# Find the equation of state (thermo)

1. Oct 28, 2008

### sprinks13

1. The problem statement, all variables and given/known data
A hypothetical substance has an isothermal compressibility k = a/v and an expansivity B = 2bT/v where a and b are constants and v is the molar volume. Show that the equation of state is v-bT2+aP = g where g is a constant.

2. Relevant equations
k= -v(dP/dv)T
B = 1/v(dV/dT)P
(dP/dT)v = -(dv/dT)P/(dv/dP)T

3. The attempt at a solution
I went about it 2 different ways; neither seem to work:
1) B = 1/v(dV/dT)P = 2bT/v
=> (dV/dT)P = 2bT
=> v = bT2 + g'

k = -v(dP/dv)T = a/v
=> (dP/dv)T = -a/v2
=> P = a/v + g"

And I don't know what to do from here...

or
2)
dP = (dP/dv)T dv + (dP/dT)V dT
= -(dv/dT)p/(dv/dP)T dT - k/v dv
= Bv/ (dv/dP)^(-1)T - k/v dV
= -Bk dT - k/v dv
= -2bTa/v^2 dT - a/v^2 dv
P = (1/v^2)baT^2 + a/v + g

Thanks

2. Oct 28, 2008

### olgranpappy

this is wrong.

3. Oct 28, 2008

### olgranpappy

should be
$$k=-\frac{1}{V}\left(\frac{dV}{dP}\right)_T$$