Question

Show that the van der Waals and Redlich-Kwong equations of state reduce to the ideal gas equation of state in the limit of low density.

Short answer

In the limit of low density, as ρ approaches zero, both the van der Waals and Redlich-Kwong equations of state converge to the ideal gas equation of state. For the van der Waals equation, we have: \[\lim_{\rho\to0} \left(P + \frac{a\rho^2}{(1/\rho)^2}\right)(\frac{1}{\rho} - b) = R T\] Simplifying the expression, we obtain \(P = R T\). Similarly, for the Redlich-Kwong equation: \[\lim_{\rho\to0} \left(\frac{R T}{1/\rho-nb} - \frac{a\rho}{\sqrt{T}\left((1/\rho)^2+(1/\rho)b\right)}\right) = P\] Simplifying the expression, we obtain \(P = R T\). Thus, both equations reduce to the ideal gas equation of state in the limit of low density.

Step-by-step solution

Analyze the van der Waals equation

To find the limit of the van der Waals equation as density approaches zero, first rewrite the equation in terms of molar density, which is given by \(\rho = \frac{n}{V}\) : \[\left(P + \frac{a\rho^2}{(1/\rho)^2}\right)(\frac{1}{\rho} - b) = R T\] Now we need to take the limit as density, ρ, approaches zero to see if it reduces to the ideal gas equation.

Take the limit of van der Waals equation as density approaches zero

The limit expression becomes: \[\lim_{\rho\to0} \left(P + \frac{a\rho^2}{(1/\rho)^2}\right)(\frac{1}{\rho} - b) = R T\] Now plug in 0 for the density: \[\left(P + \frac{a(0)^2}{(1/0)^2}\right)(\frac{1}{0} - b) = R T\] The ratios with ρ in the denominator become: \[\left(P + \frac{0}{1}\right)(\infty - b) = R T\] Simplifying the expression, we get: \[P = R T\] Thus, the van der Waals equation reduces to the ideal gas equation in the limit of low density.

Analyze the Redlich-Kwong equation

To find the limit of the Redlich-Kwong equation as density approaches zero, rewrite the equation in terms of molar density, ρ: \[P = \frac{R T}{1/\rho-nb} - \frac{a\rho}{\sqrt{T}\left((1/\rho)^2+(1/\rho)b\right)}\] Now take the limit as density, ρ, approaches zero to see if it reduces to the ideal gas equation.

Take the limit of Redlich-Kwong equation as density approaches zero

The limit expression becomes: \[\lim_{\rho\to0} \left(\frac{R T}{1/\rho-nb} - \frac{a\rho}{\sqrt{T}\left((1/\rho)^2+(1/\rho)b\right)}\right) = P\] Now plug in 0 for the density: \[\left(\frac{R T}{1/0-nb} - \frac{a(0)}{\sqrt{T}\left((1/0)^2+(1/0)b\right)}\right) = P\] The ratios with ρ in the denominator become: \[\left(\frac{R T}{\infty-nb} - \frac{0}{\sqrt{T}(1+\infty)}\right) = P\] Simplifying the expression, we get: \[P = R T\] Thus, the Redlich-Kwong equation reduces to the ideal gas equation in the limit of low density. In conclusion, both the van der Waals and Redlich-Kwong equations of state reduce to the ideal gas equation of state in the limit of low density.

Key concepts

Van der Waals Equation

The van der Waals equation is a mathematical model that extends the ideal gas law to account for the interactions between gas molecules and their finite size. It is expressed as:

• \[ \left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT \]

where \(P\) is the pressure, \(V_m\) is the molar volume, \(R\) is the gas constant, \(T\) is temperature, and \(a\) and \(b\) are constants specific to the gas.
Van der Waals introduced these constants to correct for the pressure reduction and volume availability when gas molecules are close enough to influence each other. The term \( \frac{a}{V_m^2} \) accounts for molecular attraction, while \(b\) corrects for the volume exclusion caused by finite molecular size.
In the limit of low density or high molar volume \(V_m\), these corrections become negligible and the equation simplifies to the ideal gas law \(PV = nRT\).

Redlich-Kwong Equation

The Redlich-Kwong equation improves upon the van der Waals equation by providing a better fit for gases over a wider range of pressures and temperatures. It is expressed as:

• \(P = \frac{RT}{V_m - b} - \frac{a}{T^{0.5}V_m(V_m+b)}\)

The equation introduces temperature dependence in the attraction term \(\frac{a}{T^{0.5}}\), aiming to give a more accurate portrayal of real gases, especially at higher pressures.
As with van der Waals, this equation also reduces to the ideal gas law at the low-density limit, where the influence of intermolecular forces and volume exclusion becomes minimal. This is because both \(a\) and \(b\) terms become insignificant, returning to the simple form: \(PV=nRT\).

Ideal Gas Law

The ideal gas law is the simplest equation of state for gases. It models gases as having no intermolecular forces and occupying zero volume themselves. The formula is given by:

• \( PV = nRT \)

Here, \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the temperature.
This equation assumes that the gas particles move independently of one another and collide elastically. It serves as a good approximation of real gas behavior under many conditions, especially low pressure and high temperature.
However, it fails at high-density scenarios where interactions between particles cannot be ignored. That’s when more complex equations of state, like van der Waals and Redlich-Kwong, become essential.

Low Density Limit

The concept of the low density limit involves understanding how certain equations of state simplify as the density of a gas approaches zero. This perspective is insightful as it allows more complex gas models to converge into the simpler ideal gas law.

• At low densities, intermolecular forces weaken.
• Finite molecular volume becomes less significant.
• Equations simplify to \(PV=nRT\).

Calculating limits when density approaches zero helps in verifying the applicability of the ideal gas law as a benchmark for real gas behavior under such conditions. As density decreases, contributions from terms involving \(a\) and \(b\) in van der Waals and Redlich-Kwong equations diminish.
Therefore, examining these limits confirms when it is appropriate to use the simple ideal gas law without substantial loss of accuracy, illustrating the versatility and foundational role of this core thermodynamic equation.