6.2.7

# Molecular Kinetic Theory Model 2

Test yourself

## Average molecular kinetic energy

The average molecular energy can be used to find the pressure of radiation.

### Kinetic energy

• We assume that a gas is homogeneous (the same everywhere) and isotropic (has the same value when measured in different directions).
• This means that the average speed of a molecule in the x-direction must be the same as the y and the z-directions. So we can write:
• ${v_x}={v_y}={v_Z}$
• This means that the average kinetic energy in each direction is the same.

### Momentum

• We consider a cube of length l which contains a gas.
• If we only look at the x-direction, we can say that the time taken for a particle to travel from one end to the other is:
• $t=\frac{l}{{v_x}}$
• If a particle collides with the wall of the box and is absorbed, we can write the change in momentum as:
• Change in momentum = mass x velocity
• $P=m{v_x}$

### Force

• We can write the force exerted on the wall when it absorbs a particle as:
• Force = change in momentum ÷ change in time
• $F=\frac{P}{t}$
• We can substitute in the equations for momentum and time to get:
• $F=\frac{m{v_x}^2}{l}$

### Pressure

• The pressure on the wall of the box when it absorbs one particle is given by the equation:
• Pressure = force ÷ area
• $p=\frac{F}{A}$
• We can use the value for force and that the area is length squared to get:
• $p=\frac{m{v_x}^2}{l^3}$
• Since volume is length cubed we get:
• $p=\frac{m{v_x}^2}{V}$

### Pressure 2

• We have found the pressure for one particle if N particles hit the wall the pressure would be:
• Total pressure = number of particles x pressure from one particle.
• ${p_t}=Np$
• We can substitute the pressure in to give:
• ${p_t}=N\frac{m{v_x}^2}{V}$

### Kinetic equation

• Recalling that the velocity is the same in all directions, we can write an equation for the total velocity squared as:
• ${v_t}^2= {v_x}^2+{v_y}^2+{v_z}^2$
• Because the velocities are equal, we can simplify to give:
• ${v_x}^2=\frac{1}{3}{v_t}^2$
• So the equation for pressure can be written as:
• $pV=\frac{1}{3}Nmv^2$

## Molecular Kinetic Theory - Developments Over Time

Ideas about the underlying structure of materials have changed considerably over time.

### Early ideas

• The idea that the atom, or atoms that are too small to directly view, move around is ancient. This dates from Lucretius in approx 50 BCE.
• This idea was largely ignored because of the predominance of Aristotelian ideas about elements such as fire, earth, air and water.

### Particle nature of matter

• The modern theory of the particle nature of matter was attributed to Bernoulli.
• This was prior to the ideas of conservation of energy and how collisions between particles could be elastic.
• The model did not predict anything in itself.

### More complex ideas

• Clausius developed a much more complex set of ideas.
• The observation of Brownian motion is the first direct evidence of the existence of particles.
• The kinetic theory of gases became universally accepted because of Einstein and Smoluchowski’s theoretical model, which made specific predictions about Brownian motion and diffusion.