Polytropic process vs perfect gas eq

[imsmooth]

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?

 

[256bits]

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.

 

[imsmooth]

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?

 

[Chestermiller]

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.

 

[imsmooth]

That makes sense. Thanks.