Molecular Kinetic Theory Model 2

Physics: Cambridge International A Level, topic 9.2.7

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:
$\langle{c_x}\rangle=\langle{c_y}\rangle=\langle{c_z}\rangle$
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:
$t=\frac{l}{\langle{c_x}\rangle}$
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
$P=m\langle{c_x}\rangle$

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:
$F=\frac{m\langle{c_x}^2\rangle}{l}$

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:
$p=\frac{m\langle{c_x}^2\rangle}{l^3}$
Since volume is length cubed we get:
$p=\frac{m\langle{c_x}^2\rangle}{V}$

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:
${p_t}=N\frac{m\langle{c_x}^2\rangle}{V}$

Kinetic equation

Recalling that the velocity is the same in all directions, we can write an equation for the total velocity squared as:
$\langle{c}^2\rangle= \langle{c_x}^2\rangle+\langle{c_y}^2\rangle+\langle{c_z}^2\rangle$
Because the velocities are equal, we can simplify to give:
$\langle{c_x}^2\rangle=\frac{1}{3}\langle{c_t}^2\rangle$
So the equation for pressure can be written as:
$pV=\frac{1}{3}Nm\langle c^2\rangle$

Kinetic Energy of a Molecule

Using the results we obtained from the molecular kinetic theory derivation, we can find the temperature dependence of the kinetic energy of a molecule.

Ideal gas law

Recall that the ideal gas law in terms of the Boltzmann constant, $k$ is:
$pV=NkT$
And the equation of the molecular kinetic theory is:
$pV = \frac{1}{3} Nm\langle c^2\rangle$
Equating the two gives:
$NkT=\frac{1}{3}Nm\langle c^2\rangle$

Kinetic energy of a molecule

We can cancel a factor of $N$ from each side to give:
$kT=\frac{1}{3}m\langle c^2\rangle$
Finally, multiplying each side by $\frac{3}{2}$ , we obtain an expression for the average kinetic energy for a molecule:
$\frac{1}{2}m\langle c^2\rangle = \frac{3}{2}kT$
The average kinetic energy for a molecule is proportional to the temperature.

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.

Test yourself on Molecular Kinetic Theory Model 2

How can we calculate the force of a particle when it is absorbed by a wall?
How can we calculate the pressure of one particle when it is absorbed by a wall?
If a gas is isotropic and homogeneous how does the velocity in each direction relate?
What is the constant of proportionality for the relationship between the average molecular kinetic energy and the temperature?
What did Bernoulli's model of matter predict?

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