The air we breathe is mostly made up of a mixture of various gases. Air contains 78% nitrogen, 21% oxygen, and 1% other types of gases such as argon, carbon dioxide, hydrogen, helium, neon, methane, etc.
The gaseous state of matter is easy to remember because gas particles have no fixed shape, no fixed volume, and because of the empty space that exists between them, gases are able to randomly move around the container.
- This article is about real gases.
- First, we will define real gases.
- Then we will jump into the real gas law and the van der Waals equation.
- Lastly, we will apply the real gas formula to solve problems.
Ideal Gases
The Kinetic molecular theory is a theory that makes assumptions about the behavior of gases. Gases that behave in the same way as stated in this theory are called ideal gases.
The main points that the kinetic molecular theory states about the behavior of ideal gases are:
- The particles of gases are in constant, random motion.
- Gas molecules have kinetic energy.
- The collisions between gas molecules are elastic - gases do not lose or transfer energy during collisions.
- The volume of gases is negligible because of the huge amount of empty space between gas particles.
- Gas particles have no attractive or repulsive forces between neighboring molecules.
At high temperature and low pressure, gases tend to behave like ideal gases because high temperature makes intermolecular forces become insignificant. However, no gases behave exactly as you would expect an ideal gas to behave. Ideals gases are hypothetical and are not really part of real life!
Fig. 1: Expected Behavior of an ideal gas - Isadora Santos, Vaia Originals.
Before you jump into the interesting world of real gases, you want to have a good grasp of the "Ideal Gas Law", if you are ready we can move on, otherwise read up on that first!
What is the definition of a real gas?
Under high pressure and low temperature, gases deviate from being considered an ideal gas. Instead, there are called real gases.
Real gases (also known as non-ideal gases) are gases that do not follow the Kinetic molecular theory, so they deviate from being an ideal gas.
The factors that make them real gases instead of ideal gases are:
- Real gases do occupy some volume.
- Real gases have some intermolecular (attractive) forces present.
Examples of ideal gases include noble gases (group 18 of the periodic table), diatomic molecules such as H2, O2, Cl2, and polyatomic molecules like NH3, CO2, and water vapor. Really any gases you can think of are real gases, except for the theoretical ideal gases.
Under which conditions would a real gas closely behave according to the kinetic molecular theory of gases?
A real gas closely would behave as an ideal gas when pressure is low and temperature is high.
Deviation from Ideal Gases
Gases deviate from the behavior of ideal gases when they are at low temperatures and high pressures. At high pressures, the gas particles get closer together. As pressure increases, volume decreases. When gases have low temperatures, their kinetic energy is low since temperature, and kinetic energy is directly proportional to one another. This low temperature makes it great for the attractive forces to be able to interact with the gas molecules because of their slower speed. The low temperature also makes the gas more compressible.
What are the two properties that cause the deviation of real gases from behaving ideally?
Ideal gases do not occupy volume and have no attractive/repulsive forces between neighboring molecules. Real gases have a finite volume and intermolecular forces of attraction between gas molecules.
There are two properties used to measure how much a gas deviates from the ideal gas law. One being fugacity and the other compressibility factor. Fugacity works like the activity for gases and is hard to find values for it. However, the compressibility factor (Z) is often used in industrial applications and you can easily get measured values for it. Here is an equation for this:
$$Z=\frac{R\cdot T}{p\cdot V}$$
Now if you think about this Z is suspiciously like n for ideal gases, right? Well, this is the idea here, if Z = 1, it is an ideal gas, and the bigger the deviation the less ideal the gas is. You can check that around room temperature most common gases are actually pretty close to ideal gases.
For example, the compressibility factor of hydrogen at standard temperature and pressure is around Z = 1.0006 or in percentage, there is a 0.06% difference between ideal and real hydrogen at this temp and pressure.
Real Gas Law
Johannes van der Waals, a Dutch physics professor, wanted to adapt the ideal gas law into a law that would be suitable for the behavior of real gases. So, in 1873, van der Waals derived the real gas law from the ideal gas law.
How exactly did Van der Waals do this? First, he looked at the ideal gas equation.
$$$P\cdot V=n\cdot R\cdot T$
Remember that:
- Pressure (P) is in Pascal (Pa)
- Volume (V) is in Liters
- n is the number of moles of gas
- R is the universal gas constant
- Temperature (T) is the Kelvin
Since gases do have volume, he made adjustments to correct the volume. He saw that the real volume of gases increased with the increase of the number of moles of gas present. Van der Waals proposed that suggested that the correct volume should be the volume of gas minus the number of moles (n) of gas multiplied by a constant b. So, the formula for the correct volume:
$$=V-nb$$
He also corrected pressure to account for the attractive forces, which are stronger at low pressure and higher density. Van der Waals added a new constant a for this. The greater the constant a, the greater the intermolecular forces holding the gas molecules together. The corrected pressure formula was equal to
$$P+\frac{an^{2}}{V^{2}}$$
Van der Walls then needed to modify the equation to account for the finite volume of the gas particles and the presence of intermolecular forces. So he came up with a formula for calculating real gas, which he called the van der Waals equation for real gases.
$$[P+a(\frac{n}{V})^{2}]\cdot [V-bn]=n\cdot R\cdot T$$
Van der Waals equation is an equation that relates pressure, volume, temperature, and amount of gas in real gases, accounting for finite volume and the presence of intermolecular forces.
The values for Van de Waals constants a and b depend on the type of gas we are dealing with:
| Gas | Constant a () | Constant b ( ) |
| Helium (He) | 0.0341 | 0.0237 |
| Neon (Ne) | 0.211 | 0.0171 |
| Argon (Ar) | 1.35 | 0.0322 |
| Hydrogen (H2) | 0.244 | 0.0266 |
| Nitrogen (N2) | 1.39 | 0.0391 |
| Oxygen (O2) | 1.36 | 0.0318 |
| Chlorine (Cl2) | 6.49 | 0.0562 |
| Carbon dioxide (CO2) | 3.59 | 0.0428 |
Water Vapor (H2O) | 5.46 | 0.0305 |
Which of the following reals gases have the strongest intermolecular forces present?
a) Ar
b) H2O
c) N2
d) O2
e) Cl2
The lower the value of constant a, the weaker the intermolecular forces interacting with the surrounding gas molecules. So, the gas with the strongest intermolecular force will be the gas that has the higher value for constant a. So, the correct answer choice is E.
Calculations Involving Van der Waals Equation
Know that we know a formula for real gases (non-ideal gases), we can use it to solve problems that you might encounter during your exam.
Van der waals equation:
$$[P+a(\frac{n}{V})^{2}]\cdot [V-bn]=n\cdot R\cdot T$$
You have 0.750 moles of O2 in 2.000 L at 25.0 °C. Using the Van der Waals formula, calculate the pressure for O2.
a = 1.36 atm·L2/mol2
b = 0.0318 atom/mol
R = 0.0821 atm/mol·K
1. convert 25 °C into Kelvin by using the formula: °K = °C + 273
$$^{o}K=25^{o}C+273$$
$$^{o}K=298K$$
2. Plug in all of the values into the equation to solve for pressure (P).
$$P=\frac{n\cdot R\cdot T}{V-nb}-\frac{n^{2}a}{V^{2}}$$
$$P=\frac{(0.750)(0.821)(298)}{2.000-(0.750\cdot 0.0318)}-\frac{(0.750^{2})(1.36)}{(2.000)^{2}}$$
$$P=9.09atm$$
What if the question asks you to compare the pressure of gases behaving as an ideal and a real gas? In this case, we would need to find the answer for the gas using both the ideal gas equation and the Van der Waals equation. A general rule for comparing the pressure and volume of gases behavior ideally or non-ideally is:
- The pressure of a real gas (Preal) < Pressure of an ideal gas (Pideal).
- The volume of a real gas (Vreal) > Volume of an ideal gas (Videal).
Using the same question above, find the pressure of O2 , assuming that it is behaving as an ideal gas.
We can use the Ideal gas law formula = to find the pressure of O2.
$$P=\frac{n\cdot R\cdot T}{V}$$
$$P=\frac{0.750\cdot 0.0821\cdot 298}{2.000}$$
$$P=9.17atm$$
Real Gases - Key takeaways
- An ideal gas is a gas that has no molecular volume and no interaction between molecules.
- Real gases deviate from behaving ideally at low temperature and high pressure.
- Real gases have a definite volume and the presence of intermolecular interactions between neighboring gas molecules, whilst ideal gases do not have these.
- The equation for real gases is called the Van der Waals equation.
- The Van der Waals formula is
References:
Moore, J. T., & Langley, R. (2021). AP Chemistry. McGraw-Hill.
Zumdahl, S. S., Zumdahl, S. A., & DeCoste, D. J. (2016). Chemistry. Cengage Learning
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