Second derivative of Pressure with respect to moles?

[mnnob07]

binary mixture.
Na=moles of a
Nb=moles of b

(using Peng Robinson Equation of state)

(second order partial derivative below)

d^2P/(dNa^2) holding T, molar volume, Nb constant

I can't figure out how to do this?

I know that Peng Robinson is a function of concentration of Na and Nb... but I tried to substitute all molar fractions with Na/(Na+Nb) & Nb/(Na+Nb) put the complete expression in Mathcad (was huge!) and it gives 0 as the first derivative of Na or Nb so something went wrong or it decided not to do it...

Attached is/was my Mathcad input and also the whole equation I am trying to get an expression of (this partial derivative is just part of it; don't mind the cursor on the right)

 

[Navin A S]

Thanks for any help!The equation is too large to fit here. You can find it on the Peng-Robinson website.

 

[oswaler]

I can provide some guidance on how to approach this problem. The Peng Robinson equation of state is a complicated equation that relates pressure, temperature, and molar volume to the number of moles of each component in a binary mixture. In order to find the second derivative of pressure with respect to moles, we need to use the chain rule and take the second derivative of the Peng Robinson equation with respect to moles.

First, we need to express the Peng Robinson equation in terms of the number of moles of each component, Na and Nb. This can be done by substituting the molar fractions with Na/(Na+Nb) and Nb/(Na+Nb), as you have already attempted. However, it is important to make sure that the equation is correctly substituted and that all terms are accounted for.

Next, we will use the chain rule to take the second derivative of the Peng Robinson equation with respect to Na. This involves taking the first derivative of the equation with respect to Na and then taking the second derivative of the resulting equation with respect to Na.

Finally, we can substitute the values of T, molar volume, and Nb into the second derivative equation to get the final expression for d^2P/(dNa^2) holding T, molar volume, and Nb constant.

It is possible that Mathcad was not able to solve the equation due to its complexity. In such cases, it may be helpful to break down the equation into smaller parts and solve them separately before combining them to get the final expression.

I hope this helps in understanding the process of finding the second derivative of pressure with respect to moles in a binary mixture using the Peng Robinson equation of state.