Calculate the number of moles in a real gas

[larsb]

Homework Statement

I would like to know how to calculate how much moles of gas I have in the following in a cylinder with a certain volume and pressure.

The gas in the cylinder is a mixture of air, with added oxygen and helium, the mixture is 18% Oxygen, 36,6% Nitrogen, 45% Helium and 0,4% Argon. The cylinder is 24 liters big, the pressure in the cylinder is 200bar.
I can calculate the a and b values for the gas mixture.

P = 200 bar
V = 24 L
T = 293,15 K
R = 0,083145 L bar K^-1 mol ^-1
a = 0,8746 L²bar mol^-2
b = 0,0388 L mol^-1

Homework Equations

PV = nRT ( ideal gas law)

P = nRT (V-nb)^-1 - n²a V^-2

The Attempt at a Solution

According to the ideal gas law this should be 196,93 moles, but that is not right, since if I use the Vanderwaals equation I end up at a pressure of 234,6 bar to accommodate 196,93 moles of this gasmixture.

Using the Vanderwaals equation I can't calculate the exact number of moles, this is where is end up:
200 = 24,37n (24 - 0,0388n)^-1 - 0,00152n²

So please enlighten me, how can I proceed to calculate the value for n?

 

[Char. Limit]

First, remember that the Van Der Waals pressure doesn't have to match the ideal pressure: in fact, it won't, unless a is 0. With an a as large as yours, the pressure can actually have quite a bit of variation between ideal and Van Der Waals.

For that last equation, have you considered quadratics?

 

[ehild]

You can apply some iteration procedure. Rewrite the Van der Waals equation in the form

$$n=\frac{PV}{RT}(1+\frac{n^2a}{PV^2})(1-\frac{nb}{V})$$

Plug in the data, start with n you got for the ideal gas, substitute for n at the right side, and you get a new n. Continue the procedure with the new n... You get a value near 174 mol, if I am not mistaken.

ehild

 

[larsb]

First, remember that the Van Der Waals pressure doesn't have to match the ideal pressure: in fact, it won't, unless a is 0. With an a as large as yours, the pressure can actually have quite a bit of variation between ideal and Van Der Waals.

For that last equation, have you considered quadratics?

Thanks, I know that it doesn't match the actual pressure and the ideal pressure do not have to match...
I know for sure that the pressure in the cylinder is 200 bar, or actually 201 bar, because I am reading the pressure from a gauge.

What do you mean by "considering quadratics"?

 

[larsb]

You can apply some iteration procedure. Rewrite the Van der Waals equation in the form

$$n=\frac{PV}{RT}(1+\frac{n^2a}{PV^2})(1-\frac{nb}{V})$$

Plug in the data, start with n you got for the ideal gas, substitute for n at the right side, and you get a new n. Continue the procedure with the new n... You get a value near 174 mol, if I am not mistaken.

ehild

Ok, that indeed helps me a lot! I do not have to be completely spot on, however the closer the better.