How Does Van der Waals Equation Approach Ideal Gas Law at Low Particle Counts?

[Quelsita]

QUESTION:

Van der Waals derived the expression:
(P+(a/v^2))(v-b)=RT
which is a useful approximation of the equation of state of a real gas. Here a and b are gas
dependent constants and v =V/n is the specific volume (Volume V divided by the # of
moles of gas molecules)
Show that at constant volume V and temperature T but decreasing number N=n NA (NA:
Avogadro’s constant) of particles the Van der Waals equation of state approaches the
equation of state of an ideal gas.


I know that I have to rearrange the VdW equation into the P=P(v,T) form and then use a Taylor series.

I have rearranged the eqution:

(P+(a/v²))(v-b)=RT
P=P(v,T)
P(v,T)=a₀T - b₀ln(v/v₀)

thus,

a₀T - b₀ln(v/v₀) + (a/v²))(v-b)= RT

but I'm uncertain what to do next...

Is this even correct thus far?
Any tips on where to go from here?

Thanks!

 

[jbsteele]

You are on the right track. To complete the proof, you need to use a Taylor series expansion. Rearrange the equation so that P is written in terms of v and T (as you have done). Then take the derivative of P with respect to N, and set it equal to 0. This will give you an equation for v. Next, make a Taylor series expansion of P about the point (v,T) to obtain the equation of state of an ideal gas.

 

[tomcue]

I can provide some guidance on how to approach this problem. First, let's review the definition of an ideal gas. An ideal gas is a theoretical gas that follows the ideal gas law, which states that the pressure (P), volume (V), and temperature (T) of a gas are related by the equation PV = nRT, where n is the number of moles of gas and R is the gas constant.

To show that the Van der Waals equation approaches the equation of state of an ideal gas, we can start by substituting the specific volume (v) with its definition V/n. This gives us:

(P+(a/(V/n)^2))(V/n-b)=RT

Next, let's expand the brackets and simplify the equation:

(P+aV^2/n^2-2aV/n^3+a/n^4)(V-bn)=RT

PV + aV^3/n^2 - 2aV^2/n^3 + aV/n^4 - bV + abn = RT

Now, let's assume that the number of particles (n) is decreasing, which means that n is approaching 0. This also means that V/n is approaching infinity. In this limit, we can ignore the terms with n in the denominator, as they will become negligible compared to the other terms. This leaves us with:

PV + aV^3/n^2 - bV = RT

Next, we can use the ideal gas law to replace PV with nRT:

nRT + aV^3/n^2 - bV = RT

Finally, we can rearrange this equation to get P as a function of V and T:

P = (RT + aV^3/n - bV)/V

In the limit as n approaches 0, this equation simplifies to:

P = RT/V

which is the equation of state for an ideal gas.

In conclusion, we have shown that in the limit of decreasing number of particles, the Van der Waals equation of state approaches the equation of state for an ideal gas. This demonstrates the usefulness of the Van der Waals equation in approximating the behavior of real gases.