Chain relation/ triple partial derivative rule

[Syrus]

Homework Statement

For the van der Waals equation of state, confirm the following property:

(∂P/∂T)_V (∂T/∂V)_P (∂V/∂P)_T = -1

Homework Equations

The van der Waals equation of state is:

P = nRT/(v-nb) - an²/V²

*R, n, a, b are const.

The Attempt at a Solution

I have come up with some partial derivatives, however, I cannot seem to figure out the algebra to make their product equal to -1. Perhaps my derivatives are incorrect?

(∂P/∂T)_V = nR/(v-nb)^-1

(∂T/∂V)_P = P - an²/V² + 2abn³/V³

(∂V/∂P)_T = 1/ (2an²/V³ - nRT/(v-nb)²)



Any hints or ideas?

 

[Punkyc7]

I believe there is formula that is (∂X/∂Y) =-(F[itex]_{Y}[/itex]/F[itex]_{X}[/itex])

 

[Syrus]

Can you explain what the right side of the equality represents?

 

[Punkyc7]

I am not entirely sure it was just an equation in my book, I will leave it for some one else to answer because I do not want to tell you wrong

 

[Syrus]

Anyone else?

 

[SammyS]

...

Homework Equations

The van der Waals equation of state is:

P = nRT/(V-nb) - an²/V²

*R, n, a, b are const.

The Attempt at a Solution

I have come up with some partial derivatives, however, I cannot seem to figure out the algebra to make their product equal to -1. Perhaps my derivatives are incorrect?

(∂P/∂T)_V = nR/(v-nb)^-1

Should be (∂P/∂T)_V = nR/(V-nb) or (∂P/∂T)_V = nR(V-nb)^-1​

(∂T/∂V)_P = P - an²/V² + 2abn³/V³

(∂V/∂P)_T = 1/ (2an²/V³ - nRT/(v-nb)²)

Any hints or ideas?

Show how you arrived at the last two partial derivatives. (I suggest using implicit differentiation.)

 

[Syrus]

Well, i think i figured it out. I used the reciprocal identity:

(dx/dy)_z = 1/ (dy/dx)_z (should be partial derivatives here)

to make the triple partial derivative product a double partial derivative product, and then showed it to be equal to the resulting partial derivative on the other side of the equality (which occurs when you divide -1 by one of the terms originally on the left). It worked well =)

 

[evo_vil]

Sorry for the hijack but i have a similar question:

for a recent semester test we needed to show:
[itex]\frac{\partial{P}}{\partial{V}} \frac{\partial{V}}{\partial{T}} \frac{\partial{T}}{\partial{P}} = -1[/itex]

i simply converted each partial into its implicit version and canceled terms:

[itex]\frac{\partial{P}}{\partial{V}} = \frac{-F_V}{F_P}[/itex]

[itex]\frac{\partial{V}}{\partial{T}} = \frac{-F_T}{F_V}[/itex]

[itex]\frac{\partial{T}}{\partial{P}} = \frac{-F_P}{F_T}[/itex]

resulting in

[itex]\frac{-F_V}{F_P} \frac{-F_T}{F_V} \frac{-F_P}{F_T} = -1[/itex]

yet this was marked very clearly wrong...

Why?