Kinetic Molecular Theory Ideal Gas

Gases

The Kinetic Molecular Theory

The Kinetic Molecular Theory Postulates

The experimental observations nearly the beliefs of gases discussed so far can exist explained with a elementary theoretical model known as the kinetic molecular theory. This theory is based on the following postulates, or assumptions.

Gases are composed of a large number of particles that behave similar hard, spherical objects in a state of constant, random movement.
These particles motion in a directly line until they collide with some other particle or the walls of the container.
These particles are much smaller than the distance between particles. Most of the volume of a gas is therefore empty space.
There is no force of allure between gas particles or between the particles and the walls of the container.
Collisions between gas particles or collisions with the walls of the container are perfectly elastic. None of the energy of a gas particle is lost when it collides with another particle or with the walls of the container.
The average kinetic energy of a collection of gas particles depends on the temperature of the gas and nothing else.

The assumptions behind the kinetic molecular theory can be illustrated with the appliance shown in the figure below, which consists of a drinking glass plate surrounded past walls mounted on superlative of three vibrating motors. A handful of steel ball bearings are placed on superlative of the glass plate to represent the gas particles.

When the motors are turned on, the glass plate vibrates, which makes the ball bearings move in a constant, random fashion (postulate one). Each ball moves in a directly line until it collides with some other ball or with the walls of the container (postulate 2). Although collisions are frequent, the boilerplate altitude between the ball bearings is much larger than the diameter of the balls (postulate three). At that place is no force of attraction between the private ball bearings or betwixt the ball bearings and the walls of the container (postulate 4).

The collisions that occur in this appliance are very different from those that occur when a rubber ball is dropped on the floor. Collisions between the prophylactic brawl and the flooring are inelastic, as shown in the effigy below. A portion of the energy of the brawl is lost each time information technology hits the floor, until information technology eventually rolls to a finish. In this apparatus, the collisions are perfectly rubberband. The assurance take just as much energy later a collision as before (postulate 5).

Any object in motion has a kinetic energy that is defined every bit one-half of the product of its mass times its velocity squared.

KE = ¹/₂ mv ²

At any fourth dimension, some of the ball bearings on this apparatus are moving faster than others, merely the system tin be described by an average kinetic energy. When we increment the "temperature" of the system past increasing the voltage to the motors, we discover that the average kinetic energy of the brawl bearings increases (postulate six).

How the Kinetic Molecular Theory Explains the Gas Laws

The kinetic molecular theory tin exist used to explain each of the experimentally determined gas laws.

The Link Between P and n

The pressure of a gas results from collisions betwixt the gas particles and the walls of the container. Each fourth dimension a gas particle hits the wall, information technology exerts a force on the wall. An increase in the number of gas particles in the container increases the frequency of collisions with the walls and therefore the pressure level of the gas.

Amontons' Police (PT)

The last postulate of the kinetic molecular theory states that the average kinetic energy of a gas particle depends only on the temperature of the gas. Thus, the average kinetic energy of the gas particles increases as the gas becomes warmer. Considering the mass of these particles is constant, their kinetic free energy can only increase if the average velocity of the particles increases. The faster these particles are moving when they hit the wall, the greater the forcefulness they exert on the wall. Since the forcefulness per collision becomes larger as the temperature increases, the pressure of the gas must increment as well.

Boyle'south Law (P = 1/v)

Gases can exist compressed because most of the book of a gas is empty infinite. If we shrink a gas without irresolute its temperature, the average kinetic free energy of the gas particles stays the same. There is no change in the speed with which the particles move, only the container is smaller. Thus, the particles travel from one end of the container to the other in a shorter menstruum of time. This means that they hit the walls more often. Any increase in the frequency of collisions with the walls must lead to an increment in the force per unit area of the gas. Thus, the pressure level of a gas becomes larger as the volume of the gas becomes smaller.

Charles' Police (Five T)

The average kinetic energy of the particles in a gas is proportional to the temperature of the gas. Because the mass of these particles is constant, the particles must move faster every bit the gas becomes warmer. If they motion faster, the particles will exert a greater force on the container each time they hit the walls, which leads to an increase in the pressure level of the gas. If the walls of the container are flexible, information technology volition expand until the pressure of the gas once more balances the pressure of the atmosphere. The volume of the gas therefore becomes larger equally the temperature of the gas increases.

Avogadro'due south Hypothesis (V North)

Every bit the number of gas particles increases, the frequency of collisions with the walls of the container must increase. This, in plow, leads to an increase in the pressure of the gas. Flexible containers, such as a airship, will expand until the pressure of the gas inside the airship one time again balances the force per unit area of the gas outside. Thus, the volume of the gas is proportional to the number of gas particles.

Dalton's Law of Partial Pressures (P _t = P _i + P ₂ + P _iii + ...)

Imagine what would happen if half-dozen ball bearings of a different size were added to the molecular dynamics simulator. The total pressure would increase because in that location would exist more than collisions with the walls of the container. But the pressure due to the collisions between the original ball bearings and the walls of the container would remain the same. There is so much empty space in the container that each type of ball bearing hits the walls of the container as often in the mixture equally information technology did when in that location was merely ane kind of ball begetting on the glass plate. The total number of collisions with the wall in this mixture is therefore equal to the sum of the collisions that would occur when each size of brawl bearing is nowadays past itself. In other words, the total force per unit area of a mixture of gases is equal to the sum of the fractional pressures of the private gases.

Graham'south Laws of Diffusion and Effusion

A few of the concrete properties of gases depend on the identity of the gas. One of these physical properties tin can be seen when the movement of gases is studied.

In 1829 Thomas Graham used an appliance like to the one shown in the effigy below to study the diffusion of gases the charge per unit at which two gases mix. This apparatus consists of a glass tube sealed at one end with plaster that has holes large plenty to permit a gas to enter or leave the tube. When the tube is filled with H₂ gas, the level of water in the tube slowly rises because the H₂ molecules inside the tube escape through the holes in the plaster more than rapidly than the molecules in air can enter the tube. Past studying the rate at which the water level in this apparatus inverse, Graham was able to obtain data on the charge per unit at which unlike gases mixed with air.

Graham plant that the rates at which gases diffuse is inversely proportional to the square root of their densities.

equation

This human relationship somewhen became known as Graham's law of diffusion.

To understand the importance of this discovery nosotros accept to remember that equal volumes of dissimilar gases incorporate the same number of particles. As a event, the number of moles of gas per liter at a given temperature and force per unit area is abiding, which means that the density of a gas is directly proportional to its molecular weight. Graham's law of diffusion can therefore also be written every bit follows.

equation

Similar results were obtained when Graham studied the charge per unit of effusion of a gas, which is the charge per unit at which the gas escapes through a pinhole into a vacuum. The rate of effusion of a gas is too inversely proportional to the square root of either the density or the molecular weight of the gas.

Graham's law of effusion can be demonstrated with the apparatus in the figure below. A thick-walled filter flask is evacuated with a vacuum pump. A syringe is filled with 25 mL of gas and the time required for the gas to escape through the syringe needle into the evacuated filter flask is measured with a terminate spotter.

Every bit nosotros tin can see when data obtained in this experiment are graphed in the figure below, the time required for 25-mL samples of different gases to escape into a vacuum is proportional to the foursquare root of the molecular weight of the gas. The rate at which the gases effuse is therefore inversely proportional to the square root of the molecular weight. Graham'southward observations virtually the charge per unit at which gases diffuse (mix) or effuse (escape through a pinhole) suggest that relatively light gas particles such every bit H₂ molecules or He atoms move faster than relatively heavy gas particles such equally CO_ii or SO₂ molecules.

The Kinetic Molecular Theory and Graham'south Laws

The kinetic molecular theory can be used to explain the results Graham obtained when he studied the diffusion and effusion of gases. The key to this explanation is the last postulate of the kinetic theory, which assumes that the temperature of a system is proportional to the average kinetic energy of its particles and nada else. In other words, the temperature of a organization increases if and only if in that location is an increment in the boilerplate kinetic energy of its particles.

Ii gases, such as H_ii and O_two, at the same temperature, therefore must have the aforementioned average kinetic energy. This tin be represented past the following equation.

This equation can be simplified by multiplying both sides by 2.

It can then be rearranged to give the following.

equation

Taking the square root of both sides of this equation gives a relationship between the ratio of the velocities at which the ii gases move and the square root of the ratio of their molecular weights.

equation

This equation is a modified form of Graham's constabulary. It suggests that the velocity (or rate) at which gas molecules motion is inversely proportional to the square root of their molecular weights.