Ideal Gas Equation and Polytropic Constant

[Redfire66]

Homework Statement

I'm given a initial and final pressure and temperature of an ideal gas to solve for the work done after it expans in a polytropic process (n=1.2)

Homework Equations

W = integral of P*dV
PV = nRT
PV = RT*
PV = mRT
PV^n = Constant

The Attempt at a Solution

I get W = integral of PdV = (P2V2 - P1V1)/(1-n)
After this part my solutions are completely different than how I solved for work

Initially I thought, since PV = nRT I can just substitute it in and get W = (P2V2 - P1V1)/(1-n) = nR(T2-T1)/(1-n)
but in the solution it solves for the volumes, using P1V1 =RT1
This is where I have a bunch of questions...

I haven't been explained these equations too well in lectures so I've been reading my textbook but it doesn't explain anything on PV = RT. But I believe it's using molar volume
I tried to solve for the initial volume using P1V1 = R*T1 which gave the correct value

But then I tried
P2V2 = RT2 to get V2 and substituting it into the work equation which didn't seem to work out because the volume is different if I were to use P1V1^n = P2V2^n

I understand that this equation can be used for a polytropic process but I don't get why I can't use P2V2 = nRT2 to solve for the volume instead.

The process still involves an ideal gas, so how come when I use these two different equations my volume is different?

 

[stockzahn]

PV^n = Constant
PV = nRT

I'm not sure about that, because I didn't comprehend your attempt completely, but are you aware, that the symbol "n" stands for two different values in the abovementioned equations?

But then I tried P2V2 = RT2 to get V2 and substituting it into the work equation which didn't seem to work out because the volume is different if I were to use P1V1^n = P2V2^n
[...]
I understand that this equation can be used for a polytropic process but I don't get why I can't use P2V2 = nRT2 to solve for the volume instead.

What values did you use for R and n in your equations?

 

[Redfire66]

I'm not sure about that, because I didn't comprehend your attempt completely, but are you aware, that the symbol "n" stands for two different values in the abovementioned equations?
What values did you use for R and n in your equations?

Yeah I sort of thought I didn't explain it properly. Anyway R would be the gas constant, nitrogen was used so using the table it was about .2968 I believe.
I don't really care about the values, I know it's not calculations. I just want to know what this formula is PV = RT, since there's no variable in front like the PV = nRT that I'm used to seeing.
Secondly, when attempting to find a change in volume in a polytropic process... I'm asking why can't I use the common PV = nRT given the value of n. But rather I would have to use PV^n = Constant. Also I did not understand the values for n is different in the equations. Just because there's nothing distinguishing a difference between them.

 

[stockzahn]

Secondly, when attempting to find a change in volume in a polytropic process... I'm asking why can't I use the common PV = nRT given the value of n. But rather I would have to use PV^n = Constant. Also I did not understand the values for n is different in the equations. Just because there's nothing distinguishing a difference between them.

Well that's the point:
- the n in PV = nRT is the number of mols
- the n in PV^n is the isentropic coefficient

they are completley different values it's just the same symbol

There are three common forms for the ideal gas equation:

p ⋅V = n ⋅ R_m ⋅ T ... with n is the mols and R_m is the gas constant in [J/(mol⋅K)] and the same valuefor each gas (R_m=8.314 J/(mol⋅K))
p ⋅V = m ⋅ R ⋅ T ... with m is mass and R is the gas constant in [J/(kg⋅K)] and different for each gas
p ⋅v = R ⋅ T ... with R is the gas constant in [J/(kg⋅K)] and different for each gas and v is the specific volume in [m³/kg] (= is the inverse value of the density: v = 1 / ρ)