Van der Waals Equation, Virial Expansion

[unscientific]

Homework Statement

Taken from Concepts in thermal Physics:

Homework Equations

The Attempt at a Solution

Shouldn't the van der waal's equation be:

[tex]p = \frac{RT}{V_m -b} - \frac{a}{V_m^2}[/tex]
[tex]pV_m = \frac{VRT}{V_m -b} - \frac{a}{V_m}[/tex]

 

[TSny]

Shouldn't the van der waal's equation be:

[tex]p = \frac{RT}{V_m -b} - \frac{a}{V_m^2}[/tex]

Yes. So (26.41) has a wrong sign. But it looks like they got it right in (26.42) and (26.43).

 

[Chestermiller]

Homework Statement

Taken from Concepts in thermal Physics:

Homework Equations

The Attempt at a Solution

Shouldn't the van der waal's equation be:

[tex]p = \frac{RT}{V_m -b} - \frac{a}{V_m^2}[/tex]
[tex]pV_m = \frac{VRT}{V_m -b} - \frac{a}{V_m}[/tex]

You're right. It sure looks like they made a slew of algebra mistakes. The first term on the right hand side of 26.41 isn't even dimensionally correct.

Chet

 

[TSny]

You're right. It sure looks like they made a slew of algebra mistakes. The first term on the right hand side of 26.41 isn't even dimensionally correct.

Chet

Right. I didn't notice that they should not have had the factor of V on the left side of equation 26.41.

 

[paranormal]

The van der Waals equation is often written in two different forms, as you have correctly pointed out. The first one, p = (RT/(V_m - b)) - (a/V_m^2), is known as the "pressure form" and is more commonly used. The second one, pV_m = (VRT/(V_m - b)) - (a/V_m), is known as the "volume form" and is less commonly used. Both forms are mathematically equivalent and can be derived from the same fundamental principles.

The van der Waals equation is an improvement over the ideal gas law, which assumes that gas particles have no volume and do not interact with each other. In reality, gas particles do have a finite volume and they do interact with each other, leading to deviations from ideal behavior. The van der Waals equation takes into account the volume of the gas particles and the attractive forces between them, resulting in a more accurate description of the behavior of real gases.

The virial expansion is a mathematical technique used to approximate the behavior of real gases. It involves expressing the pressure and volume of a gas as a series expansion in terms of the density of the gas. The first term in this expansion is the ideal gas law, and subsequent terms take into account the deviations from ideal behavior. The van der Waals equation is an example of a second-order virial expansion, which includes the first two terms in the series.

Overall, the van der Waals equation and the virial expansion are important tools in understanding the behavior of real gases and can be used to make more accurate predictions in various thermodynamic processes and systems.