# Equations of state -- Partial derivatives & Expansivity

## 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!

Bystander
Homework Helper
Gold Member
substituted ρ for 1/v and got stuck
"Stuck" where?

"Stuck" where?
Here, β= ρ(∂v÷∂T)

Bystander
Homework Helper
Gold Member
You're using "ρ = 1/v," so what's the next step?

• Mia_S
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.

• Mia_S
Chestermiller
Mentor
If v = 1/ρ, then, in terms of ρ and dρ, dv = ??

Chet

• Mia_S
I got it! Thank you :)