Kinetic theory of gases
In a gas, the molecules are in continuous, random motion. The kinetic theory of gases analyzes matter in terms of this molecular motion. It is built on the following assumptions:
1. A gas contains a very large number of molecules.
2. The molecules are far apart compared with their size.
3. The molecules undergo elastic collisions with one another and with the walls.
4. The molecules obey Newton's laws of motion.
5. The molecules interact only when they collide.
From these assumptions, the average kinetic energy of a molecule in a gas works out to
$\overline{KE}=\tfrac{1}{2}m\,\overline{v^2}=\tfrac{3}{2}k_B T$
where $m$ is the mass of one molecule and $T$ is the temperature in kelvin. Notice that the average kinetic energy of the molecules depends only on the temperature.
Root-mean-square speed
The root-mean-square speed of the molecules follows from the average kinetic energy:
$v_{\text{rms}}=\sqrt{\overline{v^2}}=\sqrt{\dfrac{3k_B T}{m}}$
where $m$ is the mass of one molecule. At $T=0$, $v_{\text{rms}}=0$: there is no molecular motion at absolute zero.
Example: Nitrogen gas ($\text{N}_2$, molar mass $28\text{ u}$) is at $45\,^{\circ}\text{C}$. What is the rms speed of its molecules at this temperature?
Solution: First convert the temperature to kelvin: $T=45+273=318\text{ K}$. The mass of one nitrogen molecule is
$m=28\times1.66\times10^{-27}=4.65\times10^{-26}\text{ kg}$
Now use the rms-speed formula:
$v_{\text{rms}}=\sqrt{\dfrac{3k_B T}{m}}=\sqrt{\dfrac{3(1.38\times10^{-23})(318)}{4.65\times10^{-26}}}\approx 532\text{ m/s}$
Maxwell distribution of molecular speeds
Not all molecules in a gas move at the same speed. In 1859, James Clerk Maxwell used the kinetic theory to derive the distribution of molecular speeds:
$f(v)=4\pi N\left(\dfrac{m}{2\pi k_B T}\right)^{3/2}v^2\,e^{-mv^2/(2k_B T)}$
Here $f(v)$ gives the relative number of molecules at each speed $v$. As the temperature rises, the curve broadens and its peak shifts to higher speeds, because more molecules move fast.
Kinetic theory of liquids
The kinetic theory was developed mainly for gases, but it also applies to liquids. It explains everyday phenomena such as evaporation and the fact that chemical reactions go faster at higher temperatures.
Activation energy and reaction rate
For two molecules to react, they must collide with at least a minimum kinetic energy called the activation energy, $E_A$. As the temperature rises, more molecules have energy equal to or above $E_A$, so the reaction speeds up.
Evaporation
In a liquid, the faster molecules can overcome the attractive forces of their neighbours and escape from the surface — this is evaporation. Because the fastest molecules leave, the average kinetic energy of those remaining drops, so evaporation cools the liquid. At higher temperatures more molecules have high kinetic energy, so the rate of evaporation increases.
Vapor pressure and boiling
In a closed container, molecules leave the liquid by evaporation while others return to it. When the number leaving equals the number returning, the vapor is saturated. The pressure of this saturated vapor is called the vapor pressure. Boiling occurs when the vapor pressure equals the surrounding (atmospheric) pressure.