Experimental Determination of Molecular Speeds Stephen Luzader Frostburg State University Frostburg, MD

2 Outline of topics Purpose of the experiment A schematic representation of the apparatus Some theory assuming the molecules are all moving in one direction A correction for motion in all directions Some data for our particular apparatus How to analyze the data

3 Purpose The purpose of the experiment is to verify kinetic theory predictions of how the average speed of gas molecules depends on the mass of the molecules

4 The Basic Apparatus P reservoir V reservoir N reservoir PVNPVN Molecules move from a reservoir to a chamber that is initially empty (P = 0). They flow through a small opening with area A. The gas in both chambers is at a constant temperature T.

5 P reservoir V reservoir N reservoir PVNPVN Some theory assuming the molecules are all moving from left to right. In the right hand chamber, the ideal gas law gives the relationship between the number of molecules N and the pressure P:

6 V and T are constant, so as molecules flow into the chamber, the pressure increases. The rate of change of the pressure is proportional to the rate of change of the number of molecules: If the number molecules in the chamber increases by dN, the number of molecules in the reservoir must decrease by the same amount.

7 P reservoir V reservoir N reservoir PVNPVN If all the molecules in the reservoir are moving to the right at some average speed v, the number of molecules that pass through the opening during time dt is where is the density of molecules in the reservoir.

8 We can substitute this expression into the one for the rate of change of the pressure in the chamber to find a relation between the rate of change of pressure and the average speed of the molecules:

9 We can use the ideal gas law to express the density of molecules in terms of the temperature and pressure of the gas in the reservoir: This gives very simple relation between the rate of change of the pressure in the chamber and the average speed:

10 The equation we just derived is wrong because it was based on the unrealistic assumption that all the molecules are moving in the same direction. A more detailed analysis that takes the random motions of the molecules into account gives the correct result: If we know V, A, P reservoir, and can measure the rate of pressure increase, we can calculate a value for v.

11 In our experiment, we know the following quantities: V = 60 cc plus the volume of the associated tubing P reservoir = atmospheric pressure, which we measured (units!) is determined from the experimental graph of P vs t. (units!)

12 We also need the following information about the apparatus itself, which is provided by the manufacturer: Diameter of pinhole = 12.5  m  25%

13 For your analysis, you must carry out the following steps. 1.Prepare a table with all common data, including atmospheric pressure, room temperature, data for tubing volume, hole diameter (with uncertainty) 2.Show the calculation of the total volume. 3.Calculate the area of the pinhole, including its uncertainty. 4.Tabulate the measured values of the rate of change of pressure (determined by LoggerPro). This table must include an identification of each gas. 5.Calculate experimental values of v for each gas, including uncertainty.

14 To test whether the experimental results agree with predictions made by kinetic theory, do the following. 1.Calculate expected values for v using the measured value of room temperature. These results must be in a table which includes the molecular mass of the gas. 2.Compare the experimental values of v (including uncertainty!) with the predictions. Account for discrepancies. 3.Compare the ratios of the experimental values of v with the ratios of the molecular masses of the gases and explain whether these ratios agree with predictions from kinetic theory. Again, you should consider experimental uncertainty.