2.2 Pressure, temperature, and rms speed  (Page 8/18)

In an ideal gas, we can substitute $V\text{/}N=k_{\text{B}}T\text{/}p$ to obtain

The mean free time     $\tau$ is simply the mean free path divided by a typical speed, and the usual choice is the rms speed. Then

Summary

• Kinetic theory is the atomic description of gases as well as liquids and solids. It models the properties of matter in terms of continuous random motion of molecules.
• The ideal gas law can be expressed in terms of the mass of the gas’s molecules and $\stackrel{\text{–}}{v^2},$ the average of the molecular speed squared, instead of the temperature.
• The temperature of gases is proportional to the average translational kinetic energy of molecules. Hence, the typical speed of gas molecules $v_{\text{rms}}$ is proportional to the square root of the temperature and inversely proportional to the square root of the molecular mass.
• In a mixture of gases, each gas exerts a pressure equal to the total pressure times the fraction of the mixture that the gas makes up.
• The mean free path (the average distance between collisions) and the mean free time of gas molecules are proportional to the temperature and inversely proportional to the molar density and the molecules’ cross-sectional area.

Conceptual questions