How does a gas expand according to the kinetic molecular theory? According to kinetic molecular theory, how are gases compared with liquids and solids? Use the terms volume and density to compare gases, liquids and solids in terms of the kinetic See all questions in Kinetic Theory of Gases. Impact of this question views around the world. 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.
These large molecular velocities do not yield macroscopic movement of air, since the molecules move in all directions with equal likelihood. The mean free path the distance a molecule can move on average between collisions of molecules in air is very small, and so the molecules move rapidly but do not get very far in a second.
The faster the rms speed of air molecules, the faster that sound vibrations can be transferred through the air.
The speed of sound increases with temperature and is greater in gases with small molecular masses, such as helium. Thus, the particles travel from one end of the container to the other in a shorter period 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 increase in the pressure of the gas. Thus, the pressure of a gas becomes larger as the volume of the gas becomes smaller.
Charles' Law V 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 as the gas becomes warmer. If they move 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 of the gas.
If the walls of the container are flexible, it will expand until the pressure of the gas once more balances the pressure of the atmosphere.
The volume of the gas therefore becomes larger as the temperature of the gas increases. Avogadro's Hypothesis V N. As the number of gas particles increases, the frequency of collisions with the walls of the container must increase. This, in turn, leads to an increase in the pressure of the gas. Flexible containers, such as a balloon, will expand until the pressure of the gas inside the balloon once again balances the pressure of the gas outside. Thus, the volume of the gas is proportional to the number of gas particles.
Imagine what would happen if six ball bearings of a different size were added to the molecular dynamics simulator. The total pressure would increase because there would be more 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 as it did when there was only one kind of ball bearing 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 ball bearing is present by itself. In other words, the total pressure of a mixture of gases is equal to the sum of the partial pressures of the individual gases.
Graham's Laws of Diffusion and Effusion. It applies to ideal gases close to thermodynamic equilibrium, and is given as the following equation:. Derivation of the formula goes beyond the scope of introductory physics. It can also be shown that the Maxwell—Boltzmann velocity distribution for the vector velocity [ v x , v y , v z ] is the product of the distributions for each of the three directions:.
This makes sense because particles are moving randomly, meaning that each component of the velocity should be independent. Usually, we are more interested in the speeds of molecules rather than their component velocities. The Maxwell—Boltzmann distribution for the speed follows immediately from the distribution of the velocity vector, above.
Note that the speed is:. Temperature is directly proportional to the average translational kinetic energy of molecules in an ideal gas. Intuitively, hotter air suggests faster movement of air molecules.
In this atom, we will derive an equation relating the temperature of a gas a macroscopic quantity to the average kinetic energy of individual molecules a microscopic quantity. This is a basic and extremely important relationship in the kinetic theory of gases. We assume that a molecule is small compared with the separation of molecules in the gas confined in a three dimensional container , and that its interaction with other molecules can be ignored.
Also, we assume elastic collisions when molecules hit the wall of the container, as illustrated in. 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. From the equation, we get:. 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. Recall the macroscopic expression of the ideal gas law:. Equating the right hand sides of the macroscopic and microscopic versions of the ideal gas law Eq.
It has been found to be valid for gases and reasonably accurate in liquids and solids.
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