3.2.1 Tasks For a constant amount of gas (in our case air) investigate the correlation between 1. Volume and pressure at constant temperature (Boyle-Marriotte s law) 2. Temperature and volume at constant pressure (Gay-Lussac s law) 3. Temperature and pressure at constant volume (Charles (Amonton s) law) From the correlations obtained calculate the universal gas constant as well as the coefficient of thermal expansion, the coefficient of thermal tension, and the coefficient of compressibility. Duration: approx. 1.5 hours Equipment Gas laws apparatus 4362. 1 Immersion thermostat TC1 8492.93 1 Accessory set for TC1 8492.1 1 Bath for thermostat, Makrolon 8487.2 1 Electronic weather station 87997.1 1 Lab thermometer, +11 C 3856. 1 Mercury tray 285. 1 Pinchcock, width 15 mm 43631.15 1 Support base PASS- 25.55 1 Support rod, stainl. Steel, l 1 mm 234. 1 Right angle clamp 37697. 2 Universal clamp 37715. 2 Hose clip, d 82 mm 4996.1 6 Rubber tubing, i.d. 6 mm 39282. 3 Mercury, filtered, 1 g 31776.7 1 Water, distilled, 5 l 31246.81 1 Setup During the installation of the experiment, the mercury is filled into the gas laws apparatus. This has to be done only once! Fill the mercury reservoir of the demonstration device (right tube) carefully up to about one quarter of the graduated measuring range. It is very important to read the operating instructions for correct performance. The levels of the mercury in the mercury reservoir and in the leveling container must be the same. Set up the experiment according to the following instructions and pictures: - Fill the bath for thermostat with distilled or demineralised water - Put the thermostat into the bath for thermostat and fix it at the backside with the screw (Fig. 1) Fig. 1 - Connect one end of a rubber tubing to the upper part of the left tube (measuring tube) and the other end of the tubing to the left jacket pipe of the thermostat (Fig. 2 and 3). Use the hose clips for increasing the security of the connections. Laboratory Experiments Phywe Systeme GmbH & Co. KG All rights reserved P2321 www.phywe.com 1
3.2.1 Equation of state of ideal gases Fig. 2 Fig. 3 - Now connect one end of another rubber tubing to the lower part of the measuring tube and the other end of it to the right jacket pipe of the thermostat (Fig. 4 and 5). Again make sure, that these connections are secure Fig. 4 Fig. 5 - Connect the cooling coil of the thermostat to the water supply by using two rubber tubings (Fig. 6) Fig. 6 2 www.phywe.com P2321 Phywe Systeme GmbH & Co. KG All rights reserved Laboratory Experiments
3.2.1 - Now your setup should look like the following picture: - Before beginning the measurements, remove the little rubber stopper from the mercury reservoir (right tube) (Fig. 8) Fig. 7 Fig. 8 Procedure Task 1: During this experiment, the temperature must be kept constant at 298.15 K (25 C). This is achieved by pumping water, which has the desired temperature, through the rubber tubing with the aid of the thermostat. - Switch on the thermostat and set the temperature to 298.15 K (25 C) - Wait until the temperature in the measuring tube remains constant (look at the thermometer above the measuring tube) In order to investigate the correlation between the pressure p and the volume V, the pressure in the measuring tube is varied by raising or sinking the mercury reservoir. The length l of the column of air in the measuring tube and the height difference Δh between the mercury level in the mercury reservoir and the mercury level in the measuring tube can be read off the scale of the device. Laboratory Experiments Phywe Systeme GmbH & Co. KG All rights reserved P2321 www.phywe.com 3
3.2.1 Equation of state of ideal gases - Get the air pressure p a from the Electronic weather station and note the value above Table 1 (page 6) - Move the mercury reservoir (Fig. 9) until the levels of the mercury in the mercury reservoir and in the measuring tube are at the same height ( Δh ) - Read off the scale the length l of the column of air in the measuring tube (this is the distance between the mercury level and the brown marked measuring tube segment at the top) - Perform at least 1 measurements by raising the mercury reservoir and reading off the scale the length l of the column of air and the height difference Δh between the levels of the mercury in the mercury reservoir and in the measuring tube - Note your results in Table 1 (page 6) Fig. 9 Task 2 and 3: - In order to determine the temperature dependency on pressure and volume, the two correlations are investigated at the same time for each temperature step - The initial temperature is the same as in Task 1 ( T 298. 15 K) - Move the mercury reservoir so, that the mercury levels in the mercury reservoir and in the measuring tube are at the same height - Mark this position with a marker on the measuring tube or use a piece of adhesive tape to mark the position on the scale (Fig. 1 and 11). This marking represents a constant volume of air ( V V 1 ) in the measuring tube. Fig. 1 Fig. 11 - Measure the length l of the column of air in the measuring tube and note this value as well as the value for the height difference Δh (in this initial case Δh ) in Table 2 (page 7) - Now, increase the temperature to 33.15 K (3 C) and wait until the temperature in the measuring tube remains constant 4 www.phywe.com P2321 Phywe Systeme GmbH & Co. KG All rights reserved Laboratory Experiments
3.2.1 - Task 2: To investigate the dependency of the temperature T on the volume V at constant pressure move the mercury reservoir until the two mercury levels are equal p pa, Fig. 12 - Read off the scale the new length l of the column of air in the measuring tube and note the value in Table 2 - Task 3: In order to investigate the dependency of the temperature T on the pressure p at constant volume V, move the mercury reservoir until the mercury level in the measuring tube reaches the marking again (Fig. 13) Fig. 13 - Measure the height difference Δh between the two mercury levels and note the value in Table 2 - Subsequently, increase the temperature in steps of 5 K up to 358.15 K (85 C) and proceed in each step the way described above - When you have finished your measurements, put the little rubber stopper back onto the mercury reservoir Laboratory Experiments Phywe Systeme GmbH & Co. KG All rights reserved P2321 www.phywe.com 5
3.2.1 Equation of state of ideal gases Results Task 1: External air pressure p a Note your measuring results in the following table (first and second column). Table 1 Determined length l [mm] Height difference Δh [mm] Volume V [ml] Pressure p [kpa] Now, calculate from your measuring results the Volume V and the pressure p. To do this, use the following equations: For the volume is valid: d V V1 + VR π l + V 2 2 2 11.4 mm π l + 1.1 ml (1) 2 2 and for the pressure: p pa + Δp + Δh.1333 kpa mm p a (2) Use equation (1) to calculate the Volume V. 1.1 ml is approx. the volume of the brown marked measuring tube segment, which has to be added to the volume. Take into account that you have to convert the units since you have ml and mm 3 in the same equation. 6 www.phywe.com P2321 Phywe Systeme GmbH & Co. KG All rights reserved Laboratory Experiments
3.2.1 3 The correlation between them is: 1 ml 1 mm. Record your results in Table 1 (column 3). Use equation (2) For the calculation of the pressure p in dependence of the height difference Δh. Note your results for the in Table 1, too (column 4). In our sample measurement we got the following results (Important: these are only sample results and your results may differ from them): Sample results Determined length l [mm] Height difference Δh [mm] Volume V [ml] Pressure p [kpa] 183 19.96 1. 18 16 19.38 12.12 175 33 18.87 14.4 17 56 18.36 17.46 16 12 17.34 113.6 15 158 16.32 121.6 14 213 15.3 128.39 134 251 14.69 133.46 13 282 14.28 137.59 124 328 13.67 143.72 12 366 13.26 148.79 Task 2 and 3: Note your measuring results in the table below (columns 1 3). Table 2 Temperature T [K] Determined length l [mm] Height difference Δh [mm] Volume V [ml] Pressure p [kpa] Again, calculate the Volume V and the pressure p using the equations (1) and (2) and note your results in Table 2 (column 4 and 5). In our sample measurement we got the following results: Laboratory Experiments Phywe Systeme GmbH & Co. KG All rights reserved P2321 www.phywe.com 7
3.2.1 Equation of state of ideal gases Sample results Temperature T [K] Determined length l [mm] Height difference Δh [mm] Volume V [ml] Pressure p [kpa] 298.15 183 19.69 1. 33.15 185 16 19.89 12.13 38.15 188 3 2.2 14. 313.15 193 4 2.71 15.33 318.15 196 53 21.2 17.6 323.15 199 65 21.32 18.66 328.15 22 76 21.63 11.13 333.15 25 88 21.93 111.73 338.15 28 12 22.24 113.6 343.15 211 114 22.55 115.2 348.15 215 126 22.96 116.8 353.15 219 138 23.36 118.4 358.15 222 152 23.67 12.26 8 www.phywe.com P2321 Phywe Systeme GmbH & Co. KG All rights reserved Laboratory Experiments
3.2.1 Evaluation 1. Correlation between volume and pressure at constant temperature (Boyle-Mariotte s law) In order to explain the correlation between the volume V and the pressure p of a gas one must have a closer look at the results of Task 1. In this part of the experiment the temperature T was constant at 298.15 K whereas the pressure p and the volume V of the gas varied. This process is called isothermal expansion and compression. For the change in volume is valid: where dv Vχ dp (3) χ 1 V V p T, n the coefficient of cubic compressibility. The indices T and n show that the temperature and the substance quantity (n) of the gas are constant. From integration of equation (3) with χ const. one gets V p Vp (4) 1 V const., p where V and p the volume and the pressure of the gas at T 273.15 K. Therefore V and p are constant and so the product of both is constant, too. Boyle and Mariotte first investigated this correlation and so it is called Boyle-Mariotte s law. To compare the theory with your measurements, convert your results for p into 1/p and note these values in the table below. Reciprocal of the pressure 1/p [kpa] Laboratory Experiments Phywe Systeme GmbH & Co. KG All rights reserved P2321 www.phywe.com 9
3.2.1 Equation of state of ideal gases We got the following results for our measuring values: Sample results Reciprocal of the pressure 1/p [kpa].1.979.958.931.88.826.779.749.727.696.672 Now, design a graph where you draw the reciprocal pressure 1/p against the volume V (see Table 1). It is recommended to use the software PHYWE measure to do this. It is free for download on www.phywe.com (see appendix). You should get a graph similar to the following: Fig. 14: Volume V as a function of the reciprocal pressure 1/p at constant temperature ( T 298.15 K ) and at constant substance quantity n. - Note the value for the slope of the curve in the table on page 13 2. Correlation between temperature and volume at constant pressure (Gay-Lussac s law) To investigate the correlation between the temperature T and the volume V at constant pressure ( p p a ) use your measuring results from Task 2. The process, in which the pressure p is constant and the temperature T as well as the volume V is variable, is called isobaric process. 1 www.phywe.com P2321 Phywe Systeme GmbH & Co. KG All rights reserved Laboratory Experiments
3.2.1 For the change in volume is valid: dv Vγ dt (5) where γ 1 V V T p, n the coefficient of thermal expansion. After integration of (5) with γ const. one gets V V T T V const. T (6) Gay-Lussac first investigated this correlation and that is the reason why it is called Gay-Lussac s law. Now, have a look at Table 2. Design a graph where you draw the values for the temperature T against the values for the volume V. If you measured accurately, you should be able to recognise the proportionality between V and T and your graph should look like the following. Again note the slope (page 13). Fig. 15: Dependence of the volume V on the temperature T at constant pressure ( p p a 15 kpa ) and at constant substance quantity n. 3. Correlation between temperature and pressure at constant volume (Charles (Amonton s) law) In order to determine the correlation between the temperature T and the pressure p you have to use Table 2 again. Laboratory Experiments Phywe Systeme GmbH & Co. KG All rights reserved P2321 www.phywe.com 11
3.2.1 Equation of state of ideal gases The process, in which the temperature and the pressure are variable while the volume of the gas is constant, is called isochoric process. For the change in pressure is therefore valid: dp pβ dt (7) where β 1 p p T V, n the coefficient of thermal tension. After integration of (7) with β const. one gets: p p T T p const. T (8) This correlation is called Charles (Amonton s) law. Now, use your measuring results for the temperature T and the pressure p from Table 2 and design a graph where you draw p against T. If you measured accurately, you should be able to recognise the proportionality between p and T and thus your graph should look like the following. Note the slope (page 13). Fig. 16: Correlation between the pressure p and the temperature T at constant volume V and constant substance quantity n. 12 www.phywe.com P2321 Phywe Systeme GmbH & Co. KG All rights reserved Laboratory Experiments
3.2.1 In the next step, determine the universal gas constant R. This is done with the help of the following considerations: By combining the equation (6) or (8) with equation (4) one obtains p V T p1v T 1 1 pv T (9) and from that the general (or thermal) equation of state for ideal gases: pv nrt (1) You can calculate the universal gas constant R with your measurement results and the following equations: V nrt p (1.1) 1 T, n V T p, n nr Vγ (1.2) p p T V, n nr pβ (1.3) V The three terms in these equations that are on the left of the equality sign are equal to the slopes of the corresponding graphs. That means, ( V/ p ) T,n corresponds to the slope of the graph showing the correlation between volume and pressure (Task 1), ( V/ T) p,n corresponds to the slope of the graph showing the correlation between volume and temperature (Task 2) and ( p/ T) V,n corresponds to the slope of the graph showing the correlation between pressure and temperature (Task 3). Look at the units of your measured slopes and convert the unit ml into m 3. The correlation between them is: 1 1 ml 1 3 3 3 1 mm 1cm m, 3 3 3 3 since 1 m 1 cm. This has to be done for the slopes of Task 1 and 2. The unit of the slope of Task 2 should be Pa m 3 Nm. To do this, remember, that the following is valid: 1 1 3 1 kpa. 3 kpa Measured slopes ( V/ p ) T,n ( V/ T) p,n ( p/ T) V,n After converting Laboratory Experiments Phywe Systeme GmbH & Co. KG All rights reserved P2321 www.phywe.com 13
3.2.1 Equation of state of ideal gases In our measurement we got the following results: Measured slopes After converting ( V/ p ) T,n 1.989 ml/1-3 kpa 1.989 Pa m 1.989 Nm ( V/ T) p,n.67 ml/k 8 3 6.686 1 m K ( p/ T) V,n.33 kpa/k.33 kpa K The universal gas constant is calculated by using the equations (1.1), (1.2), (1.3). But first, you have to determine the value for n by the equation: V n (11) V m where V m the molar volume of the gas (air). Under standard conditions ( T 273.15 K, 3 p 11.325 kpa ) it is.22414 m mol. In order to calculate n you have to reduce your V m measured volume V to these standard conditions. This is done by using equation (9). Insert one measured result each for V, p, T (have a look at Table 2) and the values for p and T (see above) to calculate V. Then insert this value in equation (11) to calculate n and note it below. n Now, you have all values to calculate the universal gas constant R according to the equations (1.1), (1.2) and (1.3). Note your results in the following table: R R R and calculate the mean value: R In our sample measurement we got the following results: n.7943 mol R 8.399 Nm K mol R 8.418 Nm K mol R 8.181 Nm K mol Mean value: R 8.333 Nm K The literature value is R 8.31441 Nm K mol 8.31441 J K mol. mol 14 www.phywe.com P2321 Phywe Systeme GmbH & Co. KG All rights reserved Laboratory Experiments
3.2.1 Additionally, determine the coefficients γ and β with the aid of the equations (1.2) and (1.3) and your values of the slopes and note your results below. γ β Use these values to calculate the coefficient of cubic compressibility χ with the following equation: and record the value: χ 1 γ χ p β 11.325 kpa In our sample measurement we got the following results: γ 3.756 1 K β 3 3.257 1 K χ 3 8.558 1 kpa The theoretical values for an ideal gas under standard conditions at T and p are: β γ 3 3.661 1 K 1 273.15 K 1 γ β and 9 kpa χ.872 kpa 1 11.325. Laboratory Experiments Phywe Systeme GmbH & Co. KG All rights reserved P2321 www.phywe.com 15
3.2.1 Equation of state of ideal gases Appendix Using measure for creating the graphs and evaluation of the data - Once installed, start measure. - Click Measurement and choose Enter data manually - There select the parameters that are shown in the screenshot below - Click Continue - Type in your values for V and 1/p (make sure to use commas instead of points for decimal numbers) - Click OK - Choose the first point Sort x-data - Right-click on your graph and select Display options (Symbol: ) - Under channels select interpolation and none for displaying the points only. In this menu you can also change the displayed area for the best display - For fitting the graph click Analysis and then Function fitting, choose straight and click Calculate - After clicking Add new curve the fitting curve will appear 16 www.phywe.com P2321 Phywe Systeme GmbH & Co. KG All rights reserved Laboratory Experiments
3.2.1 - Click the button Show slope ( ) and note this value for the slope - This is how you can use measure for displaying the graphs for Tasks 2 and 3, too. For further information about the program please go to the Help -menu. Laboratory Experiments Phywe Systeme GmbH & Co. KG All rights reserved P2321 www.phywe.com 17