Objective: Boyle’s Law Verify the linear relationship between pressure & inverse volume for a fixed quantity of gas. Background: As described in "Determination of Absolute Zero", kinetic theory can be used to derive the ideal gas law, PV = nRT . When written this way, pressure is given in atmospheres, volume in liters, and quantity of gas in moles. Prior to the development of this more complete theory, relationships where one variable was held constant had been developed experimentally. In 1662, Robert Boyle (1627-1691) presented an experimental confirmation of the inverse relationship of pressure and volume for a fixed quantity of gas at constant temperature, now known as Boyle’s law. æ1ö æ1ö V = nRT ç ÷ (1) P = nRT ç ÷ (2) or èPø èV ø As long as the quantity of gas and the temperature remain fixed, the relationship between one quantity and the inverse of the other quantity as given in equations (1) and (2) should be a straight line with slope equal to nRT (in appropriate units). Confirmation of Boyle’s Law In this lab we will seal a fixed amount of gas in a syringe. As we vary the pressure by hanging masses from the syringe, we will examine the relationship between V and 1 and P test whether Boyle’s law provides an appropriate model for our measurements. We will compute the quantity of gas from our graphical relationship between V and 1 . We will P compare the amount of gas determined from this relationship with the amount of gas expected in the initial volume under the conditions present in the room. We will do this for two different initial volumes and quantities of gas. ( ) ( ) Equipment: • • • • • • • 10 mL and 20 mL plastic syringes with hanger & rubber band silicone grease beaker clamp, rod and table clamp. 1 50 g slotted mass, 5 – 10 100 g slotted masses, 500 g and 1 kg masses vernier caliper thermometer to measure room temperature room barometer (shared) to measure atmospheric pressure Procedure (even though I gave you data, read this! It will help) We can seal a fixed volume of gas inside a plastic syringe. Initially, the pressure inside will equal the atmospheric pressure in the room, measurable on the wall barometer. We will hang the syringe by its handles from a test tube clamp. Once the syringe is sealed, we can examine the relationship between pressure and volume by increasing the pressure inside the syringe. We do this by hanging weights from the plunger. When making the measurements, record both an upper and lower estimate for the volume (except for the initial volume, which does not need upper and lower limits). You will need both of these measurements, not just the average, so that you can compute an uncertainty. 1. Remove the plunger from each syringe & measure the inside diameter. Grease (if necessary) and replace the plunger. 2. Suspend the 10 mL plastic syringe from the beaker clamp. The “wings” of the syringe should sit on top of the beaker clamp. The syringe should be held firmly but not compressed by the clamp. Clamp the entire rod assembly to the table. 3. Wrap the rubber band around the top of the weight hanger tightly enough to keep the hanger from falling out when weights are suspended from it. 4. Depress the plunger until it reads 10 mL and then clamp the hose as close to the end of the syringe as possible. You now have a fixed volume of gas in the cylinder, with no masses suspended from the hanger. This is your initial volume to be used in your analysis! Note: there is no need for “upper” and “lower” measurements for this first value. Read the volume as shown in figure 2. Figure 1 5. Hang increasing masses from the hanger in steps of 100 g starting from 100 g to at least 2 kg or when the pressure becomes too great to be held in the syringe. You may go to higher masses. 6. At each mass, determine the volume by pulling lightly up on the plunger and releasing and also by slightly compressing the plunger and releasing. Record both values of volume. You will need both measurements to compute an uncertainty. [Do not do this for the first, 10 mL measurement.] 7. Repeat the experiment with the 20 mL syringe. Use the hardware from the 10 mL syringe (including the small plastic tube that holds the hanger). Remember that the lower end must be open when setting the initial volume 8. Begin with a volume of between 15 mL to 20 mL and use a range of masses from 200 g to 4 kg in steps of 200 g. 9. Make sure you record uncertainty estimates for all measured variables. Measure here. Not at the tip Figure 2 Data Analysis 1. For each syringe, average the volume measurements at each mass (starting from the initial volume, for which there is only one measurement) and convert to m3. 2. Use the amount of hanging weight to compute the pressure for each trial in Pascals, using the measured mass, g = 9.80 m/s2, and the measured diameter of the syringe. (HINT: pressure is Force / Area. ) Be careful of your units. You must add the room pressure (converted from mm Hg) to each value of pressure. Show a sample calculation for each syringe, but a spreadsheet program is recommended for the bulk of the calculations. ( P ) (on the x axis). Use the average value of 3. For each syringe, plot V (on the y axis) vs. 1 V. Make a separate plot for each syringe, using an entire page for each. For error bars, use the upper and lower values of V, as measured in step 6, above. 4. Draw lines of best fit through each data set. If you do this by hand, carefully plot the straight line which does the best job of going through all the data points. Alternatively, just have the spreadsheet compute the best fit. 5. Derive n (the number of moles in the gas) from the slope of the curve. Hint: look at equation (1). What is the slope of the line equal to? 6. Use the ideal gas law to compute the number of moles expected in the given initial volume of gas at room temperature and pressure. Error Analysis 1. For each graph plot the steepest and shallowest lines that are consistent with the data: a. These lines should deviate as much as possible from the best-fit line without going outside the error bars (if possible), as shown in the example graph here b. They should NOT be parallel lines at maximum and minimum height, but lines of maximum and minimum slope. Max Slope c. Use only the points consistent with a straight line. YES! Best Fit 2. Calculate the maximum and minimum slope from these lines Slope Min J a. Using the maximum and minimum slopes, calculate Slope the maximum and minimum value of n (nmax and nmin). b. Divide the difference between nmax and nmin by 2, and that is your uncertainty, σn. c. Make sure you show how you did this in the sample calculations. NO! Best Fit Slope 3. If you plotted and fit the curve using a spreadsheet program, L you still may may have to fit maximum and minimum slope lines by hand. (if you’re really good you can do it on the spreadsheet) Conclusions In your conclusions you should discuss whether the value nRT is a constant for each syringe, based on whether the data can be fit by a straight line. You have two estimates for n for each syringe, one based on the slope and one based on the ideal gas law for the uncompressed syringe (which is the initial volume of either 10 mL or 20 mL). Discuss whether these two estimates of n are within the uncertainty derived from the maximum and minimum slope lines. As always, discuss sources of error. For those variables where you are able to estimate an uncertainty in the measurement, be sure to list the value and explain how you chose it. Do this even if you do not include the uncertainty in your calculation. What additional sources of error might reasonably contribute that you do not have numerical estimates for? Something to consider: because of the crude method for closing the syringes, there may be some extra air trapped in the tube. How does this affect your results? Explain. Summary Table Your summary table should look like this: n from slope 10 mL syringe 20 mL syringe n ± σn n from initial volume | nn – niv| n Note: if the two different estimates for each syringe (above) are very different, you have almost certainly made a calculation error. Stop and find it before turning your lab in! Every number in this table should be shown with the same exponent, even if that is not standard scientific notation. Boyle's Law Lab Data Atmospheric Pressure 766 mmHg 10 mL Syringe Diameter 15.5 cm 20 mL Syringe Diameter 18.0 cm Room Temperature (milimeters of Mercury) 22.0 Celcius Data for 10 mL Syringe Data for 20 ml Syringe uncompressed 10.0cm^3 uncompressed 20.0cm^3 M (kg) V_low (cm^3) V_high (cm^3) M (kg) V_low (cm^3) V_high (cm^3) 0.100 9.41 9.88 0.200 17.60 19.60 0.200 9.01 9.50 0.400 16.41 18.44 0.300 8.40 8.91 0.600 15.30 17.20 0.400 8.20 8.71 0.800 14.38 16.18 0.500 7.86 8.33 1.000 13.77 15.07 0.600 7.56 7.98 1.200 12.96 14.36 0.700 7.30 7.65 1.400 12.17 13.77 0.800 6.92 7.28 1.600 11.84 12.84 0.900 6.68 7.18 1.800 11.43 12.13 1.000 6.50 7.05 2.000 11.00 11.55 1.100 6.05 6.90 2.200 10.34 11.24 1.200 5.65 6.83 2.400 10.05 10.72 1.300 5.62 6.47 2.600 9.62 10.33 1.400 5.33 6.40 2.800 9.13 10.03 1.500 5.10 6.28 3.000 8.69 9.79 1.600 5.00 6.06 3.200 8.40 9.25 1.700 4.80 5.70 3.400 8.22 9.02 1.800 4.80 5.75 3.600 8.05 8.62 1.900 4.62 5.57 3.800 7.61 8.51 2.000 4.40 5.60 4.000 7.30 8.15
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