# How can I calculate the molar volume of a non ideal gas?

##### 1 Answer

Well, it depends on what equation of state you WANT to use. The easiest one to use for REAL gases is the **van der Waals equation of state** (vdW EOS):

#P = (RT)/(barV - b) - a/(barV^2)# where

#P,R,# and#T# are known from the ideal gas law,#barV -= V/n# is the molar volume of the vdW gas, and#a# and#b# are van der Waals constants accounting for the attractive intermolecular forces, and the excluded volume, respectively.

Solving for

#P = (barV^2RT)/(barV^2(barV - b)) - (a(barV - b))/(barV^2(barV - b))#

#PbarV^2(barV - b) = barV^2RT - a(barV - b)#

#PbarV^3 - bP barV^2 = barV^2RT - abarV + ab#

#PbarV^3 - (bP + RT)barV^2 + abarV - ab = 0#

This becomes a cubic equation for

#barul|stackrel(" ")(" "barV^3 - (b + (RT)/P)barV^2 + a/PbarV - (ab)/P = 0" ")|#

For this, we need

- specified pressure
#P# in#"bar"# , - temperature
#T# in#"K"# , #R = "0.083145 L"cdot"bar/mol"cdot"K"# ,- vdW constants
#a# in#"L"^2"bar/mol"^2# and#b# in#"L/mol"# .

Then this can be solved iteratively via the **Newton-Raphson method**. Of course, you can use whatever method you want to solve this cubic.

To do the Newton-Raphson method, in your calculator, let:

#b + (RT)/P = A#

#a/P = B#

#(ab)/P = C#

Then we have:

#barV^3 - AbarV^2 + BbarV - C = f(barV)#

#3barV^2 - 2AbarV + B = f'(barV)#

Each iteration acquires

#barV_(i+1) = barV_i - (f(barV_i))/(f'(barV_i))#

In your TI calculator, let

#"logical guess" -> X#

Then if you believe you chose correctly, proceed to type the following:

#(X - (X^3 - AX^2 + BX - C)/(3X^2 - 2AX + B)) -> X#

This generates an iterative loop that triggers each time you press Enter. So, press Enter **until the value you get stops changing**.

That is **ONE out of THREE** molar volumes.

- One
#barV# is of the liquid. - One
#barV# is of the gas. - One
#barV# is a so-called spurious (i.e. UNPHYSICAL) solution.

To know what you have just gotten, compare with the other