Question

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

Answer 1

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 `barV` is a hogfest all on its own...

`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 `barV`:

`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.