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

1 Answer
Mar 15, 2018

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)P=RT¯¯¯Vba¯¯¯V2

where P,R,P,R, and TT are known from the ideal gas law, barV -= V/n¯¯¯VVn is the molar volume of the vdW gas, and aa and bb are van der Waals constants accounting for the attractive intermolecular forces, and the excluded volume, respectively.

Solving for barV¯¯¯V is a hogfest all on its own...

P = (barV^2RT)/(barV^2(barV - b)) - (a(barV - b))/(barV^2(barV - b))P=¯¯¯V2RT¯¯¯V2(¯¯¯Vb)a(¯¯¯Vb)¯¯¯V2(¯¯¯Vb)

PbarV^2(barV - b) = barV^2RT - a(barV - b)P¯¯¯V2(¯¯¯Vb)=¯¯¯V2RTa(¯¯¯Vb)

PbarV^3 - bP barV^2 = barV^2RT - abarV + abP¯¯¯V3bP¯¯¯V2=¯¯¯V2RTa¯¯¯V+ab

PbarV^3 - (bP + RT)barV^2 + abarV - ab = 0P¯¯¯V3(bP+RT)¯¯¯V2+a¯¯¯Vab=0

This becomes a cubic equation for barV¯¯¯V:

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 as follows:

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

In your TI calculator, let X = "insert logical guess here" by typing:

"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 barV to see if you have found the largest one. If you did not maximize barV, try a different guess until you do.