Unizor - Physics4Teens - Energy - Heat - Kinetics of Ideal Gas

Notes to a video lecture on http://www.unizor.com

Kinetics of Ideal Gas

In this lecture we will discuss a concept of pressure of gas, enclosed in some reservoir, against the walls of this reservoir.

Usually, speaking about pressure, we have in mind an object of certain weight on a horizontal table, in which case the constant force of its weight exerts a pressure on a table equal to this weight divided by an area of the table occupied by an object or, as it's sometimes presented, the weight per unit of area.

As an introduction to kinetics of gases, let's consider a reservoir shaped as a cube with sides aligned parallel to coordinate planes and with the length of each edge L.
Assume that there is a single molecule of gas of mass mflying between the opposite walls perpendicularly to them, elastically reflecting off these walls.
Keep in mind that, according to the Third Newton's Law, the force exerted by the wall towards the molecule, which causes the reflection of a molecule from the wall, is equal to the force exerted by a molecule towards the wall, which causes pressure.

The area of each of these opposite walls equals to L², which will be used to calculate the pressure.

In this case of a single gas molecule, flying between the opposite walls of a reservoir, the force is variable. It exists during the time when a molecule hits the wall and then the force becomes zero until the next time this molecule hits the same wall.
A proper definition of pressurein this case is based on a concept of the average force during certain amount of time.

Firstly, let's evaluate the average force during a single period of oscillation of a molecule between walls from a moment of time it's near one wall to a next moment it's at this position after flying to the opposite wall and returning back.

As we know, a change of a momentum of an object equals to an impulse of the force that caused this change
m·v_end − m·v_beg = F·τ
where
m is a mass of an object
v_beg is the velocity of an object in the beginning of the action of the force
v_end is the velocity of an object at the end of the action of the force
F is the force, acting on an object
τ is the time duration of the force, acting on an object.
Keep in mind that velocity and force are vectors in the above equation.

In our case we consider one period of oscillation of a molecule as the time period of average force acted on it.
So, τ is the time between two consecutive events when a molecule is at the wall opposite to the one it collides with:
τ = 2L/|v|.
where |v| is now an absolute speed of a molecule (scalar).

Since we are considering elastic reflection of a molecule in the opposite direction to its initial trajectory,
v_end = −v_beg
and the equation above can be written in scalar form
2m·|v| = |F|·τ = |F|·2L/|v|
from which we determine the absolute average force
|F| = m·|v|²/L
Mass m of a molecule is, obviously, a constant during this process.
The average pressure of a molecule on a wall is the average force divided by the area of a wall:
p = m·|v|²/L³ = m·|v|²/V =
= 2E_kin /V
where V is the volume of a reservoir and E_kin is a kinetic energy of a molecule.

Our first step towards a comprehensive theory is to consider a case of many molecules flying parallel to each other in the same reservoir as above, but with, generally speaking, different speeds. Obviously, the pressure against the wall will be greater because each molecule contributes its own force against the wall during a collision.
If v_i is a speed of the i^thmolecule, the combined pressure will be
p = Σp_i = Σm·v_i²/V =
= 2E_tot /V
where E_tot is a total kinetic energy of all molecules moving parallel to each other.

Considering the number of molecules N remains the same, it's more convenient to express this formula in terms of an average of squares of molecular speed and average kinetic energy of molecules:
v²_ave = (1/N)Σv_i²
E_ave = (1/N)ΣE_i
Using this average of squares of molecular speed, the pressure is
p = N·m·v²_ave /V = 2N·E_ave /V
which brings us to a principle of proportionality between pressure of the gas against the walls of a reservoir and an average of squares of the molecular speed or to an average of kinetic energy of the molecules.

Granted, we showed this only for a case of all molecules moving between two opposite walls parallel each other and perpendicularly to the walls they collide with.

Let's make another step towards comprehensive theory and consider the chaotic movement of all gas molecules.
To quantitatively approach molecular movement, we will use a model of an ideal gas.
This model assumes that molecules of ideal gas are point-mass objects completely chaotically moving in all directions with equal probabilities and elastically colliding at random times among themselves or with all the walls of a reservoir that contains this gas.

We further assume that no interacting forces (like gravity or electromagnetic forces) exist between them. So, only kinetic energy of these molecules plays the role in evaluation of the characteristics of the ideal gas.

Every vector of velocity of any molecule can be represented as a sum of three vectors along the XYZ-coordinates
v = v_x+v_y+v_z

So, every movement of a molecule can be represented as simultaneous movement in three different directions. Depending on which wall of the reservoir is hit by a molecule, its pressureagainst the wall is determined only by one component - the one that is perpendicular to the surface of a wall.

If molecules inside the reservoir are moving completely chaotically, they are hitting all walls with approximately the same frequency. So, for each of the six walls of a cube we can use the same logic as above for one side, except for the walls parallel to YZ coordinate plane the pressures p_x will be proportional to the average of v_x², for the walls parallel to XZ coordinate plane the pressures p_y will be proportional to the average of v_y² and for the walls parallel to XY coordinate plane the pressures p_z will be proportional to the average of v_z².

Considering completely chaotic character of the molecular motion in the ideal gas, the three pressures p_x, p_y and p_zmust be the same. Similarly, averages of v_x², v_y² and v_z² must be also the same.

Since v = v_x+v_y+v_z, according to three-dimensional equivalent of Pythagorean Theorem,
v² = v_x² + v_z² + v_z²
and, similarly, for averages
AVE(v²) = v²_ave =
=AVE(v_x²)+AVE(v_y²)+AVE(v_z²)

From the equality of averages for the ideal gas we come to the following equalities:
v²_ave = 3·AVE(v_x²)
AVE(v_x²) = (1/3)·v²_ave

Therefore,
p_x = N·m·AVE(v_x²)/V =
= (1/3)N·m·v²_ave /V =
= (2/3)N·E_ave /V

Similarly,
p_y = N·m·AVE(v_y²)/V =
= (1/3)N·m·v²_ave /V =
= (2/3)N·E_ave /V
and
p_z = N·m·AVE(v_z²)/V =
= (1/3)N·m·v²_ave /V =
= (2/3)N·E_ave /V

And, since pressures on all walls are the same in the ideal gas,
p = (1/3)N·m·v²_ave /V =
= (2/3)N·E_ave /V = (2/3)E_tot /V

The expression of a pressure in terms of average kinetic energy and volume is more general than in terms of mass, average of squares of molecular speed and volume, as it encompasses a case with molecules of different masses.