# Gas Kinetics

Skills to Develop

- Explain the distribution of molecular speed of a gas.

### Molecular Speed of Gases

The quantitative relationship of temperature effect on chemical reaction rates is discussed in form of activation energy, depicted as E_{a} in the diagram. For reactions having large E_{a} values, high temperatures are required to have a measurable reaction rate. The reason behind this is due to the small number of molecules having sufficient energy to overcome E_{a}, the energy barrier of forming an activated reaction intermediate.

At a certain temperature, not all gas molecules are moving with the same speed, some fast and some slow. The kinetic energies of molecules are not all equal. The fast moving ones have high kinetic energy, and they may have enough energy to overcome the reaction energy barrier E_{a}.

A plot of number of gas molecules at certain speed versus speed gives a distribution curve. Careful studies of gases show that the distribution is not a bell or normal distribution, but a Maxwell- Boltzmann distribution.

A sketch of a Maxwell distribution is given here. The peaks are not symmetrical. There are more molecules at higher speed than at lower speeds. When the temperature increases, the peak shifts to the right. A more carefully plotted diagram is shown below. |

You will learn the theory and the statistics at a higher year. For the moment, you may notice how the distribution curve shifts as temperature increases.

### Questions

**Which curve has a bell shape?****Normal distribution****Maxwell-Boltzmann distribution****Chi-square distribution**

**Skill -**

Name the various distributions.**Which of the following gases has the lowest most-probable speed:**\(\ce{H2}\)**,**\(\ce{He}\)**,**\(\ce{Ne}\)**,**\(\ce{N2}\)**,**\(\ce{O2}\)**,**\(\ce{F2}\)**,**\(\ce{Ar}\)**, or**\(\ce{Xe}\)**?****Skill -**

Explain how molecular masses affect the average speed.**For**\(\ce{N2}\)**at 300 K, which of the following speeds is the highest?****Most probable speed****Average speed**-
**Root-mean-square speed**

**For**\(\ce{N2}\)**at 300 K, which of the following speeds is the lowest?****Most probable speed****Average speed**-
**Root-mean-square speed**

**For the gas**\(\ce{N2}\)**, which temperature gives the highest most probable speed: 500 C, 600 K, or 700 F?**

### Solutions

- a. Normal distributions have bell shape.
- \(\ce{Xe}\), because it is the heaviest one here.

**Discussion -**

The gas with the heaviest molecular mass has the lowest average speed. \(\ce{UF6}\) is a gas used for enrichment of the isotope \(\ce{U-235}\). - c.
**Discussion -**

The root-mean-square speed is \(\sqrt{\dfrac{3 R T}{M}}\), where*R*= gas constant,*T*= temperature,*M*= molecular mass. - a.
**Discussion -**

The formulas for the three speeds mentioned here can be derived by mathematical techniques. The most probable speed is \(\sqrt{\dfrac{2 R T}{M}}\), where*R*= gas constant,*T*= temperature,*M*= molecular mass. - 500 C
**Discussion -**

It is easy to calculate the most probable speed if you know the formula to use. Note that 500 C is higher than 700 F. Do you know what the temperature scales measure?

### Contributors

Chung (Peter) Chieh (Chemistry, University of Waterloo)