Find the equation of state (thermo)

[sprinks13]

Homework Statement

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.

Homework Equations

k= -v(dP/dv)_T
B = 1/v(dV/dT)_P
(dP/dT)_v = -(dv/dT)_P/(dv/dP)_T

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

 

[olgranpappy]

Homework Statement

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.

Homework Equations

k= -v(dP/dv)_T

this is wrong.

B = 1/v(dV/dT)_P
(dP/dT)_v = -(dv/dT)_P/(dv/dP)_T

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

 

[olgranpappy]

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