Kinetic theory model This model demos how the pressure and volume of a gas are directly linked to the velocity and therefore temperature of a gas. They exert forces on the walls which can be observed with the “lid”. It can also be used to show the effect of Brownian Motion by placing a polystyrene ball in the chamber. After using the model try the simulation to see the effects of changing pressure, temperature and number of particles. Make sure you can explain the effects seen! You could also use this as revision to plot results for Boyles law and Charles Law.

Observations Ball bearings exert a force on the cardboard disc – we need to load it to keep it down, if the speed is increased we need to add more masses to keep the volume constant – pressure law If the lid is allowed to rise, increasing the speed (temperature) increases the volume as pressure is constant – the molecules exert the same pressure by hitting the walls harder but less often Boyles Law If order to decrease the volume masses are used on the lid increasing pressure, but the speed is constant (temperature). Molecules hit walls more frequently as they have less distance to travel – increases pressure – Boyles Law

Observations inelastic collisionsgases elastic Ball bearings stop moving when the power supply switched off – inelastic collisions. In gases the molecules don’t stop moving if the temperature is constant, meaning they don’t lose kinetic energy. This proves the idea that their collisions are elastic more densely packedat bottom gravity keeps them there Bearings are more densely packed at bottom of simulation as gravity keeps them there. Higher pressure at lower height – also the case in the Earth’s atmosphere as the air molecules spread out as you gain altitude hence why it is harder to breath at the top of Everest and why the water for your tea boils more easily (lower temperature) at altitude making rubbish tea!

2.13 The Kinetic Theory of Ideal Gases Several assumptions were made by scientists about gases – producing a theoretical model A pure gas consists of a large number of identical, small, non interacting particles in continuous random motion A pure gas consists of a large number of identical, small, non interacting particles in continuous random motion Molecules never stop moving and drop to the bottom of the container – generally elastic collisions Molecules never stop moving and drop to the bottom of the container – generally elastic collisions Gases compress greatly so the molecules must occupy a very small fraction of the volume of the container (negligible). Gases compress greatly so the molecules must occupy a very small fraction of the volume of the container (negligible). Collision time is negligible compared to the time between collisions Collision time is negligible compared to the time between collisions Molecules must be relatively far apart, otherwise forces between them would need to be accounted for, during collisions is the only time we consider them to be important! Molecules must be relatively far apart, otherwise forces between them would need to be accounted for, during collisions is the only time we consider them to be important!

Distribution of Molecular speeds Pressure is caused by collisions with the surroundings Temperature is a measure of the speed of the particles average The average velocity of the particles is zero at any temperature as they are all moving in random directions and therefore cancel each other out. There is a whole range of speeds but there will be a cluster around the average value as shown on the graph. The air molecules surrounding us have an r.m.s. speed of 1200 mph (550 ms -1 ) meaning that some might be nearly stationary and others potentially 3* that speed.

Number of molecules Speed (c) c c the mean speed (average) of all the molecules c p c p the most probable speed c rms c rms the rms speed is the root mean square speed which removes any negative directions. You need to be able to use this!! Lower Temperature Higher Temperature

Question (We do not have enough particles here to be realistic but it will illustrate the point!) 6 particles have the following speeds: 600, 650, 650, 700, 725, 750ms -1. Determine the most probable speed c p, the mean speed and the root mean square speed c rms. Most probable = 650ms -1 Mean = 679ms -1 c rms. = 681ms -1

If these assumptions are correct, we should be able to prove the equation of state for an ideal gas! l l l m,c X Y Z m c Consider a particle of mass m and speed c in the direction of the x axis c

Consider the change in momentum as the particle hits the wall  2mc  = mc - -mc = 2mc t =2l/c Time between hitting the wall t = 2l/c Newton’s second law states that the Force needed to cause this change in momentum F = d  /dt = 2mc/(2l/c) = mc 2 /l Considering that Pressure = Force/Area P = (mc 2 /l )/l 2 = mc 2 /l 3 If there are N molecules in the real box, only N/3 are moving in this direction, all are moving randomly meaning we use the root mean square speed to express the “average” speed of a molecule in the box. l l l m,c X Y Z

So the total pressure is given by P = (N/3)m /l 3 is the mean square speeds of all the molecules in the box and l 3 = V P = (N/3)m /V Otherwise written as l l l m,c X Y Z Since Nm = Total mass in the box Nm/V is the density of the gas and therefore P = 1 / 3  P = 1 / 3  PV = 1 / 3 Nm PV = 1 / 3 Nm

If we consider 1 mole of a gas then P = ( 1 / 3 Nm )/V Becomes PV = ( 1 / 3 N A m ) We can take a Kinetic Energy type factor out of this equation … PV = 2 / 3 N A ( 1 / 2 m ) ( 1 / 2 m ) is the mean Kinetic Energy of one molecule Since the equation relates to a whole mole of gas it must agree with the Ideal Gas Equation PV = nRT (n=1) PV = 2 / 3 N A ( 1 / 2 m ) = RT ( 1 / 2 m ) = 3 / 2 RT/N A However we can simplify this further if we take out the 2 constants (R/N A ) which are combined into the ….. Avogadro’s number number of particles in one mole of a substance N A is Avogadro’s number, the number of particles in one mole of a substance.

Boltzman constant k = R/N A temperature in Kelvin is proportional to the average molecular Kinetic Energy. So we end up with an equation showing the temperature in Kelvin is proportional to the average molecular Kinetic Energy. 1 / 2 m = 3 / 2 kT