# Van der Waals EOS using reduced molar volume

• jbowers9
In summary, the conversation discusses the equation Z = 3Vr/(3Vr-1) - 9/(8VrTr) and its relationship to the reduced molar volume Vr. The conversation also mentions the equations {P + a/Vm^2}{Vm - b} = RT and Tc = 8a/27Rb and provides a solution for Z using the values of a and b.
jbowers9

## Homework Statement

I'm taking an online PChem course where the instructor throws out
Z = 3Vr/(3Vr – 1) – 9/(8 VrTr); where Vr is the reduced molar volume.

2. Homework Equations

{P + a/Vm^2}{Vm – b} = RT; where Vm is the molar volume
Tc = 8a/27Rb ; Vc = 3b ; Pc = a/27b^2

3. The Attempt at a Solution

(Pr + 3/Vr^2)( Vr – 1/3) = (8/3)Tr
Then
Pr = {8 Tr/(3Vr – 1) - 3/ Vr^2}
and
Z = PrVr/RTr = {8 Tr/(3Vr – 1) - 3/ Vr^2} Vr/RTr
gives
Z = {8 Vr/R(3Vr – 1) - 3/R VrTr}
I can’t figure out where to go next without chasing my tail with a’s and b’s to get the R to go away and give me the correct coefficients in the rhs of the eq. Help!

berkeman
I think you're misinterpreting Z (or the instructor is).
Let's call Zr = PrVr/Tr
Then Zr = 8Vr/(3Vr-1) - 3/VrTr
Z = PV/RT = PcVc/RTc*PrVr/Tr = ZcZr
Now Zc = 3/8, as you can get by the definitions of Pc, Vc and Tc
Hence you get the answer for Z.

berkeman

Using a = Tc 27Rb/8 & b = Vc/3 and dividing through by Vc,Tc & R it gives the desired result. Thanks. Phew! Now it's so simple. (Homer Simpson doi)

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## 1. What is the Van der Waals equation of state?

The Van der Waals equation of state is a mathematical model used to describe the behavior of real gases. It takes into account the intermolecular forces and the finite size of gas molecules, which are not considered in the ideal gas law.

## 2. How is the Van der Waals equation derived?

The Van der Waals equation is derived from the ideal gas law by adding two correction terms: one for the attractive forces between molecules and one for the volume occupied by the molecules themselves.

## 3. What is reduced molar volume in the Van der Waals equation?

Reduced molar volume is a dimensionless quantity used in the Van der Waals equation. It is calculated by dividing the molar volume of a gas by the critical molar volume, which is the molar volume at the critical point.

## 4. How is the Van der Waals equation used to predict gas behavior?

The Van der Waals equation can be used to predict the behavior of real gases at different temperatures and pressures. By inputting the values for temperature, pressure, and the gas's critical constants into the equation, the Van der Waals equation can calculate the gas's molar volume and compare it to experimental data.

## 5. What are the limitations of the Van der Waals equation?

The Van der Waals equation is an improvement over the ideal gas law, but it still has limitations. It does not take into account the variation in attractive forces between different gases, and it becomes less accurate at high pressures and low temperatures. Additionally, it does not account for the effects of molecular shape and polarity, which can greatly influence gas behavior.

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