Can a Real System Be Governed by the van der Waals or Ideal Gas Equation?

[plmokn2]

Homework Statement

BY consideration of the pressure coefficient dp/dt|V determine whether a real system can be governed by the van der Waals equation, the ideal gas equation.

Homework Equations

pV=nRT ideal gas
(p+a/V^2)(V-b)=nRT Van der Walls
dp/dT|V=dS/dV|T

The Attempt at a Solution

Using a Maxwell relation we can rewrite the pressure coefficient as dS/dV|T, which must equal 0 as T->0 from the third law. So then we need to check the behaviour of the two equations of state:

Van der Walls I get the pressure coefficient=nR/(V-b)=1/T (p+a/V^2) which goes to infinity as T goes to zero.

Similarly I get for an ideal gas the coefficient =nR/V=p/T which I don't think goes to zero either.

So it looks like neither works. I was expecting the ideal gas to fail since it doesn't take into account intermolecular forces and non-zero molecular volume, but I'm a bit surprised the van der Waal equation doesn't work, so if someone could check my working I'd appreciate it.
Thanks


ps.dp/dT|V means partial dp by dT at constant V.

 

[anathema]

Thank you for your post and for considering the pressure coefficient in determining whether a real system can be governed by the van der Waals equation or the ideal gas equation. Your approach using a Maxwell relation is correct, and your calculations for both equations of state are also correct.

As you have noticed, both equations of state do not work for a real system at very low temperatures. This is because at low temperatures, the intermolecular forces become dominant and the ideal gas assumption of negligible intermolecular forces breaks down. For the van der Waals equation, the pressure coefficient goes to infinity as the temperature approaches zero, indicating that the equation does not accurately describe the behavior of real gases at very low temperatures. Similarly, for the ideal gas equation, the pressure coefficient does not approach zero as the temperature approaches zero, indicating that this equation also fails to describe the behavior of real gases at very low temperatures.

To accurately describe the behavior of real gases at low temperatures, we need to use more sophisticated equations of state, such as the Redlich-Kwong equation or the Peng-Robinson equation, which take into account the effects of intermolecular forces and non-zero molecular volume. These equations are more accurate at low temperatures and can better describe the behavior of real gases.

I hope this helps to clarify your understanding of the behavior of real gases at low temperatures. Keep up the good work in your scientific studies!

 

[ogb p]

I would first like to clarify that the Third Law of Thermodynamics states that the entropy of a pure, perfect crystalline substance at absolute zero temperature is zero. This means that as T approaches zero, the pressure coefficient (dp/dT|V) must also approach zero.

In this homework problem, we are looking at whether a real system can be described by the van der Waals equation or the ideal gas equation. The van der Waals equation takes into account the non-zero molecular volume and intermolecular forces, while the ideal gas equation assumes that molecules have no volume and do not interact with each other.

From your calculations, it seems that neither equation works as T approaches zero. This makes sense, as both equations are based on assumptions that break down at very low temperatures. At such low temperatures, the intermolecular forces and molecular volume become significant and cannot be ignored.

In conclusion, neither the van der Waals equation nor the ideal gas equation can accurately describe a real system at absolute zero temperature. This is a limitation of these equations, and further research and development may be needed to describe the behavior of real systems at very low temperatures.