(dV2. The derivativedTPis tedious to calculate by implicit differentiation of an equation of state such asPeng-Robinson. Show that calculus permits us to find the derivative in terms of derivatives of pressure,which are easy to find. Provide the formula for(dvfor the Peng- Robinson equation of state.dT

Write the Peng-Robinson equation of state as follows: P = RTV-b -aαVV+b+bV-bP =…

Answered: The derivative dT P is tedious to…

Introduction to Chemical Engineering Thermodynamics

8th Edition

ISBN:9781259696527

Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Chapter1: Introduction

Section: Chapter Questions

Problem 1.1P

See similar textbooks

Related questions

Question

(dV
2. The derivative
dT
P
is tedious to calculate by implicit differentiation of an equation of state such as
Peng-Robinson. Show that calculus permits us to find the derivative in terms of derivatives of pressure,
which are easy to find. Provide the formula for
(dv
for the Peng- Robinson equation of state.
dT

Expert Solution

Step 1

Write the Peng-Robinson equation of state as follows:

$\begin{array}{rcl} P & = & \frac{R T}{V - b} - \frac{a α}{V (V + b) + b (V - b)} \\ P & = & \frac{R T (V (V + b) + b (V - b)) - a α (V - b)}{V (V^{2} - b^{2}) + b {(V - b)}^{2}} \\ P (V^{3} - b^{2} V + b (V^{2} + b^{2} - 2 V b)) & = & R T (V^{2} + 2 V b - b^{2}) - a α (V - b) \\ P (V^{3} + b^{3} - 2 V b^{2}) & = & R T (V^{2} + 2 V b - b^{2}) - a α (V - b) . . . . . . (1) \end{array}$

Where,

$a = 0.45724 \frac{{(R T_{c})}^{2}}{P_{c}} b = 0.0778 \frac{R T_{c}}{P_{c}} α = {(1 + m (1 - \sqrt{\frac{T}{T_{c}}}))}^{2} m = 0.37464 + 1.54226 ω - 0.26992 ω^{2}$