

Removing this I obtain n = 0.79.
 
Yes, the ^3 does not belong. You are clear on why?

Wow-- upon further review, I additionally claim that the following is Mistake #1.5
Tiberious said:
Further transposition:

n = logV1/V2 / logP1/P2
 
Presumably the unit is not required.

From #1.5, I'm at a loss.
 
The units in a ratio should be the same for top and bottom, which they are in this case. As such, they "cancel out".

Mistake #1.5:
You went from
logP1/P2 = n*logV1/V2

to
n = logV1/V2 / logP1/P2

Like saying a = xb is the same as x = b/a, which it is not true.
 
So,

logP1/P2 = n*logV2/V1

Transposes:

logP1/P2 / n = logV2/V1

Transposes:

n = logP1/P2 / logV2/V1

After values:

n = log3*10^6/5*10^6 / log0.6/0.9

n = 1.25

Awesome, Thanks!
 
  • #10
Yes, much better! I don't know if you care about digit significance--if you do, you could go with n = 1.
 

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