The Kinetic Molecular Theory of Gases Background: It is straightforward to observe that there is an inverse relationship between pressure and volume for a gas at constant temperature. Curious scientists started to wonder why. This required a working model of gases, in which people tried to imagine what gases look like so they could better understand their behavior. Learning Outcomes: Upon successful completion of this module, a student should be able to: Apply the kinetic molecular theory of gases to real situations. Predict the behavior of gases based on their mass and temperature. Use the previous information to predict the relative velocities of gases. The Kinetic Molecular Theory The Kinetic Molecular Theory is a five-part description of various assumptions that are made, which allows the ideal gas law to work. That doesn t necessarily mean that each of these are true statements in the absolute sense. It means that if these assumptions can be made, then the familiar properties of gases can be rationalized and predicted. 1. Gases consist of large numbers of molecules (or particles) that are in constant random motion.. The combined volume of all the molecules of the gas is negligible compared to the volume of the gas container. 3. Gas particles are neither attracted to nor repulsed by one another. 4. Energy can be exchanged between gas molecules when they collide, but their average kinetic energy of the molecules does not change as long as the temperature is constant. 5. The average kinetic energy of the molecules is proportional to the absolute temperature. At any given temperature, the molecules of all gases have the same average kinetic energy. Part 1: Lets look at the nd portion The combined volume of all the molecules of the gas is negligible compared to the volume of the gas container. Question 1: Do gas particles have an individual mass and volume?
Question : Is their volume small compared to the volume of a container? Question 3: Consider a container at STP. What is the volume of a single gas particle? (Remember that 1 mole = 6.0x10 3 particles) The number from the previous question should be pretty small. It is a very small fraction of the total volume of any container. Since these particles are in constant motion and don t collect in any one area, it is assumed that they collectively occupy a negligible volume even if there is a large number of particles. In the case of a real gas, the volume of individual particles is considered through a corrective term. PV nb nrt The corrective term (nb) corrects for the volume of each individual molecule (b) multiplied across all particles (n). Question 4: If the volume of individual particles does become a problem, what do you think would happen to the observed Pressure (Pobs) compared to the calculated Pressure (Pcal) of a gas. Will it be larger or smaller? Why? Part : Lets look at the 3 rd portion gas particles are neither attracted to nor repulsed by one another. The overall pressure of a collection of particles comes from the collective particles colliding with one another and hitting the walls of the container. A high-pressure gas will collide a lot and with a lot of energy. A low-pressure gas will collide less frequently and with less force. Question 5: If a particle is on its way to colliding with the wall, but feels a small attraction to a particle behind it, what is the effect?
Question 6: What would be the overall effect on the pressure if gas particles are attracted to one another? To correct for the effect of gas particles being attracted to one another (even slightly) another corrective term needs to be built in to the ideal gas law. na na P V nb nrt V The new corrective term makes a small correction in pressure that will account for V the attractions between any two gas particles. na P V nb nrt V The above equation is called the van der Waals Equation for a gas. It has all of the normal portions of the ideal gas law, it just has a couple correction terms to covert between the idea of an ideal gas (with assumptions) and a real gas (with fewer assumptions). Values of a and b are unique to every gas and would have to be looked up. Look at the graph below. It is a plot of PV/RT for 1 mole of gas against Pressure.
Question 7: If the dotted line represents ideal behavior for this gas, across all pressures, what pressure range is closest to the ideal? Question 8: If the dotted line represents ideal behavior for this gas, across all pressures, what temperature is closest to the ideal? Question 9: Based on this information, under what temperature and pressure circumstances would you expect a gas to behave like an ideal gas? Question 10: Rationalize why this might be. (In other words, why does that range work?) Part 3: Kinetic Energy. Lets look at the 5 th part At any given temperature, the molecules of all gases have the same average kinetic energy. The kinetic energy of an object is given as 1 E mv
Where m is the mass of the particle and v is its average velocity. Temperature is not a part of this equation. Another equation for kinetic energy is: 3 E RT Where R is the universal gas constant (8.31 J/mol K) and T is for absolute temperature. Question 11: Set these two E s equal to one another, and solve for velocity. This tells us that the velocity of a particle depends on the mass and temperature only. This also allows us to determine the average (root mean square) velocity for any gas at any temperature. Question 1: Determine the average velocity of Xe at 5⁰C. Question 13: Determine the average velocity of NH3 at 5⁰C. However, in a large collection of gases, like one mole or higher, not every particle is expected to have the same velocity. Like runners in the Chicago marathon (more than 40,000) not every single one has the same speed, yet there is one average speed when you consider the group as a whole. Consider two particles, m1 and m. Both of them are at the same temperature, 100K. Question 14: If particles m1 has half the mass of m, which is faster? What is the relationship between their speeds?
Question 15: Considering their masses and speeds, are their kinetic energies the same or different? For gasses at the same temperature, regardless of their identity, they will have the same kinetic energy. Question 16: Consider nitrogen gas (N) and oxygen gas (O). At 300K, what is the relationship between their kinetic energies? Question 17: Consider nitrogen gas (N) and oxygen gas (O). At 300K, what is the relationship between their average velocities? Question 18: Rate the following gases in terms of average velocities at 300K: Helium, H, water, N, O, and carbon dioxide. Effusion and Diffusion When gas particles move, we have to consider things like mass and temperature to get a sense of how fast they are moving. Those gas molecules move very quickly (hundreds of meters per second). If that were to be absolutely true, it seems that our number one gas detector (nose) should be able to detect more smells more quickly. There are two processes that we must consider. Diffusion: The process in which gas particles spread out in response to a concentration gradient. That is, a build up of gas particles in one location becoming more evenly distributed in a mixture. Effusion: The process in which a gas particle escapes from a container into an empty space through a very small opening. That is, how quickly will a gas expand into a vacuum.
The approximate rate of effusion is given below: Rate 1 MW Question 19: Consider two gases, He and N. Which one would you expect to have a higher rate of effusion? Another equation, Graham s Law of Effusion shows that the relative rates of two gases is proportional to the inverse square of their molecular weights. Rate Rate A B MW MW B A Question 0: An unknown gas effuses at a rate that is 0.46 times that of nitrogen gas (at the same temperature). What is the molar mass of the unknown gas in g/mole? Challenge Question: Consider the following reaction NH3(g) + HCl(g) NH4Cl(s) This is a classic demonstration where ammonia and hydrogen chloride gases are inserted at opposite ends of a tube (shown below). Where will the solid ammonium chloride ring form? Explain your placement. NH 3 HCl