Find the General expression for the adiabatic relationship....

[grandpa2390]

Homework Statement

edit: I figured out my mistake

Here's the entire problem. But I don't yet need help with all the steps. I'm getting off on the wrong foot (as usual)a We have shown in class that ideal gas adiabats have the form: $P_I(V_i)^γ = P_f(V_f)^γ$ with the heat capacity ratio: $γ = \frac{C_p}{C_v}$ This problem is designed to find the general expression for the adiabatic relationship between P and V for any gas.

i. The adiabatic relationship between P and V begins with the completely general form: $(\frac{∂P}{∂V})_s$

ii. Apply the permuter to this expression.

iii.Find two Maxwell relations to replace the resulting partial derivative expressions (you will need the inverter for one of them)

iv. Insert the expression we derived in class for the adiabatic relationship between V and T for one of the partials.

v. Use the permuter on the other partial derivative that came from step iii.

v. You should at this point be able to use the relationship: https://www.physicsforums.com/file://localhost/Users/simon/Library/Group%20Containers/UBF8T346G9.Office/msoclip1/01/clip_image008.png

vi. Use another Maxwell relation on the remaining partial derivative term from step v.

vii. Apply the permuter (in reverse) to the two partial derivatives to finally arrive at the general expression of the adiabatic relationship between P and V for any gas: https://www.physicsforums.com/file://localhost/Users/simon/Library/Group%20Containers/UBF8T346G9.Office/msoclip1/01/clip_image010.png

b. Show for the ideal gas that this equation integrates to the proper expression. (before integrating, I suggest you manipulate the result of https://www.physicsforums.com/file://localhost/Users/simon/Library/Group%20Containers/UBF8T346G9.Office/msoclip1/01/clip_image012.png into a form in terms of P and V only.)

Homework Equations

Maxwell Relations

The Attempt at a Solution

ii. so first I applied the permuter to $(\frac{∂P}{∂V})_s$

I got $(\frac{∂P}{∂S})_v(\frac{∂S}{∂V})_P$

iii. The adiabatic relationship derived in class between V and T is $(\frac{∂V}{∂T})_s = \frac{-C_v}{\frac{RT}{V}}$

so I inverted the partial of P with respect to S and used the maxwell relation on it to get $-\frac{∂V}{∂T}$ and plugged in the adiabatic relationship. I used the maxwell relation on the partial of S with respect to V to get the partial of P with respect to T.

this is about where I get lost because after I use the maxwell relation and use the permuter I can't plug in the relationship:
$(\frac{∂S}{∂T})_p = \frac{C_p}{T}$
because I don't get a partial of S with respect to T.

I get the partial of P with respect to V (constant T) times the partial of V with respect to T (constant P)

 

[grandpa2390]

I'm going to keep playing with it.
I may have it