Derive Vc, Pc and Tc from the Van der waals equation, then determine these values for Chlorine gas which as a=6.49 and b=.0562. Solve Algebraically without using Calculus.
Given hints (verbatim):
1. First, convert V/n = ∇ in VDW equation and P->Pc and T ->Tc
2. Reorganize VDW equation to polynomial of volume: ∇^3+?∇^2+??∇+???=0
3. Expand the following equation and compare with equation in (2): (∇-Vc)^3
Van der Waal's equation:
Ideal Gas Equation: PV=nRT
The Attempt at a Solution
Ok, I've been struggling with this problem for a few hours now and have gotten no where:
First I converted the Van der Waals equation to this: (P-a(n/v)^2)-(∇-b)=RT by diving both sides by "n" and substituting ∇ for V/n.
From here I have tried a number of things, first I tried to convert the P and T to Pc and Tc as the hints suggest. I figured that at Pc and Tc the gas has to behave ideally meaning that I can use the ideal gas equation: PV=nRT and substitute the P and T in terms of the ideal gas law (i.e Pc=nRTc/Vc). However I am unsure if this is the correct approach. From hints 2 and 3 however I assume that the coefficients of the equation (∇-Vc)^3 are equal to that of the polynomial of volume stated in hint 2. From this I assume I can obtain the critical T,V and P in terms of a,b and R. However the real problem is figuring out how I can get from the VDW equation to the polynomial of volume in the first place. Again, please no calculus (even though I feel it would be somewhat easier using Calculus, the instructions do not allow it)
Please advise me, I really only need a nudge in the right direction. I want to solve this myself (I think it's as an interesting problem). And Thank You