Kinetic theory of gases

The kinetic theory states that every substance consists of atoms or molecules and that the atom or molecule moves continuously carelessly. This assumption of kinetic theory matches the situation and condition of the atom or molecule of the gas constituent. The force of attraction between the atoms or molecules making up the gas is feeble so that atoms or molecules can move freely.

When moving, atoms or molecules have speed. Atoms or molecules also have mass. Because it has mass (m) and velocity (v), then the atom or molecule has kinetic energy (KE) and momentum (p). Kinetic energy: KE = 1/2 m v2. While momentum: p = m v. In addition to kinetic energy and momentum, there is also force (F). When moving freely, collisions must occur. So force raises because of changes in momentum when a collision occurs. Kinetic energy, momentum and impulse forces are the core of our discussion of the dynamics (the laws of motion, impulse, and momentum). We can say that the kinetic theory of gases applies dynamics at the atoms or molecules level.

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Ideal Gas Concept (based on macroscopic properties of gas)

In the discussion of the laws of gas, it has been explained about three quantities which state the macroscopic nature of real gas. The three quantities in question are Temperature (T), Volume (V) and Pressure (P). The relationship between these three macroscopic quantities is stated in Boyle’s Law, Charles’s law, and Gay Lussac’s law. Please note that these three laws only apply to real gas that has a pressure and density (density = mass/volume) that is not too large. These three laws also only apply to a real gas whose temperature is not near the boiling point.

Boyle’s law, Charles’s law, and Gay-Lussac’s law do not apply to all real gas conditions so that we can make an ideal gas model. Ideal gas does not exist in everyday life; the ideal gas is just the perfect form that is made to help our analysis, much like ideal rigid and fluid bodies. So we consider Boyle’s law, Charles law and Gay-Lussac law apply to all ideal gas conditions. The existence of an ideal gas model helps us review the relationship between macroscopic quantities of gas.

The ideal gas law is expressed in two equations, PV = nRT (ideal gas law in the number of moles) and PV = NkT (ideal gas law in the number of molecules). We assume that the ideal gas meets these two equations. In other words, the ideal gas law applies to all ideal gas conditions, whether the ideal gas pressure or mass is very large or when the ideal gas temperature approaches the boiling point. In contrast, the ideal gas law does not apply to all real gas conditions. The ideal gas law only applies when the pressure and density of the real gas are not too large. The ideal gas law also only applies when the temperature of the real gas is not near the boiling point. Based on this brief description, we can say that real gas has similar properties with ideal gas only when the real density and gas pressure are not too large and when the real gas temperature is not near the boiling point.

The concept of the ideal gas that has been described above is reviewed based on macroscopic properties. Although an ideal gas is only an ideal model, an ideal gas is still considered a gas that consists of atoms or molecules that move freely. Therefore, it would be nice if we also discuss the ideal gas concept from a microscopic perspective.

Ideal Gas Concept (based on microscopic properties of gas)

The following are some brief descriptions that describe the microscopic conditions of the ideal gas, which are based on the Gas Kinetic Theory:

1. An ideal gas consists of particles, called molecules. The ideal gas molecules can include of one atom or several atoms. Each molecule has mass (m) and moves randomly in any direction at a specific rate (v).

2. The distance between each molecule is higher than the diameter of each molecule.

3. These molecules obey the laws of motion and interact with each other when collisions occur.

4. Collisions between molecules and molecules or between molecules with a container wall are the perfect collision, and each collision happens in a concise time.

In the perfect collision, the law of conservation of energy applies (energy before collision = energy after collision) and the law of conservation of momentum (momentum before collision = momentum after collision).

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The review of impulse-collision for the kinetic theory of gases

Review the quantitative relationship between the macroscopic quantities and the microscopic quantities of the gas. The quantities that state the macroscopic nature of gas are temperature (T), volume (V) and pressure (P). While the quantities that say the microscopic nature of gas are velocity or velocity (v), momentum (p), force (F) and kinetic energy (EK) of atoms or molecules making up the gas.

Kinetic theory of gases 1We review some gas molecules in a closed container. The side length of the box = l and the cross-sectional area = A.

Molecules have mass (m) and when moving the molecule has velocity (v). Because the container is closed, there is a possibility of collisions between molecules and the wall of the vessel that has a surface area of A.

To simplify the analysis, we only the collision that occurs on the left wall (the wall that is parallel to the z-axis). First, we discuss the collisions experienced by one molecule. Call it molecules 1. The molecular mass = m1 and the speed of movement = v1. The direction of movement to the left is set to be negative, while the direction of movement to the right is set to be positive.

We can assume that before colliding the container wall, the molecular motion is parallel to the x-axis and the direction of the movement to the left. Therefore there is a component of speed on the x-axis which is negative (-v1x). Because it has mass (m1) and speed (-v1x), the molecule has momentum (p1 = -m1 v1x). This is the initial momentum. When colliding with a wall, the molecule provides an action force on the wall. Because there is an action force, the wall provides a reaction force. The reaction force from the wall causes the molecule to be reflected on the right. Because of the direction of movement to the right, the molecular velocity component is positive (v1x). The molecular momentum after the collision is p2 = m1 v1x. This is the final momentum.

The magnitude of the change in momentum due to collisions is:

Total momentum = final momentum – initial momentum

p total = p2 – p1

p total = m1 v1x – (-m1 v1x)

p total = 2m1 v1x

2m1 v1x = total momentum for one collision. Because molecular collisions are the perfect collision, the collisions occur not only once but repeatedly. In the perfect collision, the law of conservation of energy and the law of conservation of momentum applies. Energy and momentum before collision = energy and momentum after the collision. Therefore the molecule will never stop moving (endless energy). Molecular speed also never decreases (continued momentum).

After colliding with the left wall, the molecule moves to the right until it hits the right wall. After colliding the right wall, the molecule moves back to the left and hit the left wall again. Because the length of the side of the box = l, then after colliding the left wall for the first time, the molecule will travel a distance of 2l before colliding the left wall a second time. When moving as far as 2l, molecules must require a certain time interval (Δt). The amount of time interval (Δt) that the molecule needs to move as far as 2l, is mathematical:

Kinetic theory of gases 2

Δt is the time interval between each collision. When colliding a wall, the molecule provides an action force on the wall. Because of the action force, the wall provides a reaction force. The existence of this reaction force makes the molecule move again to the right. In this case, the direction of molecular movement changes. At first, the molecule moves to the left (-v1x), after colliding the wall, the molecule moves to the right (v1x). Changes in direction of movement cause changes in momentum (final momentum – initial momentum = m1 v1x – (-m1 v1x) = 2m1 v1x). We can say that the change in momentum occurs because of the total force is given by the wall. The amount of total force given by the wall, mathematically:

Kinetic theory of gases 3

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In the box above, only one molecule is described. This does not mean that only one gas molecule in the box. In reality, there are a lot of gas molecules. The amount of total force for all gas molecules in the box, mathematically:

F = F1 + F2 + F3 + ….. + Fn

F1 = total force for molecule 1

F2 = total force for molecule 2

F3 = total force for molecule 3

…… = and so on

Fn = total force for molecules 4

n = the last molecule.

Kinetic theory of gases 4

m1 = molecular mass 1, m2 = molecular mass 2, m3 = molecular mass 3, mn = mass of the last molecule. m1 + m2 + m3 + ….. + mn = m (the mass of gas in the box). l = length of the side of the box.

Kinetic theory of gases 5

v12x = molecular speed 1, v22 x = molecular speed 2, v33 x = speed of molecule 3, vn2 x = speed of the last molecule. The speed of each molecule varies, so we need to calculate the average velocity of all molecules. To calculate the average velocity of a molecule, we can divide the speed of all molecules by the number of molecules. In the kinetic theory of gas, the number of molecules is usually given the symbol N. Mathematically, the average speed of all molecules is written:

Kinetic theory of gases 6

We combine the equation b with the equation a :

Kinetic theory of gases 7

F = force, m = mass of gases, l = length of the side of the box, N = number of molecules.

In the previous explanation, assumed the movement of molecules parallel to the x-axis. This presupposition is made to easy our analysis. In reality, all gas molecules in a box do not move in any direction at random. Because the motion occurs randomly, besides having an average velocity component on the x-axis, the molecule also has an average velocity component on the y-axis or z-axis. Thus, the average velocity of a gas molecule = the total number of components of the average velocity on the x-axis, y-axis, and z-axis. Mathematically:

Kinetic theory of gases 8Kinetic theory of gases 8

Because molecules move randomly, the velocity component on the x-axis, y-axis, and z-axis has the same magnitude. Mathematically:

Kinetic theory of gases 9

We combine the equation 2 with the equation 1 :

Kinetic theory of gases 10

We combine the equation 3 with the equation c :

Kinetic theory of gases 11

F = the amount of force exerted by gas molecules on the wall of the container which has a surface area of A.

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The relationship between pressure (P) and microscopic magnitude

Pressure (P) is a quantity that states the macroscopic nature of gas. Review Pressure based on the microscopic properties of the gas. The amount of pressure given by the gas molecule on the wall that has cross-sectional area A is:

Kinetic theory of gases 12

P = pressure, N = number of gas molecules, m = mass, v = Average speed of a molecule, V = volume of container