[Cooper]

Hi,

I am confused about this question...

For a gas, T_c = 303.34 K, p_c = 48.08 atm, V_c = 0.1480 dm^3 mol^-1. Calculate the van der Waals perimeters (a,b) for the gas.

Given:

[tex]p_c=\frac{a}{27b^2}\\T_c=\frac{8a}{27Rb}[/tex]

My professor said to use the formulas for the critical pressure and temperature to solve for b and then plug it into an equation for a.

When you rearrange the critical pressure for a and substitute this into the critical temperature equation, you get (after rearranging):

[tex]b=\frac{RT_c}{8p_c}=0.06514\frac{dm^3}{mol}[/tex]

However...

If you derive the critical volume with the van der Waals equation you find it is equal to V_c = 3b. I figured it would be easier to solve for b just using this, but when I do I get...

[tex]b=\frac{V_c}{3}=0.04933\frac{dm^3}{mol}[/tex]

Why are the answers different??

Thanks

[Borek]

If you derive the critical volume with the van der Waals equation you find it is equal to V_c = 3b.

I am not sure I understand where you got it from, please elaborate.

[Cooper]

If you derive the critical volume with the van der Waals equation you find it is equal to V_c = 3b.

I am not sure I understand where you got it from, please elaborate.

It's a drawn out derivation, but if, using the van der Waals equation, you take the second derivative [tex]\frac{d^2p}{dV^2}[/tex] and set it equal to zero (since the critical point is at an inflection point), you can prove V_c = 3b. At around 6:35 in this video (https://www.youtube.com/watch?v=VjVQxzxxLVw) it is derived.

Maybe I am not using it in the correct circumstance? I am not really sure...

[Cooper]

Furthermore, I just found a problem in my textbook asking the same question with different numbers and the first thing the answer key does to solve for b is divide V_c by 3.