Equations of state -- Partial derivatives & Expansivity

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Homework Help Overview

The discussion revolves around the coefficient of volume expansion, specifically how it can be expressed in terms of density and temperature while keeping pressure constant. The original poster presents an equation and attempts to manipulate it but encounters difficulties in progressing further.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to substitute density for volume in the equation for volume expansion but becomes uncertain about the next steps. Other participants question the specific point at which the original poster feels stuck and suggest using the chain rule of derivatives to progress.

Discussion Status

The conversation is active, with participants providing guidance on how to approach the problem. Suggestions have been made regarding the use of derivatives and the chain rule, indicating a productive direction in the discussion. However, there is no explicit consensus or resolution at this point.

Contextual Notes

The original poster is working under the constraints of a homework assignment, which may limit the information they can use or the methods they can apply. The discussion also reflects uncertainty about the manipulation of partial derivatives in the context of the problem.

Mia_S
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Homework Statement



Show that the coefficient of volume expansion can be expressed as

β= -1÷ρ (∂ρ÷∂T) keeping P (pressure) constant
Where rho is the density
T is Temperature

Homework Equations


1/v =ρ
β= 1/v (∂v÷∂T) keeping P (pressure ) constant

The Attempt at a Solution


I started with the original equation (β= 1/v (∂v÷∂T) ) ,substituted ρ for 1/v and got stuck . What should I do next? What's the solution?

Thank you!
 
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Mia_S said:
substituted ρ for 1/v and got stuck
"Stuck" where?
 
Bystander said:
"Stuck" where?
Here, β= ρ(∂v÷∂T)
 
You're using "ρ = 1/v," so what's the next step?
 
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Bystander said:
You're using "ρ = 1/v," so what's the next step?
I don't know :( ...Should I change the partial derivative to 1/(∂T÷∂v) ?
 
May I suggest you use the chain rule of derivatives since V=1/rho=(rho)^-1. Take the derivative of the outside function times the derivative of the inside.
 
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If v = 1/ρ, then, in terms of ρ and dρ, dv = ??

Chet
 
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I got it! Thank you :)
 

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