5.8 Kinetic Theory

How gas particles interact

Ideal Gas:

  • Large number of molecules
  • Undergoing random rapid motion
  • Particles have negligible volume
  • All collisions are elastic (KE is conserverd)
  • No forces between particles except during collisions

Derivation - Pressure exerted by an ideal gas

Consider a square container of side length $L$ A gas molecule is colliding with the side of the containers and bouncing back and forth. At time $t = 0$ the particles has velocity $v$, and $t = 1t$ is has velocity $-v$. The gas particles collide, the momentum change $p$ is as follows: $mv-(-mv)=2mv$. In two trips from one side of the container to the other, the molecule travels a distance $2L$.

Time between collisions with the edge of the container is $1t$. $t = \frac{d}{s} = \frac{2L}{v}$ Force on the right side of the container is $dp/t = 2mv/t$ $f_\text{shaded} = \cfrac{2mv}{\cfrac{2L}{v}} = \frac{mv^2}{L}$

$\text{Pressure} = \frac{F}{A} = \frac{mv^2}{L}$

The equation

$PV = \frac{1}{3}Nm\bar{c}^2$

For many molecules:

In 3D, on average $\frac{1}{3}$ of the molecules are travelling in the $x$ direction. For many molecules:

$x\frac{1}{3}N$ $P = \frac{1}{3}N\frac{mv^2}{V}$ $4PV = \frac{1}{3}Nmv^2$

Where $P$ is the pressure, $V$ is the volume, $N$ is the number of particles, $M$ is the mass of one particle and $v$ is the velocity.

$P = \frac{1}{3}\rho v^2$

Where $\rho$ is the density and $v$ is the velocity. The average velocity of the molecules is $\bar{c}$

$\mathbb{\text{However}}$, $v$ varies with temperature (molecules have more kinetic energy at higher temperatures)

Using root mean squared to find the average speed:

$\bar{c} = \sqrt{\frac{v_1^2+v_2^2+v_3^2…}{N}}$