- #1

nobosity

- 5

- 0

## Homework Statement

The Equation of State and the expression for the entropy for a sample of salt water is given by:

[tex] V = V_{0}(1 + \beta(T - T_{0}) - \gamma(P - P_{0})) [/tex] [tex] S = S_{0} + C_{v}ln(T - T_{0}) + \frac{\beta}{\gamma}(V - V_{0})[/tex]

where the subscript 0 denotes a reference state, the coefficients [tex] \beta [/tex] and [tex] \gamma [/tex] are constants and [tex] C_{v} [/tex] is the heat capacity of the salt water at constant volume.

Derive an expression for the gradient of an adiabat in a PV diagram.

## Homework Equations

Listed above.

## The Attempt at a Solution

I struggle to write the attempts I've made trying to answer this question. I understand that in this case we have V(T,P) and S(T,V), the gradient will be (dP/dV) and using the fact that an adiabat occurs when there is no change in heat energy. Also aware of the fact that Cv can be written as a differential in terms of (dU/dT), which is possibly relevant.

The real issue is I have no idea of the best way to put all of this information together and find a logical pathway to answer. Do I want to get to: [tex] dP = \frac{\partial{P}}{\partial{T}}dT + \frac{\partial{P}}{\partial{V}}dV [/tex] and substitute a concoction of the above information to get to the gradient?

Been banging my head against this problem for a couple of weeks, and would be very grateful for someone to point me in the right direction!

EDIT:

Probably should have included that in my explanation. I get that for an adiabatic change, dS is 0 assuming reversibility. Similarly, I got to the point where I have:

[tex]dV = \frac{\partial{V}}{\partial{T}}dT + \frac{\partial{V}}{\partial{P}}dP[/tex]

and

[tex]dS = \frac{\partial{S}}{\partial{T}}dT + \frac{\partial{S}}{\partial{V}}dV = 0[/tex]

However, none of these approaches seems to lead to a place where I can rearrange to get to dP/dV. Thats the part I'm a bit stuck on.

Last edited: