Ideal GasesIdeal Gases

One mole is defined as the amount of substance that contains as many elementary entities as there are atoms in 12g of carbon-12. This number is called the Avogadro constant, N_A, and has been measured as 6.02 × 10²³.

$N =ntimes N_{A}$

The kinetic theory of gases is a model used to describe the behaviour of atoms or molecules in an ideal gas. It involves a number of assumptions.

Large number of molecules in random, rapid motion
Particles (atoms or molecules) occupy a negligible volume compared to the volume of gas
All collisions are perfectly elastic
The time of the collisions is negligible compared to the time between collisions
Negligible forces exist between particles except during a collision (no intermolecular forces – have KE but no PE)

6. Particles travel in straight lines/at uniform velocity between collisions and force of gravity on molecules is negligible Using these assumptions and Newton’s laws of motion, it is possible to explain the origin of pressure.

1.Many molecules collide with the walls

2. A change in momentum occurs when molecule(s) collide with (and rebound from) the walls of container

3. Force is rate of change of momentum

4. The force exerted by the molecule(s) on wall is equal to force exerted by the wall on the molecule(s) (by Newton’s third law)

5. $P = frac{F}{A}$ .

Pressure’s equal on all faces of the container because there’s a very large number of molecules moving randomly.

Various laws can be used to describe a gas:

is constant $therefore p_{1}v_{1}=p_{2}v_{2}$ (Boyle’s Law) – for a fixed mass of gas at a constant temperature
$frac{P}{T}$ is constant $therefore frac{P_{1}}{T_{1}}=frac{P_{2}}{T_{2}}$ (Amonton’s Law) – for a fixed mass of gas occupying a constant volume

The gas laws can be combined to yield the equation of state of an ideal gas pV = nRT

The graph of against produces a straight line, gradient . This allows you to estimate absolute 0 by extrapolating and finding the intercept.

Because a gas contains a vast number of particles moving in random directions, the average velocity of all the particles in a gas would average to 0ms^-1. So to describe the typical motion of particles inside a gas, we instead use the root mean square speed (r.m.s speed). To find this, square all the speeds of the gas particles and average them. Then take the square root of the result. This yields $sqrt{C^{2}}$ or .

Mean square speed appears in the equation for pressure and volume of a gas: $Pv = frac{1}{3}Nmc^{2}$

The range of speed of particles in a gas at a given temperature is known as the Maxwell-Boltzmann distribution.

Meanwhile, the Boltzmann constant, k, is given by: $k=frac{R}{NA}$

It follows that $pV = nkN_{a}T and so pV = NKT$ . Therefore:

$frac{1}{3}Nmc^{2} =NkT$

$frac{1}{3}times frac{1}{2}mc^{2}=frac{1}{2}=kT$

$frac{1}{2}mc^{2}=frac{3}{2}kT$

Since $frac{1}{2}mc^{2}$ gives the mean kinetic energy of the particles in a gas, it follows that $E_{k}propto T$ .

The internal energy of a gas is the sum of the kinetic and potential energies of the particles inside it. But one of the assumptions of an ideal gas is that electrostatic forces between particles are negligible except during collisions. It follows that there is no EPE in a gas, so all the internal energy is in the form of the kinetic energy of its particles.