Derivation of Critical Volume, Temperature and Pressure for the VDW equation

[shihabdider]

Homework Statement

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

Homework Equations

Van der Waal's equation:

(P-a(n/v)^2)-(V-nb)=nRT

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

 

[sankalpmittal]

Homework Statement

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

Homework Equations

Van der Waal's equation:

(P-a(n/v)^2)-(V-nb)=nRT

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

Vander waals equation is given by :

(P+an²/V²)(V-nb) = nRT

Find for 1 mole of chlorine gas. Replace n=1 , expand and simplify.By doing so , replace V = V_molar. Since you are not allowed to use calculus, calculations will be tough but you will have to make an attempt. Prepare a cubic equation.

Then At T=T_critical , P=P_critical , find the condition for roots to be repetitive. Compare the coefficients.

Sorry , I can not give more hints than this.

 

[shihabdider]

First of all, thank you for your reply. However I feel that you did not read my attempt at the problem. I already found and expanded the formula to P∇-Pb-a/∇-ab=RT (set n=1 is the same as diving by n). However my real issue was with Pc and Tc and "preparing the cubic" which was not made any clearer from your comments. I understand that I have to prepare a cubic, I just don't know HOW to do so. Furthermore you say "Then at T=Tc and P=Pc find the condition for the roots to be repetitive". But what does T actually equal when it is Tc, is it as I have assumed equal to the T in the ideal gas law? You never actually affirm or deny this.

Again thanks for your reply.

 

[shihabdider]

Update:

So after struggling with this some more, I made the VDW equation into a cubic one:

Pc∇^3 - ∇^2RTc + ∇(RTcb-a) + ab = 0 (I don't know if this is correct)

Then I compared it to the expanded form of the equation: (∇-Vc)^3 Since they are both equal to zero they are equal to each other.

From this I compared the coefficients and found these 4 equations:

Pc=1
RTc=3Vc
RTcb-a=3Vc^2
ab=Vc^3

And now again I am stuck, I tried moving the equations around and substituting (for example subbing RTc for 3Vc in the third equation) but have gotten nowhere. Can anyone help please?

Thank You

 

[sankalpmittal]

Update:

So after struggling with this some more, I made the VDW equation into a cubic one:

Pc∇^3 - ∇^2RTc + ∇(RTcb-a) + ab = 0 (I don't know if this is correct)

Firstly , this isn't correct.

Then I compared it to the expanded form of the equation: (∇-Vc)^3 Since they are both equal to zero they are equal to each other.

From this I compared the coefficients and found these 4 equations:

Pc=1
RTc=3Vc
RTcb-a=3Vc^2
ab=Vc^3

And now again I am stuck, I tried moving the equations around and substituting (for example subbing RTc for 3Vc in the third equation) but have gotten nowhere. Can anyone help please?

Thank You

You did the wrong calculation and got wrong values for P_c , V_c and T_c. Also I hope you noticed that you were using the wrong VDW equation. If you want me to check , where you went wrong , then you will have to post your whole procedure along with working , highlighting all the steps , and also detailing your logical process. Just to clarify : You're using ∇ for V_molar right ?

Did you notice this before ?

Your form : (P-an²/V²)-(V-nb)=nRT

Correct form : (P+an²/V²)(V-nb) = nRT

 

[shihabdider]

1. THANK YOU for pointing out I was using the wrong VDW equation, I feel like an idiot but at least, having made the mistake now, I won't be making the same mistake on a test :D
2. Yes, I realize now that n=1 is the only way to go, and no I don't need the crit values in terms of n, I need their pure values (a, b and R only)
3. Yes ∇ is supposed to be Vmolar
4. I need to redo my calculations (with the correct VDW equation) then I will post those calculations

Again, thank you so much for helping me

 

[sankalpmittal]

1. THANK YOU for pointing out I was using the wrong VDW equation, I feel like an idiot but at least, having made the mistake now, I won't be making the same mistake on a test :D
2. Yes, I realize now that n=1 is the only way to go, and no I don't need the crit values in terms of n, I need their pure values (a, b and R only)
3. Yes ∇ is supposed to be Vmolar
4. I need to redo my calculations (with the correct VDW equation) then I will post those calculations

Again, thank you so much for helping me

Ok , however one point to note down : Dividing both sides by n will also do. You can replace V/n as ∇ , then. Thinking deeply , both the methods will lead you to same path. Both carry same logic as well.

Go well with your studies.