# Kinetic Model for Gases

## What is a Gas?

The material that makes up the atmosphere is mostly in the form of a gas. A gas is one of three basic forms or "states" of matter.

1. Gas - substance in which individual molecules are not chemically bonded together. Even though many gases are not visible, it does not mean there is nothing there. Gases are composed of the same molecules that make up liquids and solids, the difference being that individual gas molecules are not chemically bonded together. Water vapor is an example of a type of gas.
2. Liquid - substance in which individual molecules are chemically bonded together, but the bonds are continuously breaking and reforming, so the substance flows. Liquid water is an example of a liquid.
3. Solid - substance in which individual molecules are tightly bonded togther in an orderly fashion. Bonds are difficult to break and the substance does not flow. Ice is an example of a solid. Schematic Diagram representing the packing (density) of molecules that make up three different states of matter.

### Kinetic Model of Matter

The kinetic model concept is used to help us to understand and visualize how matter behaves at the level of individual molecules. A model is necessary because molecules are too small to observe individually; however, in order to understand some of the properties of matter, we need to consider how individual molecules behave. The kinetic model can be used to differentiate the behavior of solids, liquids, and gases. Look at slides 1-8 in this Kinetic Model Slide Show for a more visual explanation of the Kinetic Theroy of Matter.

We will use the kinetic model concept to help us to understand and visualize how gases behave. In the kinetic model for gases, the individual molecules that make up a gas are treated like tiny spheres, all moving in random directions. As in the atmosphere of Earth, the gas molecules (spheres) are quite small compared to the average distance between molecules (spheres). The spheres collide with each other and any solid or liquid that happens to be in the way, but they remain separate, i.e., they do not stick together. Please refer to this NASA link describing the Kinetic Model for Gases.

As described in the previous link, the energy state of a gas is determined by its

1. Temperature
2. Pressure
3. Density
Each of these properties of gases can be measured by instruments. The temperature and pressure of gases are very easy to measure using thermometers and barometers, while density is more difficult to measure. The instruments are said to measure the macro scale or large-scale properties of the gas. The instruments are sensitive to the motions of a very large number of individual molecules. The motion of individual gas molecules is said to take place at the micro scale. Macro scale observations are based on what we can measure, while micro scale motions are too small to measure easily (at the level of individual molecules), thus we rely on a simple model, the kinetic model, to help us understand what is going on at micro scales. Keep in mind though that macro scale and micro scale are just two different ways of viewing the same thing, i.e., the behavior of gases.

#### Temperature of a Gas

Temperature is determined by the average speed of the molecules making up a substance. The higher the temperature, the faster they move. For gases, this is the random motion of the individual molecules that make up the gas. Random motion is disordered, i.e., individual molecules are equally likely to be moving in any direction. At temperatures common in Earth's atmosphere, the average speed of each molecule is approximately 1000 mi/hr. This is different from what we call "wind" which is ordered movement of air at the macro scale (basically the ordered movement of a fluid in a given direction). For example, when the windspeed is 10 mi/hr, it means that "blobs" of air are moving at 10 mi/hr, but individual molecules are still moving at an average speed of about 1000 mi/hr. An analogy can be made with a ball of angry bees. The random motion of the individual bees is like the random motion of individual gas molecules. If the entire ball remains stationary, this would be the no wind analogy (only random motion, no ordered motion). If the ball of bees moves off as a unit, this would be the wind analogy (ordered motion on top of the random motion).

Using the concept of energy, the higher the temperature, the more energy that is possessed by the gas. It should make sense, then, that the higher the temperature, the higher the energy, and the faster the speed at which the molecules are moving.

We sense temperature by touch. Thermoreceptor nerve cells in our body are sensitive to the average speed at which air molecules are moving. Similarly, when air molecules strike a thermometer, energy is transferred between the thermometer and the air. The reading on the thermometer is calibrated to read the average thermal motion of all of the air molecules that collide with it.

#### Number Density of a Gas

The number density of a gas is defined as the number of gas molecules per unit volume. In Earth's atmosphere, near sea level there are about 2.7x1019 molecules per cm3(cubic centimeter) or 4.4x1020 molecules per inch3(cubic inch). While this may seem like a lot of molecules in a small volume, molecules are very small. At this number density, there is acually much more "empty" space than the space occupied by gas molecules. By comparison, the number density for solids and liquids is much higher.

An important property of gases is that they are easily compressed, i.e., squeeze a gas together and its number density increases. In other words, we say gases are compressible because they can easily be squeezed into a smaller volume. Solids and liquids on the other hand are not easily compressed. Keep in mind that this works both ways. Gases will expand to fill any size volume, while solids and liquids have fixed volumes and do not expand.

#### Pressure

From a microscopic point of view, gas pressure is caused by the collisions of gas molecules on a surface. Each individual collision provides a tiny push (or force) on the surface that it contacts. The sum total of all of these tiny forces determines the gas pressure. The physical units for pressure is force per area. More generally, all fluids (liquids and gases) exert pressure on the surfaces of solids that are immersed in them, which is simply the force of the molecules of the fluid bouncing off the solid surface. When gas is placed in a sealed container, the collisions of the randomly moving gas molecules with the sides of the container is the gas pressure.

The Earth's atmosphere is contained by the force of gravity, i.e., without gravity the atmosphere would just spread out into outer space. This results in a layer of gas surrounding the planet and a gas pressure (or air pressure) is exerted on all objects immersed in the atmosphere (including us). If you are having trouble understanding air pressure, consider an analogy with liquid water in a pool. The water is held down in the pool by gravity. If you dive down into the water, water pressure pushes in on you from all directions. We are immersed in air and air pressure pushes in on us from all directions.

## Ideal Gas Law

The temperature, number density, and pressure of a gas are related to each other through the gas law equation:

 pressure = temperature x number density x constant

The equation above is equivalent to the gas law equation you may remember from chemistry class, PV = nRT.

We will use the ideal gas law and the kinetic model representation of a gas to explain the behavior of air. It much simplier to understand this material if one of the three state variable (temperature, pressure, or number density) is held constant, while the other two are allowed to change.

### A. Hold number density constant

Suppose we put some air (gas) in a sealed, rigid container. No gas can enter or leave the container, so the number of gas molecules in the container cannot change. The container is rigid so that the size and shape of the container cannot change, in other words the volume of the container cannot change. Therefore, the number density (number of molecules divided by the volume of the container) cannot change.

Now suppose we heat the air in the container. This raises the temperature of the air in the container. At a higher temperature, the average speed of the individual gas molecules increases. Therefore, gas molecules hit the walls of the container harder and more often, increasing the pressure in the container. The reverse happens if you cool the air in the container. In summary, if the number density of a gas is held fixed, increasing the temperature of the gas, increases its pressure and decreasing the temperature of the gas, decreases its pressure.

### B. Hold pressure constant

This situation is most applicable to understanding some of what happens in Earth's atmosphere. In this case, suppose we put some air (gas) in a sealed, flexible container like a balloon. Gas cannot enter or leave the container, but the size (volume) of the container adjusts so that the air pressure inside the container equals the air pressure outside of the container. In such a flexible container, the gas pressure inside the container (pushing outward) must always be equal to the gas pressure surrounding the container (pushing inward). If the inside pressure is greater than the outside pressure, the flexible container will pushed outward by the pressure difference, expanding the container. On the other hand, if the outside pressure is greater than the inside pressure, the flexible container will be pushed inward by the pressure difference, compressing the container. The change in volume will continue until the inside and outside pressures are equal.

Now suppose we heat the air in the container. The average speed of individual molecules increase, so they hit the walls of the container harder and more often, initially causing the pressure inside to increase. This forces the container to get larger until the air pressure inside the container again equals the air pressure outside the container. In this case, the number density of the air in the container has decreased because we have the same number of molecules in a larger volume. The reverse happens if you cool the air in the container. In summary, if the pressure of a gas does not change, increasing the temperature of the gas causes the gas to expand (decrease number density) and decreasing the temperature of the gas causes the gas to contract (increase number density).

## Relationship between 500 mb height and air temperature

We will now apply B above to explain why the height of the 500 mb pressure surface is related to the temperature of the air in the vertical column from the ground surface to 500 mb. Consider a vertical column of air that extends from the ground surface upward to the top of the atmosphere. Assume that no air is allowed to enter or leave the column. This means that no matter what we do to the temperature of the air in the column, the air pressure at the ground surface will not change The reason for this is that the air pressure at any point in the atmosphere is determined by the weight of the air above that point and since the amount of air above the surface has not changed, the pressure does not change.

If the column of air is heated, it expands upward making the column taller. The air pressure at the ground does not change, but the rate at which air pressure decreases with altitude is now slower. The result is that the height of the 500 mb pressure level is now higher. The reason is that warmer air has a lower number density, i.e., it is more spread out. Since the surface pressure did not change (constant pressure) and the air temperature increased, this means that the number density of the air at the surface is lower (applying rule B above). In fact since we assumed the entire column of air was warmed, the density at any given pressure will be lower. Conversely, if the column of air is cooled, it contracts, and the 500 mb pressure level becomes lower.

General Rule: Air pressure decreases more slowly with increasing altitude in a warm column of air compared with a colder column of air. This explains why higher 500 mb heights are associated with warmer air and lower 500 mb heights are associated with colder air (see figure).

## Thickness: A better estimate for air temperature near the ground

You will not be specifically tested on material from this section. However, you should be able to understand it.

Although we will continue to use the 500 mb height to estimate the pattern of air temperature in the lower troposphere (just above the ground surface where we live), it is not exact. In this section, we will discuss a couple of reasons why the 500 mb height is not completely determined by the air temperature in the lower troposphere. In other words, we will point out why the 500 mb height can sometimes be misleading with regard to air temperature just above the ground.

One issue is that the 500 mb height does depend on the sea level air pressure, i.e., the air pressure at ground level. While the average air pressure at sea level is 1013 mb, the actual sea level pressure at a given location and time varies, typically within the range from 983 - 1043 mb. Thus relatively high sea level pressure tends to raise the 500 mb height since the pressure drop from sea level pressure to 500 mb is greater compared to a case where the sea level pressure is lower. To overcome this dependence on sea level pressure, meteorologists often use a measurement called 1000 mb to 500 mb thickness. This is a measure of the vertical distance between the 1000 mb pressure surface and the 500 mb pressure surface. The 1000 - 500 mb thickness is directly related to the average air temperature between 1000 mb and 500 mb without being influenced by variations in surface pressure (see 1000 - 500 mb thickness figure). Therefore, thickness is a better indicator of how warm or cold the air is in a vertical column above a given location. Contour plots of 1000 - 500 mb thickness are routinely produced by weather forecast models. We will not look at thickness maps in this course.

Another potential problem for interpreting surface air temperature even if using the 1000 - 500 mb thickness is that the 1000 - 500 mb thickness is generally in the range from 4800 to 5700 meters (or about 3 to 3.5 miles). This entire vertical extent need not be uniformly warm or cold, i.e., there can be sublayers of smaller vertical extent that are both relatively warm and relatively cold. For surface temperature, we need to know how warm or cold it is at the bottom of the air column. A better estimate of surface temperature could be obtained by looking at the thickness of a shorter column of air just above the ground surface. For this purpose, meteorologists will look at the 1000 - 850 mb thickness, which is more closely related to the air temperature just above the ground (see 1000 - 850 mb thickness figure). Contour plots of 1000 - 850 mb thickness are also produced by weather forecast models. We will not look at thickness maps in this course.

In general, the vertical height between any two pressure levels in the atmosphere is related to the temperature of the air between the two pressure levels. The warmer the air, the lower the air number density, and the greater the vertical distance between the two pressure levels.