Write the van der Waals EOS in the reduced coordinatesTr Pr

Write the van der Waals EOS in the reduced coordinates(T_r, P_r and V_r): T_r T/T_c P_r P/P_c V_r V/V_c You should result to the following equation: P_r = 8T_r/3V_r - 1 - 3/V^2_r Using the van der Waals equation of state: P = RT/V - b - a/V^2 Derive the parameters alpha and b: You should find a = 27/64 R^2T^2_c/P_c b = 1/8 RT_c/P_c USE YOUR BOOK and the Lecture notes! Notice that the parameters are functions of critical temperature and pressure. So, start by writing the Van der Waals equation at the critical temperature and pressure and use the fact that at the critical point (V-V_c)^3 = 0

Solution

a)

Vander waals equation of state is

P = RT/V-b - a/V²     which can be reduced to

PV³ - (Pb+RT)V² + aV-ab = 0

P(V³ - ( b+RT/P)V² +aV/P - ab/P ) = 0

since at critical point we can write (V-V_c)3 = V³-3V²V_c+3V(V_c)² - (V_c)³ = 0

by comparison we can write 3V_c = b+RT/P ; 3(V_c)² = a/P : (V_c)³ = ab/P

here P and T corresponds to critical pressure and temperatures

from last 2 equations we can write V_c = 3b substitute this value in 1st equation

8b= RT/P divide this equation with 2nd equation from this we can write T_c = 8a/27Rb and P_c = a/27b²

now

P = P_r*P_c = Pr*a/27b² , V = V_r*V_c = 3bV_r , T = T_r*T_c = 8aT_r/27RB , put these values in below vanderwaals equation

(P+a/V²) (V-b) = RT ( Pr*a/27b² +a/(3bV_r)^2) *( 3bV_r -b) = R*8aT_r/27RB

after simplication we can write

(P_r +3/V_r^2) (3V_r-1) = 8T_r

P_r = 8T_r/3V_r-1 - 3/V_r^2

B)

V_c = 3b , T_c = 8a/27Rb , P_c = a/27b²

use above equations to get

RT_c/8P_c = b and 27R²(T_C)²/64P_C    = a