Kinetic Theory of Pressure and Temperature

Matter is described as a collection of small particles in constant, random motion.
Higher temperature correlates to molecules moving faster.
Higher pressure correlates to more frequent molecule collisions with their container.
Molecular states are defined by particle behavior:
- Solids: Particles are held together but vibrate.
- Liquids: Particles move across each other within a fixed volume.
- Gases: Particles move across all available volume.

Temperature ( $T$ ) measures the average kinetic energy ( $K_{avg}$ ) of individual molecules.
The relationship is defined by the equation: $K_{avg} = \frac{3}{2} k_B T$ .
Boltzmann's constant ( $k_B$ ) is $1.38 \times 10^{-23}$ .
Molecules in a gas travel at different speeds; $K_{avg}$ is based on the average of all particles.
At Absolute Zero ( $0\,K$ ), kinetic energy is zero and molecules stop moving completely.
The root-mean-square speed ( $v_{RMS}$ ) represents the average speed of molecules: $v_{RMS} = \sqrt{\frac{3k_B T}{m}}$ .
If temperature increases from $T$ to $3T$ , the average kinetic energy triples ( $3K$ ) and the root-mean-square speed increases by a factor of $\sqrt{3}$ .
In thermal equilibrium at $700\,K$ , if molecule A has one-fourth the mass of molecule B and an rms speed of $150\,m/s$ , the rms speed of molecule B is $300\,m/s$ .

This distribution describes the range of molecular speeds in an ideal gas.
Gases at higher temperatures exhibit a distribution that shifts to the right and becomes broader, indicating a larger range of speeds.
In comparing two samples (A and B) where Sample B atoms move faster on average, it cannot be concluded that a specific random atom from B is faster than one from A, only that it is more likely.

Pressure ( $P$ ) is the force exerted on a surface per unit area: $P = \frac{F}{A}$ .
The unit of pressure is the Pascal ( $Pa$ ), where $1\,Pa = 1\,N/m^2$ .
Force results from the rate of change in momentum ( $p$ ) as molecules collide with container walls.
In an elastic collision, a single molecule of mass $m$ hitting a surface at speed $v$ and rebounding at speed $v$ has a momentum change of $\Delta p = mv - m(-v) = 2mv$ .
If $N$ molecules collide with a surface in time $t$ , the total force is $F = \frac{2Nmv}{t}$ .
Pressure derived from molecular collisions is expressed as: $P = \frac{2Nmv}{At}$ .

Atmospheric pressure ( $P_{ATM}$ ) is caused by air molecules colliding with the exterior of a container.
Standard atmospheric pressure is approxmiately $1\,atm = 1.0 \times 10^5\,Pa$ .
In a cylindrical system with a piston:
- Force of the gas ( $F_{GAS}$ ) acts outward.
- Force of the atmosphere ( $F_{ATM}$ ) and the weight of the piston ( $Mg$ ) act inward.
For a piston with area $A = 0.05\,m^2$ , mass $M = 4.0\,kg$ , and $2.0 \times 10^{26}$ molecules (mass $3.0 \times 10^{-26}\,kg$ ) colliding at $500\,m/s$ each second:
- $F_{GAS} = 6000\,N$ .
- $F_{ATM} = 5000\,N$ .
- Net acceleration ( $a$ ) is determined by Newton's 2nd Law: $a = \frac{F_{GAS} - F_{ATM} - Mg}{M} = 24\,m/s^2$ upwards (using $g = 10\,m/s^2$ ).