Equations of state -- Partial derivatives & Expansivity

  • #1
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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!
 

Answers and Replies

  • #2
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substituted ρ for 1/v and got stuck
"Stuck" where?
 
  • #3
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"Stuck" where?
Here, β= ρ(∂v÷∂T)
 
  • #4
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You're using "ρ = 1/v," so what's the next step?
 
  • #5
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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) ?
 
  • #6
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.
 
  • #8
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I got it! Thank you :)
 

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