Kinetic Theory & Ideal Gas

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1 1 of 6 Thermodynamics Summer 2006 Kinetic Theory & Ideal Gas The study of thermodynamics usually starts with the concepts of temperature and heat, and most people feel that the temperature of an object is somehow a measure of the object s heat content. The higher the temperature, the hotter the object. The Fahrenheit and Celsius temperature scales are geared to humans, and are really comfort indexes. We instinctively know that a 90 0 F afternoon is uncomfortably hot, a 10 0 F winter s night uncomfortably cold, and a 70 0 F room the definition of comfort (STP). We live in a sea of gases (predominantly nitrogen and oxygen), and the way these gases interact with us gives us the sensation of temperature. Pressure and temperature are both a product of the collisions the gas molecules have with our bodies and other objects. To explain temperature and pressure, we need to think of a gas as a collection of hard spheres that endlessly collide elastically with each other and objects immersed in the gas. Elastic collisions imply that no energy is lost in each collision. If air molecules didn t collide elastically, the air we breathe would settle to the ground, and we d suffocate! On a pool table we can assume that the balls collide elastically, but we know that eventually the balls come to a stop. Imagine playing pool with the molecules of the air as the balls. The action on the table would never die down after the first shot, and the balls would roll forever. Another aspect of the hard sphere approximation for gases is that, except for collisions, the spheres don t influence each other. They are also considered dimensionless, point particles. This is a usually unstated assumption for the ideal gas law. However, as there are countless billions of molecules, it would be practically impossible to calculate temperatures and pressures by applying the laws for elastic collisions to each collision as there are an almost infinite number of collisions happening at a fantastic rate. Therefore thermodynamics uses statistical mechanics to deal with the great numbers of molecules found in a gas.

2 2 of 6 Thermodynamics Summer 2006 Exercise A: Temperature In theory temperature and the average kinetic energy of the molecules of a gas are related by the equation kt = mv avg, (Eqn. 1) 2 2 where k is 1.38x10-23 J/molecule-K 0 and T is the temperature in degrees Kelvin. 1.) Double click on the Atoms in Motion icon, and the program will pop up on your screen already running. You should see a 3-D box with a lot of atoms and molecules colliding with each other and the walls. There will probably be two different types of molecules or atoms combined in your volume of gas. Initially we ll want a gas made up of only one type of molecule. Select the Atoms menu and deselect the less numerous molecule by putting zero in the Number box. There should be at least 20 molecules of the single type. Next select the Display menu and make sure the RMS Speed is displayed on the screen while the program is running. The RMS speed is the average speed of each molecule. There is a menu item listed as Kinetic Energy (KE), but it gives the total KE of the system, and not the average KE of a single molecule. Using the RMS speed and mass of a molecule, calculate the temperature using Equation 1. Compare your predicted temperature with that shown on the screen. 2.) Select the KE menu item, and start decrementing the kinetic energy of the system in steps of 20x10-21 J. Starting from room temperature (around K), record the total system energy and temperature for ten steps. Plot the table with total system energy on the horizontal axis, and the temperature on the vertical axis. Find the equation of the curve, and extrapolate it to 0. Does it go through the origin? Reduce the system s KE to zero in the Atoms in Motion program. What s happening in the 3-D cube?

3 3 of 6 Thermodynamics Summer 2006 Exercise B: Pressure 1.) Open and run the program lab_collide_a.am from the file menu. The program contains a pair of identical molecules in a 3-D cube that move only in the y-z plane. The molecules are started in such a way that they will follow the same path and have the same collisions with the walls and each other continuously. Under the colliding molecules are their shadows on the floor of the box. If you watch the movement of the shadows, you will see that they only move along the y-axis, showing the y-component of motion (v y ). Stop the action and select one of the molecules so that you can record its velocity components. To select a molecule, you put the cursor on it and click the mouse while holding down the Shift key. A yellow box will appear around the selected molecule. Clicking on the Vel menu item will give you the velocity components of the selected molecule. Let the program run for a while, and check the molecule s speed occasionally. You will see that it doesn t change its magnitude, only the direction. 2.) Click on the Display button, and have the pressure shown on the screen. Initially the pressure will read zero, but if you click on the Average button, the pressure reading will go to around 1.3 or 1.4x10 5 N/m 2. Again click on the Display button, and have the box s volume shown on the screen. The box is a cube, and all sides are the same length. Click on the Atom button and record the mass of each molecule. 3.) We will make the assumption that the pressure the screen shows is the pressure exerted on the walls by the molecules colliding with them. With each collision there is a change in momentum of the colliding molecule. The rate of momentum change is a force, and pressure is the force per unit area. Knowing the y-component of the velocity and the mass of a molecule, you can calculate the change in momentum for each collision. Knowing the length of the box and v y, you can also find the time between collisions. This information coupled with the change in momentum for

4 4 of 6 Thermodynamics Summer 2006 each collision will allow you to calculate the impulsive force each molecule exerts on the wall as it collides with it. Knowing the area of a wall, you can find the pressure. Compare the pressure you calculated with that displayed on the screen. They should be close to one another. Ideal Gas and Absolute Zero Most gases that we encounter can be considered ideal. To be considered ideal the molecules and atoms in the gas are modeled as point particles that interact only through elastic collisions with each other and their surroundings. (A plasma can t be considered ideal because the particle in it are charged and will therefore exert electrical forces on each other at a distance.) Although the requirements to qualify as an ideal gas are placed on the individual atoms and molecules, we treat the gas as a homogenous whole and deal with quantities (pressure, temperature and volume) of the bulk gas. The ideal gas law can be stated by the equation: PV = nrt, (Eqn. 1) Where P is the pressure, V the volume, n the number of moles of gas, R the gas proportionality constant ( liter-atm/mole-deg), and T the temperature. In dealing with an ideal gas it s common practice to hold the pressure, temperature or volume constant, and vary the other two variables. In this experiment we will hold the volume constant and watch how the pressure varies with temperature. As n and R are also constant, the pressure ought to vary linearly with the temperature. Both the pressure and the temperature in the ideal gas law are absolutes. This is so because the pressure and volume of an ideal gas are predicted to be zero when the temperature is zero. This obviously isn t so with the Fahrenheit and Centigrade temperature scales. Absolute zero is a very difficult temperature to achieve in the laboratory, but we can point to it (extrapolate) using hot water and the ideal gas law. Procedure: Set up the equipment as shown in the photo below. Do not plug in the immersion heater until the beaker has been filled with water. Be careful that you don t overfill the

5 5 of 6 Thermodynamics Summer 2006 beaker. The water level in the beaker will rise considerably when you immerse the gage s bulb. The immersion heater should be lowered into the beaker so that it is completely surrounded by water without touching the sides of the beaker or the copper bulb, but don t plug it in yet. The end of the thermocouple should also be surrounded by water without touching anything else in the beaker. Turn the thermocouple thermometer on, and set it so that it reads in degrees Centigrade. The pressure gage reads relative pressure, not absolute pressure. It reads relative to atmospheric pressure, which you may assume is 1.01x10 5 N/m 2. Because of the good heat conduction and the large surface area of the copper bulb, you can assume that the water temperature is the same as the gas temperature in the bulb. Open the valve of the pressure gage and place the bulb in the water, completely immersing it. After a couple of minutes close the valve. The gage should read zero, but remember that this is a relative pressure, and that the absolute pressure is actually one atmosphere. You are now ready to take data. Constant Volume Pressure Gage Water Bath with Immersion Heater Thermocouple Thermometer Plug in the immersion heater and record the gage pressure and temperature every five degrees as the temperature rises. Let the temperature rise from around room temperature

6 6 of 6 Thermodynamics Summer 2006 to approximately 60 0 C. Record the temperature and pressure for every 5 0 C rise. Because of the thermal inertia of the water, this could take some time. Plot the temperature in degrees Centigrade on the vertical axis, and the absolute pressure in N/m 2 on the horizontal axis. The plot should be linear. Obtain an equation of the line and place it on the graph. The y-intercept should be absolute zero ( K).

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