# Rearranging variables Van Der Waals EoS into new variables

• KDS4
In summary, we can re-arrange the Van Der Waals equation of state, Eq. (1), by substituting dimensionless combinations v, p, and t for V, P, and T respectively. This results in P appearing as 27b2P/a and T appearing as 27bKbT/8a, as well as V appearing as V/(3Nb). This rescaling is based on the critical constants of the Van Der Waals equation.
KDS4
Moved from a technical forum, so homework template missing
The question I'm stuck on is:

P = NKBT/(V-Nb) - aN2/(V2) -----> (1)

Re-arrange variables in the Van Der Waals equation of state, Eq. (1), so that V always appears in the equation as V/(3Nb) and P appears as 27b2P/a. Then T should appear in the combination 27b kBT/(8a). Call these dimensionless combinations v, p, and t (also called reduced variables), and express the van der Waals equation in terms of v, p, and t.

I managed to get P to appear as 27b2P/a and T to appear as 27bKbT/8a using algebra and getting some pretty ugly terms along the way that are floating around but with V appearing multiple times in the initial equation I can't for the life of me get it to appear the way the question wants me to.

Any insight would be greatly appreciated!

KDS4 said:
I managed to get P to appear as 27b2P/a
That should be ##27 b^2 P / (8a)## as the critical pressure is ##p_c = 8a/(27 b^2)##.

DrClaude said:
That should be ##27 b^2 P / (8a)## as the critical pressure is ##p_c = 8a/(27 b^2)##.

My assignment says otherwise but it also doesn't reference critical pressure.

http://imgur.com/a/Q29yi

That'll teach me to look up things on the internet

Indeed, the source I used was wrong, and there is no 8 there. You should look up "critical constants" (maybe here http://cbc.arizona.edu/~salzmanr/480a/480ants/vdwcrit/vdwcrit.html) to get a deeper understanding of that rescaling of the VdW equation.

As for you problem, you'll have to post some details of the derivation if you want more help.

The easiest way to work this problem is to substitute:$$V=3Nbv$$
$$P=\frac{a}{27b^2}p$$
$$T=\frac{8a}{27bK_B}t$$

## 1. What is the purpose of rearranging variables in the Van Der Waals equation of state?

Rearranging variables in the Van Der Waals equation of state allows for a more simplified and convenient form of the equation, making it easier to use and understand.

## 2. How do you rearrange the Van Der Waals equation of state into new variables?

The process of rearranging the Van Der Waals equation of state involves manipulating the original equation by dividing both sides by the pressure term and then substituting variables such as compressibility factor and reduced temperature.

## 3. What are the benefits of using new variables in the Van Der Waals equation of state?

The use of new variables in the Van Der Waals equation of state can provide a more accurate representation of real-world gas behavior, as well as make it easier to compare gases with different properties.

## 4. Can the Van Der Waals equation of state be rearranged into other forms besides the commonly used one?

Yes, the Van Der Waals equation of state can be rearranged into various forms depending on the desired variables or properties that need to be represented. Some examples include the reduced form, the reduced and compressibility form, and the reduced and reduced compressibility form.

## 5. Are there any limitations to rearranging the Van Der Waals equation of state into new variables?

While rearranging the Van Der Waals equation of state can provide a more simplified and useful form, it may not accurately represent all gas behavior in all situations. It is important to understand the limitations and assumptions of the equation when using it in calculations or experiments.

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