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

In summary, the problem aims to find the general expression for the adiabatic relationship between pressure and volume for any gas. This is done by applying the permuter to the general expression for the adiabatic relationship between pressure and volume, and then using Maxwell relations to replace the partial derivative expressions. By manipulating the resulting expressions and applying another Maxwell relation, the final general expression is obtained. The next part of the problem involves showing that this equation integrates to the correct expression for an ideal gas.
  • #1
grandpa2390
474
14

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)
 
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  • #2
I'm going to keep playing with it.
I may have it
 

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

1. What is the general expression for the adiabatic relationship?

The general expression for the adiabatic relationship is PVγ = constant, where P is the pressure, V is the volume, and γ is the ratio of specific heats.

2. How is the adiabatic relationship derived?

The adiabatic relationship is derived from the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. By assuming that the process is adiabatic (no heat transfer), the first law simplifies to the expression PVγ = constant.

3. What is the significance of the adiabatic relationship?

The adiabatic relationship is important in thermodynamics as it describes the behavior of a gas during an adiabatic process. It helps us understand the changes in pressure and volume of a gas as it undergoes an adiabatic expansion or compression.

4. Can the adiabatic relationship be applied to all gases?

The adiabatic relationship only applies to ideal gases, which follow the ideal gas law (PV = nRT). Real gases have more complex behavior and may deviate from the adiabatic relationship at high pressures or low temperatures.

5. How is the adiabatic relationship used in practical applications?

The adiabatic relationship is used in many practical applications, such as in the design of engines and compressors. It also helps in understanding atmospheric processes, such as the adiabatic lapse rate in the Earth's atmosphere. Additionally, it is used in the study of thermodynamics and heat transfer in various engineering fields.

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