We have developed macroscopic definitions of pressure and temperature. Pressure is the force divided by the area on which the force is exerted, and temperature is measured with a thermometer. We gain a better understanding of pressure and temperature from the kinetic theory of gases, which assumes that atoms and molecules are in continuous random motion.
Figure 1. When a molecule collides with a rigid wall, the component of its momentum perpendicular to the wall is reversed. A force is thus exerted on the wall, creating pressure. Because a huge number of molecules will collide with the wall in a short time, we observe an average force per unit area. These collisions are the source of pressure in a gas. As the number of molecules increases, the number of collisions and thus the pressure increase.
Similarly, the gas pressure is higher if the average velocity of molecules is higher. The actual relationship is derived in the Making Connections feature below. What can we learn from this atomic and molecular version of the ideal gas law? We can derive a relationship between temperature and the average translational kinetic energy of molecules in a gas.
Figure 2. Gas in a box exerts an outward pressure on its walls. A molecule colliding with a rigid wall has the direction of its velocity and momentum in the x-direction reversed. This direction is perpendicular to the wall. The components of its velocity momentum in the y- and z-directions are not changed, which means there is no force parallel to the wall. Figure 2 shows a box filled with a gas.
We know from our previous discussions that putting more gas into the box produces greater pressure, and that increasing the temperature of the gas also produces a greater pressure. But why should increasing the temperature of the gas increase the pressure in the box? A look at the atomic and molecular scale gives us some answers, and an alternative expression for the ideal gas law.
The figure shows an expanded view of an elastic collision of a gas molecule with the wall of a container. Calculating the average force exerted by such molecules will lead us to the ideal gas law, and to the connection between temperature and molecular kinetic energy. We assume that a molecule is small compared with the separation of molecules in the gas, and that its interaction with other molecules can be ignored.
There is no force between the wall and the molecule until the molecule hits the wall. During the short time of the collision, the force between the molecule and wall is relatively large. It is the time it would take the molecule to go across the box and back a distance 2 l at a speed of v x.
This force is due to one molecule. We multiply by the number of molecules N and use their average squared velocity to find the force.
We would like to have the force in terms of the speed v , rather than the x -component of the velocity. We note that the total velocity squared is the sum of the squares of its components, so that. This gives the important result. This calculation produces the result that the average kinetic energy of a molecule is directly related to absolute temperature. It is another definition of temperature based on an expression of the molecular energy.
Before substituting values into this equation, we must convert the given temperature to kelvins. The temperature alone is sufficient to find the average translational kinetic energy. Substituting the temperature into the translational kinetic energy equation gives.
Finding the rms speed of a nitrogen molecule involves a straightforward calculation using the equation. Using the molecular mass of nitrogen N 2 from the periodic table,. Substituting this mass and the value for k into the equation for v rms yields. Note that the average kinetic energy of the molecule is independent of the type of molecule. The average translational kinetic energy depends only on absolute temperature.
The kinetic energy is very small compared to macroscopic energies, so that we do not feel when an air molecule is hitting our skin. The rms velocity of the nitrogen molecule is surprisingly large. The volume of the container has decreased, which means that the gas molecules have to move a shorter distance to have a collision.
There will therefore be more collisions per second, causing an increase in pressure. What will happen to the pressure of a system where the temperature is increased and the volume remains constant?
Again, this type of problem can be approached in two ways:. Temperature is located in the numerator; there is a direct relationship between temperature and pressure. Therefore an increase in temperature should cause an increase in pressure. Temperature is increased, so the average kinetic energy and the rms speed should also increase. This means that the gas molecules will hit the container walls more frequently and with greater force because they are all moving faster.
This should increase the pressure. The physical behaviour of gases is explained by the kinetic molecular theory of gases. The number of collisions that gas particles make with the walls of their container and the force at which they collide determine the magnitude of the gas pressure. Temperature is proportional to average kinetic energy. Questions State the ideas of the kinetic molecular theory of gases.
Using the kinetic molecular theory, explain how an increase in the number of moles of gas at constant volume and temperature affects the pressure. Answers Gases consist of tiny particles of matter that are in constant motion.
Gas particles are constantly colliding with each other and the walls of a container. These collisions are elastic; that is, there is no net loss of energy from the collisions. Gas particles are separated by large distances. The size of gas particles is tiny compared to the distances that separate them and the volume of the container. There are no interactive forces i.
The average kinetic energy of gas particles is dependent on the temperature of the gas. We begin by recalling that the pressure of a gas arises from the force exerted when molecules collide with the walls of the container. This force can be found from Newton's law. To evaluate the derivative, which is the velocity change per unit time, consider a single molecule of a gas contained in a cubic box of length l.
For simplicity, assume that the molecule is moving along the x -axis which is perpendicular to a pair of walls, so that it is continually bouncing back and forth between the same pair of walls. After the collision the molecule must travel a distance l to the opposite wall, and then back across this same distance before colliding again with the wall in question.
As noted near the beginning of this unit, any given molecule will make about the same number of moves in the positive and negative directions, so taking a simple average would yield zero.
To avoid this embarrassment, we square the velocities before averaging them. This result is known as the root mean square rms velocity. We have calculated the pressure due to a single molecule moving at a constant velocity in a direction perpendicular to a wall. The average translational kinetic energy is directly proportional to temperature:. The Boltzmann constant k is just the gas constant per molecule. Chem1 Virtual Textbook. The Model The five basic tenets of the kinetic-molecular theory are as follows: A gas is composed of molecules that are separated by average distances that are much greater than the sizes of the molecules themselves.
The volume occupied by the molecules of the gas is negligible compared to the volume of the gas itself. The molecules of an ideal gas exert no attractive forces on each other, or on the walls of the container. The molecules are in constant random motion , and as material bodies, they obey Newton's laws of motion. This means that the molecules move in straight lines see demo illustration at the left until they collide with each other or with the walls of the container.
Collisions are perfectly elastic ; when two molecules collide, they change their directions and kinetic energies, but the total kinetic energy is conserved.
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