How to derive pv^gamma=constant

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Homework Help Overview

The problem involves demonstrating the relationship between pressure and volume for an ideal gas in a reversible adiabatic condition, specifically showing that \( pV^\gamma = \text{constant} \), where \( \gamma = \frac{C_{p,m}}{C_{v,m}} \). The context includes the comparison to the isothermal condition where \( pv = \text{constant} \).

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants discuss the derivation of the adiabatic condition and question the validity of certain algebraic manipulations. There are inquiries about the definitions and implications of adiabatic processes, as well as the roles of \( C_p \) and \( C_v \). Some participants express confusion regarding the logical structure of the original poster's argument.

Discussion Status

The discussion is ongoing, with participants providing guidance on the necessary steps to approach the problem. There is a recognition of errors in the original attempt, and suggestions for starting from fundamental principles such as the first law of thermodynamics are being explored.

Contextual Notes

Participants note the importance of understanding the definitions of adiabatic processes and the relationships between internal energy, heat, and work in this context. There is also mention of the need to clarify the distinction between the isothermal and adiabatic conditions.

milan talaviya
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Homework Statement


The relationship between pressure and volume of an ideal gas is expressed as pv=constant in a reversable isothermal condition. Show that the relationship between pressure and volume of the same gas is expressed as pV^gamma=constant in a reversible adiabatic condition where gamma=Cp,m/Cv,m.

Homework Equations



gamma=Cp,m/Cv,m.
pV^gamma=constant, rev. adiabatic
pv=constant, rev. ideal

The Attempt at a Solution


p1v1^(Cpm/Cvm)=p2v2^(Cpm/Cvm) take ln and mult both sides by the Cvm/Cpm
ln p1v1 = ln p2v2 e to both sides
p1v1=p2v2
p1v1/p2v2 = 1
 
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please answer it if you have better than these.
your's thankfully milan talaviya
 
Hello Milan, welcome to PF.

You have to show that pVγ= constant assuming a reversible adiabatic process on an ideal gas.

What is an adiabatic process? How does the internal energy change in an adiabatic process? What are Cp and Cv?

ehild
 
milan talaviya said:

The Attempt at a Solution


p1v1^(Cpm/Cvm)=p2v2^(Cpm/Cvm) take ln and mult both sides by the Cvm/Cpm
ln p1v1 = ln p2v2 e to both sides
p1v1=p2v2
p1v1/p2v2 = 1
You haven't proved anything here. For one, you've made an algebra mistake. The more serious error is that you started with what you're supposed to be proving. That sort of argument isn't valid logically.
 
milan talaviya said:

Homework Statement


The relationship between pressure and volume of an ideal gas is expressed as pv=constant in a reversable isothermal condition. Show that the relationship between pressure and volume of the same gas is expressed as pV^gamma=constant in a reversible adiabatic condition where gamma=Cp,m/Cv,m.




Homework Equations



gamma=Cp,m/Cv,m.
pV^gamma=constant, rev. adiabatic
pv=constant, rev. ideal



The Attempt at a Solution


p1v1^(Cpm/Cvm)=p2v2^(Cpm/Cvm) take ln and mult both sides by the Cvm/Cpm
ln p1v1 = ln p2v2 e to both sides
p1v1=p2v2
p1v1/p2v2 = 1
The adiabatic condition that you have to prove is not:

(PV)^{\frac{C_p}{C_v}} = \text{constant}

Rather, it is :

PV^{\frac{C_p}{C_v}} = \text{constant}

which means:

\frac{P_1}{P_2} = (\frac{V_2}{V_1})^{\frac{C_p}{C_v}}

1. Start with the first law and find an expression for dU in terms of PdV (hint: what is dQ if it is adiabatic?).

2. Then express dU in terms of dT and substitute your answer in 1. for dU.

3. Finally express dT in terms of d(PV). (hint: use R = Cp-Cv).

AM
 

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