Calculate the specific volume for carbon dioxide gas at 450

Calculate the specific volume for carbon dioxide gas at 450 K and 4000 kPa using (a) ideal gas equation, (b) compressibility chart, and (c) van der Waals equation. (d) Provide the percent difference for each method to the exact value of 0.020296 m^3/kg

Solution

1. using ideal gas equation v_i = specific volume = RT/P

where P = pressure = 4*10⁶ Pa

T= Temperature of gas = 450K

R = characteristic gas constant = Universal gas constant /molecular weight of CO₂ = 8314 /44 = 189 J/kgK

v_i = RT/P = (189*450) /4*10⁶ = 0.02126m³/kg

2. using compressibility chart specific volume = v_c = Z*R*T/P

Z= compressibility facor of CO₂ = 0.9640 at 40bar, 450K

v_c = 0.9640*189*450/4*10⁶ = 0.02049 m³/kg

3. Using Van der Waals equation specific volume = vv = nR³T³/(PR²T²+aP²) + nb

n = number of moles = 1

a = 3.96 x 10^-1 Pa m³= emperical constants

b= 42.69 x 10^-6 m³/mol =  emperical constants

v_v = nR³T³/(PR²T²+aP²) + nb = 0.0212 m³/kg

the percent difference of specific volume using ideal gas =( (v-v_i)/v)*100

   = ( (0.020296- 0.02126)/0.020296)*100

   = -5.17%

the percent difference of specific volume using compressibility =( (v-v_c)/v)*100

= ( (0.020296- 0.02049)/0.020296)*100

= -1.38%

the percent difference of specific volume using van der waals eq. =( (v-v_v)/v)*100

= ( (0.020296- 0.0212)/0.020296)*100

= -4.88%