Thermodynamics: Isothermal Compressibility and Expansivity Equations

[mmmboh]

Homework Statement

Hi, I got a set of problems to do and there are a few I am having trouble with: (edit: I had more questions before but I figured them out)

1. A hypothetical substance has an isothermal compressibilty $$\kappa$$=a/v and an expansivity $$\beta$$=2bT/v, where a and b are constants, and v is the specific volume.
a) Show that the equation of state is given by v-bt² + aP=constant
b) If at a pressure P₀ and temperature T₀, the specific volume is v₀, evaluate the constant.

The Attempt at a Solution

1. Ok I figured out 2a), but I don't know how to evaluate the constant for 2b)...what I did was just replace v, T, and P with v₀, t₀, and P₀, but I don't think that is right.

I think it may be something with an exact differential equation, I am not sure though. I have ($$\beta\nu$$/$$\partial$$P)$$\partial$$T = -($$\kappa\nu$$/$$\partial$$T)$$\partial$$P
Help for that one please



Btw how do you make the kappa symbol on the same level as everything else?

 

[mmmboh]

Ok number 3 I figured out

 

[mmmboh]

Any help?