Question

Calculate the molar volume of ammonia gas at 273K and 1 bar using the van der Waals equation of state.

(a) First, use the critical point data Tc = 406K and Pc = 113 bar to obtain the van der Waals parameters a and b.

(b) Next use the following iterative algorithm to estimate the molar volume: (1) Re‐write the vdW EOS in the form Vm = b + RT/(P+a/Vm2), (2) Using a guess for Vm on right hand side obtain a new value of Vm (left hand side). (3) Repeat this process until Vm is converged to a few decimal places.

Answer 1

(a)

Given that,

Tc = 406K

Critical point temperature is,

Tc = 8a / 27Rb

406 = 8a / 27*8.314*b

a / b = 11392.9 .......(1)

Given that, Pc = 113 bar = 113*10^5 Pa

Critical point pressure is,

Pc = a / 27b^2

113*10^5 = a / 27b^2

a / b^2 =3.051*10^8 .......(2)

From equation (1) and (2)

(a / b) / b = 3.051*10^8

11392.9 / b = 3.051*10^8

b = 3.73*10^(-5) m^3 / mol

a = 0.4254 Pa*m^6 / mol^2

(b)

At T = 273 K , P = 1 bar = 10^5 Pa

from ideal gas equation,

PV = nRT

ldeal molar volume of ammonia gas,

Vm = RT / P

Vm = 8.314*273 / 10^5 = 0.0227 m^3 / mol

From Vm =  b + RT / (P+a / Vm2)

By putting values and calculating,

Vm = 0.0225 m^3 / mol

so, LHS = 0.0227

RHS = 0.0225

we will choose a lower molar volume

Hence, take LHS = RHS

molar volume of ammonia gas,

Vm = 0.0225 m^3 / mol