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PHYWE Systeme GmbH & Co. KG · D - 37070 Göttingen Laboratory Experiments Physics 143
Equation of state of ideal gases with Cobra3 3.2.01-
Ideal and Real Gases Thermodynamics
Principle:
The state of a gas is determined by temperature, pressure and amount of substance. For the limiting case of ideal gases, these state variables are linked via the general equation of state. For a change of state under isochoric conditions this equation becomes Amontons’ law. In this experiment it is investigated whether Amontons’ law is valid for a constant amount of gas (air).
Dependence of the pressure on the temperature under isochoric conditions.
What you can learn about …
Thermal tension coefficient General equation of state for ideal gases Universal gas constant Amontons’ law
Cobra3 Basic-Unit, USB 12150.50 1 Power supply 12V/2A 12151.99 1 Data cable 2 x SUB-D, plug/socket, 9 pole 14602.00 1 Measuring module pressure 12103.00 1 Measuring module temperature NiCr-Ni, 330°C 12104.00 1 Thermocouple NiCr-Ni, sheathed 13615.01 1 Cobra3 measuring module converter 12150.04 1 Temperature sensor, semiconductor type 12120.00 1 Software Cobra3 Gas Laws 14516.61 1 Glass jacket 02615.00 1 Gas syringes, without cock, 100 ml 02614.00 1 Heating apparatus 32246.93 1 Power regulator 32288.93 1 H-base -PASS- 02009.55 1 Support rod, stainless steel 18/8, l = 250 mm, d = 10 mm 02031.00 2 Right angle clamp 37697.00 3 Universal clamp 37715.00 2 Universal clamp with joint 37716.00 1 Magnet rod, l = 200 mm, d = 10 mm 06311.00 1 Magnetic stirring rod, cylindrical, l = 30 mm 46299.02 1 Beaker, DURAN®, tall form, 250 ml 36004.00 1 Funnel, glass, d = 50 mm 34457.00 1 Hose connector, reducing, d = 3-5/6-10 mm 47517.01 1 Silicone tubing, d = 2 mm 39298.00 1 Silicone tubing, d = 7 mm 39296.00 1 Rubber caps 02615.03 1 Hose clip, d = 8-12 mm 40996.01 2 PC, Windows® XP or higher
What you need:
Complete Equipment Set, Manual on CD-ROM included
Equation of state of ideal gases
with Cobra3 P
Tasks:
For a constant amount of gas (air) investigate the correlation of
- Volume and pressure at constant temperature (Boyle and Mariotte’s law)
- Volume and temperature at con- stant pressure (Gay-Lussac’s law)
- Pressue and temperature at con- stant volume (Charles’ (Amontons’ law))
From the relationships obtained cal- culate the universal gas constant as well as the coefficient of thermal ex- pansion, the coefficient of thermal tension, and the coefficient of cubic compressibility.
Related concepts Thermal tension coefficient, General equation of state for ideal gases, Universal gas constant, Pressure, temperature, volume, coefficient of thermal expansion, coefficient of thermal tension, ideal gas law, universal gas constant, Gay-Lussac’s law, Charles’ (Amontons’) law, coefficient of cubic compressibility, Boyle and Mariotte’s law.
Principle The state of a gas is determined by temperature, pressure, vol- ume and amount of substance. For the limiting case of ideal gases, these state variables are linked via the general equation of state. For a change of state under isochoric conditions this equation becomes Amontons’ law. In the case of isothermal process control this equation converts to Boyle and Mariotte’s law. In the case of isobaric conditions the ideal gas law converts to Gay-Lussac’s law.
Tasks For a constant amount of gas (air) investigate the correlation of
- Volume and pressure at constant temperature (Boyle and Mariotte’s law)
- Volume and temperature at constant pressure (Gay-Lussac’s law)
- Pressure and temperature at constant volume (Charles’ (Amonton’s law))
From the relationships obtained calculate the universal gas con- stant as well as the coefficient of thermal expansion, the coeffi- cient of thermal tension, and the coefficient of cubic compress- ibilty.
Equipment Cobra3 Basic-Unit 12150.00 1 Power supply 12 V/2 A 12151.99 1 Data cable, RS232 14602.00 2 Measuring module, pressure 12103.00 1 Temperature measuring module, NiCr-Ni 12104.00 1 Thermocouple, NiCr-Ni, sheated 13615.01 1 Software Cobra 3 Gas laws 14516.61 1 Module converter 12150.04 1 Cobra3 sensor, -10…120°C 12120.00 1 Glass jacket 02615.00 1 Glass syringe, 100 ml 02614.00 1 Heating apparatus 32246.93 1 Power regulator 32247.93 1 H-base -PASS- 02009.55 1 Support rod, l = 250 mm 02031.00 2 Right angle clamp 37697.00 3 Universal clamp 37715.00 2 Universal clamp with joint 37716.00 1 Magnet, l = 200 mm, d = 10 mm 06311.00 1 Magnetic stirrer bar, l = 30 mm 46299.02 1 Glass beaker, 250 ml, tall 36004.00 1 Funnel, do = 55 mm 34457.00 1 Rubber caps 02615.03 1 Silicone tubing, di = 7 mm 39296.00 1 Tubing adaptor 3-5 / 6-10 mm 47517.01 1 Silicone tubing, di = 2 mm 39298.00 1 Hose clips, d = 8…12 mm 40996.01 2 Motor oil Water, distilled 31246.81 1 PC, Windows®^ 95 or higher
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen P2320115 (^1)
3.2.
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Equation of state of ideal gases with Cobra
Fig. 1. Experimental set-up for Gay-Lussac’s law
For the limiting case of an ideal gas (sufficiently low pressure, sufficiently high temperature), the integration of a differential equation resulting from (1) and (2), where g 0 = constant, yields
and
According to this correlation, which was discovered by Gay- Lussac, the graphic presentation of the volume as a function of the temperature provides ascending straight lines (Fig. 3) where V = 0 for T = 0. From (2) and the ideal gas law
R Universal gas constant
the following is true for the slope of these linear relationships
From this, the thermal coefficient of expansion g 0 and the uni- versal gas constant R are experimentally accessible for a known initial volume V 0 and a known amount of substance n. The con- fined constant amount of substance n is equal to the quotient of the volume V and the molar volume Vm
which is V 0 = 22.414 l · mol-1^ at T 0 = 273.15 K and p 0 = 1013.25 hPa at standard conditions. A volume measured at p and T is therefore first reduced to these conditions using the relationship obtained from (4):
Fig. 4 shows the quantity pV/T appearing to be constant.
Data and results The investigation of the correlation between volume and tem- perature with a constant quantity of gas of n = 2.23 mmol, cal- culated according to the relations (6) and (7), confirms the valid- ity of the Gay-Lussac’s first law, with the linear relationship demonstrated in Fig. 3.
From the corresponding slope (∂V/∂T)p,n = 0.18 ml/K and for the initial volume V 0 = 50 ml, the following values are obtained for the universal gas constant R and the coefficient of thermal expansion g 0. R (exp.) = 8.07174 Nm · K-1^ · mol- g 0 (exp.) = 3.04 · 10-3^ K- The theoretical values for an ideal gas are R (lit.) = 8.31441 Nm · K-1^ · mol-1^ = J · K-1^ · mol- g 0 (lit.) = 3.661 · 10-3^ K-
II. Amonton’s law
Set-up and procedure Set up the experiment as shown in Fig. 5. Install the gas syringe in the glass jacket as described in the operating instructions supplied with the glass jacket. Pay partic- ular attention to the air-tightness of the gas syringe. As an exception here, because no air is to be allowed to leak out even at higher pressures, lubricate the plunger with a few drops of multigrade motor oil, so that the glass plunger is covered with an uninterrupted clear film of oil throughout the entire experiment; but avoid excess oil. Fill the glass jacket with water via the fun- nel and place a magnetic stirrer bar in it. Attach a piece of sili- cone tubing to the hose nipple of the tubular sleeve through which the water that expands during heating can drain into a beaker. Insert the thermocouple and place it as close to the syringe as possible. After adjusting the initial volume of the gas syringe to exactly 50 ml, connect the nozzle of the gas syringe to a reducing adaptor via a short piece of silicone tubing, where- by the reducing adaptor should directly abut on the glass tubu- lar sleeve after the tubing has been slipped over it. Secure the tubing on both the gas syringe’s nozzle and on the reducing adaptor with hose clips. Connect the reducing adaptor to the measuring module by means of a short piece of silicone tubing (di = 2 mm). Keep the tubing connections as short as possible.
Connect the measuring module to the Cobra3 Basic-Unit using a module converter and a data cable. Call up the “Measure” pro- gramme in Windows and enter as gauge. Set the measuring parameters as shown in Fig. 6. Under <Start / Stop> choose . Under select Digital Display 1 for , Digital Display 2 for and Diagram 1 with Channel , X bounds <from 1 to 15> and Mode . Now calibrate your sensors under by entering temperature and pressure values measured with a ther- mometer respectively manometer and pressing . After having made these settings, press to reach the field for the recording of measured values. Arrange the displays as you want them.
Record the pressure corresponding to the initial temperature by pressing . Subsequently, adjust the heating appa- ratus to slow heating with the power regulator. Mix the water in the glass jacket by moving the magnetic stirrer bar with the aid of a bar magnet. After each temperature increase of 5 K, push the plunger rapidly into the gas syringe until the gas volume is compressed to the initial volume of V = 50 ml and take the next value. After the temperature has reached approximately 370 K or if there is an evident loss of air during compression, switch off the heating apparatus and terminate measurement by pressing . Save the measurement with <Save the mea- surement as…>. To have the plot of Pressure versus Temperature carried out, make the following alterations. Under the menu prompt , choose Temperature for the x-axis and Pressure for the y-axis. Fig. 7 shows the graph as it is then presented by the programme. If you choose pV/T as y-axis, you obtain the graph as it is shown in Fig. 8.
Theory and evaluation The state of a gas is a function of the state variables tempera- ture T, pressure p, volume V and the amount of substance n, which interactively determine each other. Thus, the dependence
p 0 V 0 T 0
p 1 V 1 T 1
p V T
const.
n
V
Vm
a
0 V
0 T
b V 0 g 0
nR p
pV nRT
V const. · T
V 0
T 0
V
T
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen P2320115 (^3)
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Equation of state of ideal gases with Cobra
of pressure on the temperature, volume and amount of sub- stance variables is described by the total differential
For a given amount of substance (n = const., dn = 0; enclosed quantity of gas in the gas syringe) and isochoric change of state (V = const., dV = 0) this relationship simplifies to
The partial differential quotient (∂p/∂T)V,n corresponds geomet- rically to the slope of a tangent to the function p = f(T) and thus characterises the dependence of the pressure on the tempera- ture. The degree of this dependence is determined by the initial pressure. Therefore, one defines the thermal coefficient of ten- sion b 0 as a measure of the temperature dependence by refer- ring it to p or p 0 at T 0 = 273.15 K.
For the limiting case of an ideal gas (sufficiently low pressure, sufficiently high temperature), the integration of a differential equation resulting from (8) and (9), where b 0 = const., yields
and (10.2)
According to this correlation, which was discovered by Charles and Amontons, the graphic presentation of the pressure as a function of the temperature results in an ascending straight line (Fig. 7) where p = 0 at T = 0.
p const. · T
p 0 T 0
p T
b 0
p 0
a
0 p 0 T
b V,n
dp a
0 p 0 T
b V,n
dT
dp a
0 p 0 T
b V,n
dT a
0 p 0 V
b T,n
dV a
0 p 0 n
b T,V
dn
4 P2320115^ PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
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Equation of state of ideal gases with Cobra
Fig. 6: Measurement parameter
Fig. 5. Experimental set-up for Amonton’s law
and Pressure for the y-axis. Fig. 11 shows the graph as it is then presented by the programme. If you choose 1/V as x-axis, you get the graph as it is shown in Fig. 12. Under menu prompt Analysis you can let the programme show the slope.
Theory and evaluation The state of a gas is a function of the state variables tempera- ture T, pressure p, volume V and the amount of substance n, which reciprocally determine one another. Thus, the depen- dence of pressure on the temperature, volume and amount of substance variables is described by the total differential
Analogously, the following is true for the change of pressure with T, V and n:
This relationship simplifies for a given amount of substance (n = const., dn = 0; enclosed quantity of gas in the gas syringe) and isothermal change of state (T = const., dT = 0) to
and
The partial differential quotient (∂V/∂p)T,n resp. (∂p/∂V)T,n corre- sponds geometrically to the slope of a tangent to the function V = f(p) or p = f(V) and therefore characterises the mutual depen- dence of pressure and volume. The degree of this dependence is determined by the initial volume or the initial pressure. One thus defines the cubic compressibility coefficient x 0 by referring it to V or V 0 at T 0 = 273.15 K.
For the limiting case of an ideal gas (sufficiently low pressures, sufficiently high temperatures), the correspondence between the state variables p, V, T and n is described by the ideal gas law:
R Universal gas constant
For cases of constant quantity of substances and isothermal process control this equation changes into the following equa- tions:
p V const. (18.1)
pV nRT
x 0
V 0
a
0 V
0 p
b
T,n
dp a
0 p
0 T
b
V,n
dT
dV a
0 V
0 T
b
T,n
dp
dp a
0 p
0 T
b
V,n
dT a
0 p
0 V
b
T,n
dV a
0 p
0 n
b
T,V
dn
dV a
0 V
0 T
b
p,n
dT a
0 V
0 p
b
T,n
dp a
0 V
0 n
b
T,p
dn
6 P2320115^ PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
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Equation of state of ideal gases with Cobra
Fig. 9. Experimental set-up for Boyle and Mariotte’s law
and
(18.2)
According to this correlation, which was determined empirically by Boyle and Mariotte, a pressure increase is accompanied by a volume decrease and vice versa. The graphic representation of the functions V = f(p) or p = f(V) results in hyperbolas (Fig. 11). In contrast, plotting the pressure p against the reciprocal volume 1/V results in straight lines where p = 0 at 1/V = 0 (Fig. 12). From the slope of these linear relationships,
it is possible to determine the gas constant R experimentally when the enclosed constant quantity of air n is known. This is equal to the quotient of the volume V and the molar volume Vm,
which is V 0 = 22.414 l · mol-1^ at T 0 = 273.15 K and p 0 = 1013.25 hPa at standard conditions. A volume measured at p and T is therefore first reduced to these conditions using the relationship obtained from (18):
Data and results Figs. 11 and 12 confirm the validity of Boyle and Mariotte’s law. From the slope obtained for n = 2.086 mmol and T = 295.15 K, (∂p/∂V-1)T,n = 4.6464 kPa/m-3^ = 4.6464 Nm of the linearised cor- relation between p and 1/V (Fig. 4), the universal gas constant can be calculated to be R = 7.547 Nm · K-1^ · mol-1. The deviation from the literature value (R = 8.31441 Nm · K-1^ · mol-1^ = 8.31441 J · K-1^ · mol-1) is due to the unavoidable lack of gas-tightness with increasing deviation from atmospheric pressure through compression or expansion, whereby the condition dn = 0 is violated and the observed slope (∂p/∂V-1)T is diminished in comparison with the value measur- able with a constant quantity of substance.
p 0 V 0 T 0
p 1 V 1 T 1
p V T
n
V
Vm
a
0 p 0 V^1
b T,n
n R T
p const. ·
V
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen P2320115 (^7)
3.2.
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Equation of state of ideal gases with Cobra
Fig. 11: Correlation between the volume V and the pressure p at constant temperatures (T = 295.15 K) and constant quantity of substance (n = 2.086 mmol)
Fig. 10: Measurement parameters
Fig. 12: Pressure p as a function of the reciprocal volume 1/V at constant temperature (T = 295.15 K) and constant quantity of substance (n = 2.086 mmol)
