Van der Waals Equation of State for Real Gases and its Derivation

In the year 1873, a scientist named Johannes Diderik Van der Waals gave an equation. Johannes Diderik was a theoretical physicist and a Dutch thermodynamic scientist. In 1910, he won the noble prize in physics. 

The equation of Van der Waals of state for real gases does not follow ideal gas law. By the ideal gas law, that is, PV = nRT.

This equation was the van der Waals equation. It’s a higher version of the ideal gas. This equation is composed of non-interacting point particles which satisfies ideal gas law. According to this equation, there are some small masses in gases. These masses go through accurate elastic collisions. The real gas law isn’t able to explain the pattern of real gases. Thus, van der Waal explains the behavior of a real gas and the proportionality between particle surface and number density. 

Is Van Der Waals equation a thermodynamic equation?

The Van der Waals equation is a state of the thermodynamic equation. The theory behind the equation of Van der Waals is that it says the particles that have a non-zero volume or the proportionality between particle surface and number density are non-zero. They make fluids. This equation also talks about the compression of gases to liquid phases. The impact of this equation made other scientists like James Maxwell say that this equation will reach heights in molecular science as this equation tells the proportionality between particle surface and number density.

The van der Waal equation is (P+an²/V²)(V−nb)=nRT.

Derivation

The equation of Van der Waals is composed of non-interacting point particles which satisfies ideal gas law. 

This equation for a real gas

The volume is given as (Vm – b)

Here ‘b’= volume occupied per mole

Thus, when we substitute the ideal gas law with Vm – b= V, it comes as

Nrt = P(Vm-b)

the value of P is modified due to intermolecular attraction as

RT= (P+a/Vm2)(Vm-b)

nRT = (P+an2/V2)(V-nb)

Here, 

Vm:  The molar volume of gas

R: The universal gas constant

T: The temperature

P: The pressure

V: The volume

Therefore, the Equation of Van der Waals of proportionality between particle surface and number density can be reduced to the ideal gas law as PVm = RT.

The derivation of Van der Waals equation for one mole of gas

To deduce or derive the equation of Van der Waals for the volume of one mole of a gas. We need to follow these easy steps.

RT / V = (RT / v) = p 

RT / Vm  – b = p

Na – Vm = C   

a’C²= a'(Na/Vm)² = a/Vm²

p = RT/(Vm-b)-a/ Vm²  

Next, 

[p+(a/Vm²)[p+(a/Vm²) ] Vm−b = RT 

Next, 

[p+(n²a/V²)[p+(n²a/V²)]V−nb = nRT

(Then we substitute nVm  = V )

This is the derivation of van der Waals equation for one mole of gas.

Significance or insignificance of Van Der Waals Equation of State

Significance or merits of van der Waals equation

• The equation of Van der Waals can tell the behavior of gas easily in a better and more accurate way than this equation named equation of ideal gas.
• Equations of Van der Waals are also used and are accurate to fluids, not just to gases.
• The pattern arrangement of the equation of Van der Waals is in the way of a cubic equation in terms of volume. 
• The equation of cube in volume gives the three volumes.
• These three volumes are for measuring the volume at critical temperature or below the given critical temperature.

Insignificance or demerits of van der Waals equation

• The equation of Van der Waals can get correct or accurate answers only for the real gases, also only those which are above or at the critical temperature.
• The gases below the critical temperature, these gases results get accepted.
• But when gas is in the phase of transition, the equation of this is termed as wrong or a failure to an extent.

Conclusion

Van der Waal equation works on the interactive molecular forces and molecular size. These forces can be repulsive as well as attractive. We also call this equation the equation of state. This article will help students and aspirants with every question related to the van der Waal equation. This article covers what is van der Waal equation, how to derive it, and the proportionality between particle surface and number density. 

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