Chapter 10 Gas Laws
Express the ideal gas law in terms of molecular mass and velocity. Describe the relationship between the temperature of a gas and the kinetic energy of atoms. Collisions between gas particles or collisions with the walls of the container are If we compress a gas without changing its temperature, the average kinetic . gives a relationship between the ratio of the velocities at which the two gases. Get an answer for 'Describe the relationship between temperature and kinetic Temperature is a measurement of the average kinetic energy of the molecules in an object or a system. What's the difference between speed and velocity?.
All gases at a given temperature have the same average kinetic energy. Lighter gas molecules move faster than heavier molecules. As velocity increases so does kinetic energy.
Of course the inverse is also true, that as kinetic energy increases so does velocity. You can see from this relationship how a molecule with a higher temperature will be moving faster. Thermal energy is the total kinetic energy of all the particles in a system.
Temperature, thermal energy, and the speed of a molecule are all directly related. In order to further understand of kinetic theory, let us review some of its applications. Say you have a given amount of particles in a box. If you want to add more particles, but you do not want to increase the pressure, you must make the container larger.
This is consistent with the predictions of Boyle's law. Boyle's law for a box of varying volume. The particles have the same energy temperature throughout. As the box gets smaller, they have a smaller distance to travel before they collide with the walls, and thus the time between collisions gets increasingly smaller.
In a given amount of time the partials hit the walls more, which results in a greater amount of pressure. The amount of moles is clearly constant, as we are not adding or subtracting particles from the box. Another way of looking at this is that as the pressure increases, it drives the particles together. These compacted particles now occupy less volume.
According to Charles' law, gases will expand when heated. The temperature of a gas is really a measure of the average kinetic energy of the particles. As the kinetic energy increases, the particles will move faster and want to make more collisions with the container.
However, remember that in order for the law to apply, the pressure must remain constant. The only way to do this is by increasing the volume. This idea is illustrated by the comparing the particles in the small and large boxes. The higher temperature and speed of the red ball means it covers more volume in a given time.
You can see that as the temperature and kinetic energy increase, so does the volume. Also note how the pressure remains constant.
Both boxes experience the same number of collisions in a given amount of time. As the temperature of a gas increases, so will the average speed and kinetic energy of the particles. At constant volume, this results in more collisions and thereby greater pressure the container. It is assumed that while a molecule is exiting, there are no collisions on that molecule.
The kinetic energy of a collection of gas particles is given by where m is the mass of a molecule of the gas and u is its root-mean-square velocity. The root-mean-square velocity is not exactly equal to the average velocity, but they are close.
The rms velocity is the speed of a molecule that has the average molecular kinetic energy. This relationship will be important in later discussions of some interesting properties of gases. Right now it is important to recognize that a collection of gas particles has a range of speeds, not a single speed. When a collision occurs energy can be transferred.
So if the white gas particle is moving slowly and it collides with a fast moving particle, transfer of energy can result in the white particle moving faster, and the other particle slower after the collision.
Total energy of the collision is conserved--elastic collision. Now the usefulness of a model depends on its ability to reproduce experimental observations as well as make predictions which can be verified by further experiment. Lets begin by verifing some of the experiments we performed earlier.
Boyle's Law related pressure to volume of an ideal gas at constant mol and temperature. So if set the temperature at K and 4 mol of gas we can observe how the pressure is effected by a decrease in volume. Initially the pressure is 2.
The Kinetic Molecular Theory
Lets observe what happens as the number of moles of gas are changed at constant temperature and volume. Now lets consider how changing the temperature of a gas effects the pressure at constant volume and constant moles. To help recognize how we use the Kinetic-molecular model, watch carefully what happens as the temperature is lowered. Now lets consider how changing the temperature of a gas effects the volume at constant pressure and constant moles. We will change to the 'V' mode to calculate volume.
To help recognize how we use the Kinetic-molecular model to explain Charles' Law, watch carefully what happens as the temperature is lowered. Diffusion and effusion The last postulate says the average kinetic energy for molecules is the same for all gases at the same temperature. It should be noted that this says if you have two samples of gas, one with a large mass and one with a small mass, the average speed of the sample with the large mass is slower than the average speed of the light gas.
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 A few of the physical properties of gases depend on the identity of the gas. One of these physical properties can be seen when the movement of gases is studied. In Thomas Graham used an apparatus similar to the one shown in the figure below to study the diffusion of gases the rate at which two gases mix.
This apparatus consists of a glass tube sealed at one end with plaster that has holes large enough to allow 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 rapidly than the molecules in air can enter the tube.
By studying the rate at which the water level in this apparatus changed, Graham was able to obtain data on the rate at which different gases mixed with air. Graham found that the rates at which gases diffuse is inversely proportional to the square root of their densities.
This relationship eventually became known as Graham's law of diffusion. To understand the importance of this discovery we have to remember that equal volumes of different gases contain the same number of particles. As a result, the number of moles of gas per liter at a given temperature and pressure is constant, 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 as follows. Similar results were obtained when Graham studied the rate of effusion of a gas, which is the rate at which the gas escapes through a pinhole into a vacuum. The rate of effusion of a gas is also 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.
Kinetic Theory of Gases
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 stop watch. As we can see when data obtained in this experiment are graphed in the figure below, the time required for mL samples of different gases to escape into a vacuum is proportional to the square 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's observations about the rate at which gases diffuse mix or effuse escape through a pinhole suggest that relatively light gas particles such as H2 molecules or He atoms move faster than relatively heavy gas particles such as CO2 or SO2 molecules.
The Kinetic Molecular Theory and Graham's 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 nothing else. In other words, the temperature of a system increases if and only if there is an increase in the average kinetic energy of its particles.
Two gases, such as H2 and O2, at the same temperature, therefore must have the same average kinetic energy.