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

[Tiberious]

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.

 

[lewando]

Mistake #1:

Transposing:

P1/P2 = (V1/V2)^n

Mistake #2:

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.

 

[Tiberious]

Transposing:

P1/P2 = (V2/V1)^n

Value's in reverse = 2.38.

Still, that does not seem right.

 

[lewando]

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.

 

[Tiberious]

Would Mistake #2 be the volume ^3.

Removing this I obtain n = 0.79.

 

[lewando]

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

Further transposition:

n = logV1/V2 / logP1/P2

 

[Tiberious]

Presumably the unit is not required.

From #1.5, I'm at a loss.

 

[lewando]

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.

 

[Tiberious]

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!

 

[lewando]

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