Molar Volume and the Universal Gas Constant
Experiment 54 Molar Volume and the Universal Gas Constant Problem How can the value of the universal gas constant be verified experimentally?
Introduction There are a number of gas laws described in Chapter 13. Some, like the laws of Avogadro, Boyle and Charles, concern how one of the four variables – pressure, volume, absolute temperature, and number of moles – varies when another is changed, with two of the four being held constant. Two other relationships are of a more general nature. One, the Combined Gas Law, represents a combining of the simpler laws. Given that volume is inversely proportional to pressure (Boyle's Law), and directly proportional to temperature (Charles' Law), you can write an expression that incorporates both sets of observations: P1V1/T1 = P2V2/T2 Going one step further and introducing the law of Avogadro, which says that the volume occupied by a gas is directly proportional to the number of moles of gas in the system, gives the expression P1V1/ n1T1 = P2V2/n2T2 In which P1, V1, n1 and T1 refer to an initial set of conditions and P2, V2, n2 and T2 refer to the final conditions, after the change has occurred. This "combined" gas law allows you to do calculations involving changes in any or all of the four variables. What the equation tells us is that the value of the expression, PV/nT, does not change; if that is the case, then we can determine a value for PV/nT that should be valid under any set of conditions as long as the sample remains a gas. In other words we can state that PV/nT = constant = R This relationship (PV/nT = R ) can be rearranged to the form known as the Ideal Gas Law: PV = nRT The constant, R, is known as the universal gas constant. One of your objectives in this experiment is to determine the value of R experimentally. Notice that the units of R must reflect pressure times volume, divided by number of moles times temperature. R is most often expressed in units of L atm/mol·K (“liter-atmospheres per mole-kelvin”). In this experiment you will calculate the value of R for each of three trials, as well as an average result which will then be used to determine your percentage error. You will first determine the molar volume of a gas. You will then use the molar volume at laboratory conditions to determine what volume one mole of gas would occupy at STP (0°C and one atmosphere pressure). You can make this conversion by using the combined gas law. Once that is
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done, you will use your experimental values of P, V, n, and T to calculate an experimental value for R. As with the molar volume calculation, you will determine individual values for each trial, along with an average value, which you will compare with the accepted value of R: 0.0821 L atm/mol.K.
Prelaboratory Assignment Read the Introduction and Procedure before you begin. Answer the Prelaboratory Questions. 1. Explain how the mass of a piece of magnesium ribbon may be calculated from its length and the mass of a 100.0-cm strip of the same type of ribbon. 2. What two physical properties of hydrogen gas make it possible for you to collect it by displacement of water in your graduated cylinder? 3. Use Dalton’s Law of Partial Pressures to explain why the pressure of hydrogen gas in the cylinder will be less than the observed barometric pressure in the laboratory. How will you determine the pressure of the hydrogen you produce? 4. Calculate the mass of baking soda, NaHCO3, needed to neutralize 3.0 mL of 3.0 M hydrochloric acid, HCl(aq). Show your work; the answer alone is not enough. (Hint: the amount needed is less than 1 gram.) 5. The accepted value for the universal gas constant is 0.0821 L atm/mol.K. What would it be if the pressure was measured in torr and the volume in milliliters? Show your calculations.
Materials Apparatus 10-mL graduated cylinder #00 1-hole rubber stopper Copper wire, 10 cm length Thermometer Safety goggles Lab apron Reagents Magnesium ribbon, 3 pcs, ~0.9 cm 3 M HCl (aq) NaHCO3(s) (baking soda)
Safety 1. Wear safety goggles and a lab apron at all times in the laboratory.
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2. Hydrochloric acid is corrosive to skin and clothing. Clean up all spills thoroughly. Note:
If acid spills on the lab bench, use a bit of baking soda to neutralize it before cleaning up. Do not neutralize acid that spills on skin or clothing; flood the affected area with water.
3. If you are using a mercury thermometer; be very careful. Mercury vapor is very poisonous. If the thermometer breaks, notify your teacher right away.
Procedure 1. Fill a 400-mL beaker about three-fourths full of tap water; the water should be at or near room temperature. 2. Obtain a short (0.80–1.00 cm) piece of magnesium ribbon, then measure and record its length to the nearest 0.01 cm. Use a 10-cm piece of copper wire to make a cage for your magnesium, by folding the magnesium over the wire then rolling the wire around the magnesium. Fit the wire cage into a 1-hole #00 rubber stopper. The cage should be about 2–3 cm from the small end of the stopper to hold the magnesium in place. See Figure 1.
Figure 1 3. Carefully pour about 3 mL of 3 M HCl(aq) into a 10-mL graduated cylinder. Using a wash bottle or beaker, carefully add distilled or deionized water to the graduated cylinder until it is completely full. Try to direct the water down the side of the graduated cylinder to prevent mixing of the water and the acid. Insert the stopper assembly (see Fig. 1, above) into the top of the graduated cylinder. Water should escape through the hole in the stopper; if it does not, remove the stopper and carefully add more water, then replace the stopper assembly. This will keep air from being trapped in the graduated cylinder. 4. Place your finger over the hole in the stopper and invert the graduated cylinder, lowering it into the beaker of water. Remove your finger when the stopper is below the level of water in the beaker. The hydrochloric acid is more dense than pure water, so it will slowly sink toward the stopper and the magnesium; observe and record evidence of reaction. (See Figure 2.)
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Figure 2 5. When the reaction is complete, allow the system to stand for two or three minutes, tapping the sides of the graduated cylinder to dislodge any gas bubbles that may be clinging to the glass wall. Make sure that there are no little pieces of unreacted magnesium on the wall of the graduated cylinder. If a small piece remains, gently shake the graduated cylinder up and down to wash the metal back into the acid solution, allowing it to finish reacting. Be careful not to lift the graduated cylinder completely out of the water in the beaker. 6. Adjust the position of the graduated cylinder so that the water levels inside and out are even; this ensures that the total pressure on the gases inside the graduated cylinder is the same as the barometric pressure in the room. Record the volume of gas trapped in the graduated cylinder, and the temperature of the water near the mouth of the graduated cylinder. The temperature of the escaping solution may be assumed to be the same as the temperature of the trapped gases. Enter these data in the Data Table.
Cleaning Up 1. Take apart the apparatus. Dispose of the copper wire in the solid waste container or as your teacher directs. 2. Use baking soda to neutralize the acidic solution remaining in the beaker. The neutralized solution can be flushed down the drain safely. As part of the Prelaboratory Assignment, you calculated the mass of baking soda needed. Use a plastic spoon to add approximately that amount of the solid (a little at a time to minimize foaming) to the beaker. Stir well and flush the solution down the drain. 3. Clean all glassware and return it to its proper location. 4. Wash your hands before leaving the laboratory.
Analysis and Conclusions Complete the Analysis and Conclusions section for this experiment either on your Report Sheet or in your lab report as directed by your teacher.
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All of the calculations are to be shown for your first trial; you may simply report the results for the other two. Enter the results for all trials in a Summary Table. 1. Calculate the mass of magnesium for each trial, using the mass of 1.00 m of magnesium ribbon supplied by your instructor. 2. Calculate the number of moles of magnesium used. This is the same as the number of moles of hydrogen generated. (Why?) 3. Because you collected hydrogen over water, a small portion of the gas in the cylinder at the end of the reaction is water vapor; we say the hydrogen gas is “wet.” The amount of water that evaporates is dependent only on the temperature, so it is a simple matter to determine the partial pressure of the water vapor in the graduate. Use the table of vapor pressures, found in Appendix A of your lab manual, to find the pressure due to water in the graduated cylinder. Subtract this value from the barometric pressure to get the pressure exerted by the “dry” hydrogen (hydrogen without the water vapor). Be sure to report the pressure to the correct degree of precision. Enter the results in Summary Table 1. 4. You have measured the volume occupied by a very small fraction of a mole of hydrogen, under a specific set of conditions of pressure and temperature. The volume occupied by one mole of gas is called the molar volume of the gas, and it is the same for all gases (behaving ideally) at a particular pressure and temperature. For each trial: a. Calculate the volume that 1.00 mole of hydrogen would occupy at your experimental temperature and pressure (called “laboratory conditions”). Record your answers in the Summary Table. b. Use the combined gas law to calculate the volume that 1.00 mole of hydrogen would occupy at 1.00 atm and 273 K (Standard Temperature and Pressure, STP). 5. Determine the value of PV/nT for each trial. P is the pressure of dry hydrogen, V is the volume of gas collected, T is the Kelvin temperature, and n is the number of moles of hydrogen generated. 6. Determine the average of your three values for PV/nT. Also determine the deviation for each of your three trials and the average deviation. 7. Your average for PV/nT represents your experimental value for the universal gas constant, R. Calculate the percent error in your determination of the value of R.
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Report for Experiment 54
Name___________________________ Section___________ Date___________
Molar Volume and the Universal Gas Constant Prelaboratory Questions 1. Explain how the mass of a piece of magnesium ribbon may be calculated from its length and the mass of a 100.-cm strip of the same type of ribbon.
2. What two physical properties of hydrogen gas make it possible for you to collect it by displacement of water in your graduated cylinder?
3. Use Dalton’s Law of Partial Pressures to explain why the pressure of hydrogen gas in the cylinder will be less than the observed barometric pressure in the laboratory. How will you determine the pressure of the hydrogen you produce?
4. Calculate the mass of baking soda, NaHCO3, needed to neutralize 3.0 mL of 3.0 M hydrochloric acid, HCl(aq). Show your work; the answer alone is not enough. (Hint: the amount needed is less than 1 gram.)
5. The accepted value for the universal gas constant is 0.0821 L.atm /mol.K. What would it be if the pressure was measured in torr and the volume in milliliters? Show your calculation.
Data/Observations
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Mass of 100. cm of Mg ribbon
__________ g
Barometric pressure atm
__________ atm
Experimental Data
Trial 1
Trial 2
Trial 3
Length of Mg ribbon
__________ cm
__________ cm
__________ cm
Volume of hydrogen collected
__________ mL
__________ mL
__________ mL
Temperature of water
__________ °C
__________ °C
__________ °C
Vapor pressure of water at experimental temperature
__________ atm
__________ atm
__________ atm
Analysis and Conclusions All of the calculations are to be shown for your first trial; you may simply report the results for the other two. Enter the results for all trials in the Summary Tables. 1. Calculate the mass of magnesium for each trial, using the mass of 1.00 m of Mg ribbon supplied by your instructor. Show your calculations here for Trial 1 (only) and enter the results in Summary Table 1.
2. Calculate the number of moles of magnesium used; enter the value for each trial in Summary Table 1. This is the same as the number of moles of hydrogen generated. (Why?)
3. Because you collected hydrogen over water, a small portion of the gas in the cylinder at the end of the reaction is water vapor; we say the hydrogen gas is “wet.” The amount of water that evaporates is dependent only on the temperature, so it is a simple matter to determine the partial pressure of the water vapor in the graduate. Use the table of vapor pressures, found in Appendix A of your lab manual, to find the pressure due to water in the graduated cylinder. Subtract this value from the barometric pressure to get the pressure exerted by the “dry” hydrogen (hydrogen without the water vapor). Be sure to report the pressure to the correct degree of precision. Enter the results in Summary Table 1.
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4. You have measured the volume occupied by a very small fraction of a mole of hydrogen, under a specific set of conditions of pressure and temperature. The volume occupied by one mole of gas is called the molar volume of the gas, and it is the same for all gases behaving ideally at a particular pressure and temperature. For each trial: a. Calculate the volume that 1.00 mole of hydrogen would occupy at your experimental temperature and pressure (called “laboratory conditions”). Record your answers in Summary Table 1.
b. Use the combined gas law to calculate the volume that 1.00 mole of hydrogen would occupy at 1.00 atm and 273 K. (STP). Record your results in Summary Table 1.
Summary Table 1 Trial 1
Trial 2
Trial 3
Mass of magnesium ribbon used
__________g
__________ g
__________g
Moles of magnesium used
__________mol
__________ mol
__________mol
Pressure of dry hydrogen
__________ atm
__________ atm
__________ atm
__________ L/mol
__________ L/mol
__________L/mol
__________ L/mol
__________ L/mol
__________L/mol
Molar volume of H2 (lab conditions) Molar volume of H2 (STP)
5. Determine the value of PV/nT for each trial. P is the pressure of dry hydrogen, V is the volume of gas collected, T is the Kelvin temperature, and n is the number of moles of hydrogen generated. Enter these values in Summary Table 2.
6. Determine the average of your three values for PV/nT and enter this in the last box of the first column of Summary Table 2. In similar fashion, determine the deviation for each of your three trials
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and the average deviation.
Summary Table 2 PV/nT
Deviation
Trial 1
______________
______________
Trial 2
______________
______________
Trial 3
______________
______________
Averages
______________
______________
7. Your average for PV/nT represents your experimental value for the universal gas constant, R. Calculate the percent error in your determination of the value of R.
______________ % error
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Experiment 54 Molar Volume and the Universal Gas Constant Introduction Intent This small-scale version of a lab that may originally have been created by Michael Faraday, allows the student to do three trials of the molar volume determination, then use the data to calculate an experimental value for the Universal Gas Constant, R. Objectives 1. To measure the volume of gas generated by complete reaction of a piece of magnesium ribbon, of known length. 2. To calculate the molar volume of the gas under experimental conditions and at STP for each of three trials. 3. To calculate an average value for the Universal Gas Constant, R, based on experimental data. 4. To compare the experimentally-determined value with the accepted value.
Materials (for each lab team of 2 students) Apparatus 10-mL graduated cylinder #00 1-hole rubber stopper Copper wire, 10 cm length Thermometer, −10° to +110°C Safety goggles (2) Lab apron (2) Reagents Magnesium ribbon, 3 pcs, ~0.9 cm each 3 M HCl (aq), 10 mL NaHCO3(s) (baking soda), 2 g
Preparation Hints Use #0000 steel wool to clean the magnesium as well as possible, then cut a piece that is exactly 1.00 meter in length, and weigh it. Give students the mass of the 1-meter piece (this is the linear density). Cut the piece into lengths of about 3 cm; students can make the final cuts themselves. One 3-cm piece per lab group, plus a couple of extras, will do for a class, so the 1-meter length will serve two sections. The 3 M
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HCl is most conveniently dispensed in dropper bottles; 6 bottles of 30-mL each will easily do for two sections. Several groups can share the same acid bottle. Place several small (1 lb) boxes of baking soda around the lab. 3 M hydrochloric acid, HCl(aq): Add 250 mL concentrated acid to 750 mL water.
Prelaboratory Discussion The experiment is written to serve either as an introduction to the universal gas constant, or as a verification of its value. Leave most of the theoretical discussion for the post-lab, concentrating on the mechanics of the procedure. Show students the apparatus, emphasizing that the graduate is to be completely filled (not just to the 10-mL line) with water after the acid is added. Water should come out of the hole when the stopper is inserted in the cylinder. Explain that the acid will drift down to the trapped piece of magnesium, where they will react. Explain the need to get the water levels at least close to equal (It is best to have the inner level about 0.5 cm above the outer level, for ease of reading. This small difference has no real effect on accuracy). Remind students that the contents of the beaker will be acidic once the reaction is complete.
Procedure Hints 1. Keep track of students' first volumes of gas; many will be over 10.0 mL. Suggest slightly shorter pieces of Mg. Help them estimate the volume for the first trial; if there is time, they can do a fourth. 2. Although students have calculated the mass of baking soda needed to neutralize the system, it is not necessarily time-efficient for them to actually measure the amount (about .76 g). If they just stir half a plastic teaspoon-full into the beaker, that will suffice. 3. The most challenging part of the procedure is trapping the magnesium piece in the copper wire. Have the student fold the Mg over the wire then wrap the wire a couple of turns around the folded Mg. 4. When the magnesium escapes the copper wire, the students must watch to see that it doesn't become trapped on the wall of the graduate, above the water line. If it does, have the students shake the graduate vertically. This should wash the escaped metal back into the acid solution.
Disposal Students neutralize the contents of the beaker in Cleaning Up; no further precautions are needed.
Postlab Discussion It is almost always necessary to walk the class through a sample round of calculations, such as the ones shown in the Report Sheet that accompanies this guide. As you do, explain the purpose of making the water levels (almost) equal, inside and out, and of adjusting the pressure to correct for water vapor (The bulk of the text's discussion of vaporization is found in Chapter 14, Liquids and Solids).
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Question 4a has the students adjust the number of moles from the experimental value (about 4 × 10−4) to 1.00 mole; 4b has them adjust to STP. No mention is made of the fact that 22.4 L/mol is the accepted molar volume at STP; you may want to incorporate that into your discussion. Illustrate the use of experimental values of n, P, V, and T to calculate a value for R. Although they are not directed to do so, many will change the pressure to atm so that they can compare their result with 0.0821 atm L/mol K. If they do not, the accepted value is 62.4 torr L/mol K, as they can deduce from the final Prelaboratory Question. Anticipate values for R that are within 10% of the accepted value; many will be closer. More students report high, rather than low values.
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Report for Experiment 54
Name___________________________ Section___________ Date___________
Molar Volume and the Universal Gas Constant Prelaboratory Questions 1. Explain how the mass of a piece of magnesium ribbon may be calculated from its length and the mass of a 100.-cm strip of the ribbon. The length of the small piece is multiplied by the value provided for the mass of 100.0 cm. 2. What two physical properties of hydrogen gas makes it possible for you to collect it by displacement of water in your graduated cylinder? It is insoluble in water. It is less dense than water. 3. Use Dalton’s Law of Partial Pressures to explain why the pressure of hydrogen gas will be less than the observed barometric pressure in the laboratory. How will you determine the pressure of the hydrogen you produce? The total pressure in the graduate is equal to the barometric pressure because the water levels are equal, inside and out. The total pressure, according to Dalton's Law is equal to the sum of the pressures of hydrogen and water vapor, so the hydrogen pressure is less than the barometric pressure by the amount of pressure from H2O(g). 4. Calculate the mass of baking soda, NaHCO3, needed to neutralize 3.0 mL of 3.0 M hydrochloric acid, HCl(aq). Show your work; the answer alone is not enough (Hint: the amount needed is less than 1 gram). 0.0030L × 3.0 mol HCl/L × 1 mol NaHCO3/1 mol HCl × 84.0 g NaHCO3/mol = 0.76 g NaHCO3 5. The accepted value for the universal gas constant is 0.0821 atm.L/mol.K. What would it be if the pressure was in torr, and the volume in milliliters? Show your calculations. 0.0821 atm.L/mol.K × 760 torr/1 atm × 1000 mL/1 L = 62,400 torr.mL/mol.K
Data/Observations Mass of 100. cm of Mg ribbon
_____0.970 g
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Barometric pressure atm
_____0.976 atm
Experimental Data
Trial 1
Trial 2
Trial 3
Length of Mg ribbon
0.94 cm
__________ cm
__________ cm
Volume of hydrogen collected
9.50 mL
__________ mL
__________ mL
Temperature of water
210 °C
__________ °C
__________ °C
__________ atm
__________ atm
Vapor pressure of water at experimental temperature
0.025 atm
Analysis and Conclusions All of the calculations are to be shown for your first trial; you may simply report the results for the other two. Enter the results for all trials in the Summary Tables. 1. Calculate the mass of magnesium used in each trial, using the mass of 1.00 m of ribbon, supplied by your instructor. Show your calculations here for Trial 1 (only) and enter the results in Summary Table 1. 0.92 cm × 0.970 g/100. cm = 0.00912 g Mg (The subscript in 0.00912 indicates that the 2 is not a significant digit).
2. Calculate the number of moles of magnesium used; enter the value for each trial in Summary Table 1. This is the same as the number of moles of hydrogen generated (Why?). 0.0091 g × 1 mol Mg/24.31 g Mg = 3.75 × 10−4 = 3.8 × 10−4 mol Mg The equation shows that 1 mol Mg produces 1 mol H2, so mol H2 = mol Mg
3. Because you collected hydrogen over water, a small portion of the gas in the cylinder at the end of the reaction is water vapor; we say the hydrogen gas is “wet.” The amount of water that evaporates is dependent only on the temperature, so it is a simple matter to determine the partial pressure of the water vapor in the graduate. Use the table of vapor pressures, found in Appendix A of your lab manual, to find the pressure due to water in the graduated cylinder. Subtract this value from the barometric pressure to get the pressure exerted by the “dry” hydrogen (hydrogen without the water vapor). Be sure to report the pressure to the correct degree of precision. Enter the results in Summary Table 1. At 21°C, the vapor pressure of hydrogen is 0.025 atm. The barometric pressure is 0.996 atm, so: P(H2) = 0.996 − 0.025 = 0.971 atm
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4. You have measured the volume occupied by a very small fraction of a mole of hydrogen, under a specific set of conditions of pressure and temperature. The volume occupied by one mole of gas is called the molar volume of the gas, and it is the same for all gases at a particular pressure and temperature. For each trial: a. Calculate the volume that 1.00 mole of hydrogen would occupy at your experimental temperature and pressure (called laboratory conditions). Record your answers in Summary Table 1. P and T are both held constant, so the combined gas law simplifies to: V1/n1 = V2/n2: 0.00950 L/3.8 × 10−4 mol = V2/1.00 mol V2 = 0.00950 L × (1.00 mol/3.8 × 10−4 mol) = 25 L/mol
b. Use the combined gas law to calculate the volume that 1.00 mole of hydrogen would occupy at 1.00 atm and 273 K. Record your results in Summary Table 1.
V2 = 22.5 L = 23 L/mol at STP
Summary Table 1 Trial 1
Trial 2
Trial 3
Mass of magnesium ribbon used
0.0091 g
__________ g
__________g
Moles of magnesium used
3.8x10−4 mol
__________ mol
__________mol
Pressure of dry hydrogen
0.971 atm
__________ atm
__________ atm
Molar volume of H2 (lab conditions) Molar volume of H2 (STP)
25
L/mol
__________ L/mol
__________L/mol
23
L/mol
__________ L/mol
__________L/mol
5. Determine the value of PV/nT for each trial; P is the pressure of dry hydrogen, V is the volume of gas collected, T is the Kelvin temperature, and n is the number of moles of hydrogen generated. Enter these values in Summary Table 2.
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6. Determine the average of your three values for PV/nT, and enter this in the last box of the first column of Summary Table 2. In similar fashion, determine the deviation for each of your three trials and the average deviation. Summary Table 2 Value for PV/nT
Deviation
Trial 1
______________
______________
Trial 2
______________
______________
Trial 3
______________
______________
Averages
______________
______________
7. Your average for PV/nT represents your experimental value for the universal gas constant, R. Calculate your percentage error in the determination of R.
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