Worksheets/Van Der Waal Equation

Background

Critical points of gas, supercritical behavior

The Van Der Waal equation for a gas accounts for non-ideal behavior caused by strong intermolecular forces of attraction or repulsion:

$ (P + \dfrac{a}{V^2})(V-b) = RT $

where P is the pressure, V is the molar volume (volume of a certain number of moles), a and b are constants that depend on the molecules, k is the Boltzmann constant, and T is temperature.

Now the critical points can be found: https://www.youtube.com/watch?v=VjVQxzxxLVw

Critical point is the saddle point of the above equation, and is defined as the point where:

$ \dfrac{\partial P}{\partial V} \bigg|_{T=T_c,V=V_c} = 0 $

and

$ \dfrac{\partial^2 P}{\partial V^2} \bigg|_{T=T_c,V=V_c} = 0 $

Worksheet Questions

Question 1: Derivation

Show that the critical point $ (P_c, V_c, T_c) $ is given by:

$ P_c = \dfrac{a}{27b^2} $

$ V_c = 3b $

$ T_c = \dfrac{8a}{27bR} $

Start by rearranging to get P as a function of T and V.

Find the two expressions, now you have two equations and two unknowns Vc and Tc.

Solve for these two quantities.

Finally, you know Pc as a function of Vc and Tc.

Question 2: Carbon Dioxide

Carbon dioxide has interesting properties when supercritical, and can be used as an environmentally benign solvent in semiconductor manufacturing.

Look up the values of a and b for carbon dioxide, and use them to calculate the critical point for carbon dioxide. Cite your source of information.

Question 3: Steam

In this question, you'll be investigating steam, the vapor form of water. You will compute its properties using the Van Der Waals equation, and compare the same calculation using the Ideal Gas equation, to estimate the error in assuming steam is an ideal gas.

If steam has a temperature of 150 C (423.15 K) and a molar volume of 7.002 L/mol, what is the resulting pressure using the Van Der Waals equation?

What is the resulting pressure using the Ideal Gas equation?

What is the relative (percent) error in the ideal gas approximation?