# Fall 2018: Lecture 32 - Kinetic Theory and Real Gases

The Kinetic Theory of Gases represents the first statistical theory in physics, and is thus extremely significant. It also does a pretty good job of describing ideal gases.

## Ideal gas law from a molecular perspective

We can work out the pressure exerted by an ideal gas on it's container by starting from the change in momentum of a molecule when it strikes the container wall

The average force due to one molecule is then

$F=\frac{\Delta(mv)}{\Delta t}=\frac{2mv_{x}}{2l/v_{x}}=\frac{mv_{x}^{2}}{l}$

The net force on the wall will be the sum of the forces from all $N$ molecules

$F=\frac{m}{l}\Sigma_{i=1..N} v_{xi}^{2}$

$\frac{\Sigma_{i=1..N} v_{xi}^{2}}{N}=\bar{v_{x}^{2}}$ → $F=\frac{m}{l}N\bar{v_{x}^{2}}$

$v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}$ → $\bar{v^{2}}=\bar{v_{x}^{2}}+\bar{v_{y}^{2}}+\bar{v_{z}^{2}}=\bar{3v_{x}^{2}}$

$F=\frac{m}{l}N\frac{\bar{v^{2}}}{3}$

$P=\frac{F}{A}=\frac{1}{3}\frac{Nm\bar{v^{2}}}{Al}=\frac{1}{3}\frac{Nm\bar{v^{2}}}{V}$

$PV=\frac{2}{3}N(\frac{1}{2}m\bar{v^{2}})=NkT$

$\bar{K}=\frac{1}{2}m\bar{v^{2}}=\frac{3}{2}kT$

## Maxwell Boltzmann Distribution

If we look at a simulation of particles moving according to the kinetic theory we can get some idea of the distribution of the speeds of the particles.

The Maxwell-Boltzmann Distribution gives the probability $f(v)$ that a particular particle in an ideal gas has a given speed $v$. It was originally derived by James Clerk Maxwell based on symmetry arguments, later Ludwig Boltzmann derived it on a more general basis. We will follow Maxwell's reasoning.

$4\pi A^{3}=\frac{4B^{3/2}}{\sqrt{\pi}}$

$B=\frac{m}{2kT}$

$f(v)dv=4\pi v^2A^{3}e^{-Bv^{2}}dv$

$f(v)=4\pi (\frac{m}{2\pi k T})^{\frac{3}{2}}v^{2}e^{-\frac{1}{2}\frac{mv^{2}}{kT}}$

Now that we have this we can find various useful quantities, such as the average velocity

$\bar{v}=\int_{0}^{\infty}vf(v)dv=2\sqrt{\frac{2}{\pi}}\sqrt{\frac{kT}{m}}\approx 1.6 \sqrt{\frac{kT}{m}}$

or the most probable speed of a particle, from the condition $\frac{df}{dv}=0$ which gives

$v=\sqrt{\frac{2kT}{m}}\approx 1.41 \sqrt{\frac{kT}{m}}$

A third measure of velocity that we could already obtain before deriving the Maxwell Boltzmann distribution, is the root-mean-square, or rms velocity. Here we take the square root of the average of the squared velocity

$\frac{1}{2}m\bar{v^{2}}=\frac{3}{2}kT$ → $\sqrt{\bar{v^{2}}}=\sqrt{\frac{3kT}{m}}\approx 1.73 \sqrt{\frac{kT}{m}}$

## Temperature dependence

The distribution of speeds is highly temperature dependent, as can be seen in simulations and is predicted by the Maxwell distribution.

The plot below is for He atoms at various temperatures.

## Mass dependence

The speed distribution is also a function of the mass of the molecules, the plot below is for the first 4 noble gases at room temperature.

## Phase Diagram of Water

Real gases deviate from ideal gas behavior because of interatomic forces, especially when the average distance between molecules becomes small, at lower temperatures and higher pressure.

The phase diagram of water is fairly complicated, particularly at high pressures, but we are not going to pay attention to all the different kinds of ice that can exist!

## A general phase diagram for a liquid-gas-solid system

The lines represent conditions where 2 phases co-exist, referred to as a phase boundary. Where two lines intersect we have a triple-point. A critical point is a point where a phase boundary disappears. At temperatures above the critical point the liquid and vapor phases are indistinguishable, we call this a supercritical fluid, or more simply a gas. The dotted line shows the behavior of water, which is anomalous (ie. it expands upon freezing).

A phase region on a phase diagram represents the equilibrium state of the material under the stated conditions. Materials can, however, exist in non-equilibrium states.

## Evaporation

Within a liquid there will be a distribution of velocities for the molecules and the most energetic ones can leave the liquid and join the vapor phase, which is evaporation. The reverse process can also happen where lower energy molecules in the vapor phase join the liquid through condensation. In a closed system at equilibrium these processes occur at the same rate. The amount of vapor in the air under this condition is the saturated vapor pressure, which will depend on the temperature and the overall pressure.

## Boiling

Boiling occurs when the saturated vapor pressure equals the external pressure. Normally we boil water by increasing the temperature…but it can also be done by decreasing the pressure. See here for the vapor pressure of water.

## Partial Pressure and Humidity

Air can contain a varying amount of water, with the total pressure of the air being due to many different kinds of molecule. The partial pressure of a particular gas is the pressure due to a particular kind of gas molecule.

We define humidity in terms of the partial pressure of H2O compared to the saturated vapor pressure of H2O.

$\mathrm{Humidity=\frac{\textrm{partial pressure of } \mathrm{H_{2}O}}{\textrm{saturated vapor pressure of }\mathrm{H_{2}O}}\times 100\%}$. 