Worksheets/Van Der Waal Equation

Read This Before You Begin

In this worksheet, students are getting some experience applying the concept of a partial derivative in a setting other than (x,y,z) space. Thermodynamics is an important area of application for vector calculus.

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The Story: Gases

Ideal gas law: what might we want to describe about gases? pressure, volume, and temperature. This is useful for many reasons - if you're designing a rocket, you're vaporizing lots of fuel into a gas, so you want to understand the dynamics of the gas. If you're burning gasoline in a combustion engine, the gas dynamics determine how much work you can extract from the engine. Gas behavior is important in weather phenomena: cloud formation, precipitation, fog, water cycle. In outer space, gas dynamics determine the formation of stars, planets, and atmospheres.

Models for gases: ideal gas is nice and simple, treats molecules as perfectly elastic balls bouncing around and perfectly repellent

In reality, molecules are often polar, and experience intermolecular forces of attraction or repulsion. Example: water has dielectric, meaning positive-negative imbalance across molecule

This means water molecules tug on each other, and is the same reason water has such a high boiling point and why ice expands when it freezes.

The Ideal Gas Equation

The model for an ideal gas (perfectly elastic molecules) leads to the equation:

$PV=RT$

where P is the pressure, V is the molar volume (volume of a certain number of moles), R is the ideal gas constant, and T is the temperature.

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This equation works amazingly well for all sorts of different gases, but as temperatures get higher, molecules get bigger, or intermolecular forces become stronger, this equation just doesn't cut it.

Van Der Waals Equation

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The Van Der Waal equation for a gas can model this kind of non-ideal behavior, by adding terms for 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, R is the ideal gas constant, and T is temperature.

We can use this to find properties of the gas: given the volume and temperature, for example, we can find the pressure.

Critical Points

The ability to account for non-ideal behavior allowed the Van Der Waals equation to describe a real, and very mysterious phenomena: supercritical fluids.

It turns out that each material has a special temperature, pressure, and molar volume, at which the material becomes critical - meaning, it behaves like a fluid with zero viscosity, allowing it to do strange things like climb up walls or dissolve other materials. When the temperature or pressure are increased, the material becomes supercritical.

This special critical point corresponds to the saddle point of the Van Der Waals equation. By rewriting the Van Der Waals equation to express pressure P as a function of molar volume V and temperature T, $P(V,T)$, the following conditions are true at the Van Der Waals equation saddle point:

${\dfrac {\partial P}{\partial V}}{\bigg |}_{T=T_{c},V=V_{c}}=0$

and

${\dfrac {\partial ^{2}P}{\partial V^{2}}}{\bigg |}_{T=T_{c},V=V_{c}}=0$

These can be solved to find the critical point $(P_{c},V_{c},T_{c})$ as a function of the gas properties a, b, and the ideal gas constant R.

Worksheet Questions

Question 1: 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?

Question 2: Critical Point 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 3: Carbon Dioxide Critical Point

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.