02.03 Heat Capacity of Gases Thermochemistry/Calorimetry

02.03 Heat capacity of gases

What you can learn about 1st law of thermodynamics Universal gas constant Isobars Isotherms Isochors and adiabatic changes of state

Principle and tasks Heat is added to a gas in a glass vessel by an electric heater which is switched on briefly. The temperature increase results in a pressure increase which is measured with a manometer. Under isobaric conditions a temperature increase results in a volume dilatation that can be read from a gas syringe. The molar heat capacities Cv and Cp are calculated from pressure and volume change.

What you need: Experiment P3020311 with Cobra3 Basic-Unit Experiment P3020301 with digital counter Precision manometer 03091.00 1 1 Barometer/Manometer, hand.held 07136.00 1 1 Cobra3 Basic-Unit, USB 12150.50 1 Power supply12 VDC/2 A 12151.99 1 Cobra3 current probe 6A 12126.00 1 Software Cobra3 Universal writer software 14504.61 1 Digital counter, 4 decades 13600.93 1 Digital multimeter 07128.00 2 Mariotte flask, 10 l 02629.00 1 1 Gas syringe, 100 ml 02614.00 2 2 Stopcock, 1.way, straight 36705.00 1 1 Stopcock, 3.way, T.shaped, capillary 36732.00 1 1 Pressure change ⌬p as a func tion of the heat-up time ⌬t. U = 4.59 V, Rubber Stopper 26/32, 3 holes 39258.14 1 1 I = 0.43 A. Rubber Stopper 50.5/59.5, 1 hole 39268.01 1 1 Rubber tubing, d = 6 mm 39282.00 2 2 Nickelel ectrode, d = 3 mm, with socket 45231.00 2 2 Nickelel ectrode, 76 mm x 40 mm 45218.00 1 1 Chrome-nickel wire, d = 0,1 mm 06109.00 1 1 Scissors, straight, blunt, l = 140 mm 64625.00 1 1 Two-way switch, single pole 06030.00 1 Push-button switch 06039.00 1 Connecting cord, 32 A, 500 07361.01 1 1 Connecting cord, 32 A, 500 07361.02 2 Connecting cord, 32 A, 750 07362.01 1 1 Universal clamp 37715.00 2 2 Connecting cord, 32 A, 500 07361.04 3 4 Right angle clamp 37697.00 2 2 Connecting cord, 32 A, 750 07362.04 1 PC, Windows® XP or higher Tripod base -PASS- 02002.55 1 1 Retord stand, 210 x 130 mm, h 37694.00 2 2 Heat capacity of gases P3020301/11

24 Laboratory Experiments Chemistry PHYWE Systeme GmbH & Co. KG · D-37070 Göttingen LEC Heat capacity of gases 02.03

Related topics Scissors, straight, blunt, l = 140 mm 64625.00 1 Equation of state for ideal gases, 1st law of thermodynamics, Two-way switch, single pole 06030.00 * 1 universal gas constant, degree of freedom, mole volumes, iso- Connecting cord, l = 750 mm, blue 07362.04 * 1 bars, isotherms, isochors and adiabatic changes of state. Connecting cord, l = 500 mm, red 07361.01 1 Connecting cord, l = 500 mm, yellow 07361.02 * 2 Principle Connecting cord, l = 750 mm, red 07362.01 1 Heat is added to a gas in a glass vessel by an electric heater Connecting cord, l = 500 mm, blue 07361.04 * 3 which is switched on briefly. The temperature increase results in Tripod base -PASS- 02002.55 1 a pressure increase, which is measured with a manometer. Retort stand, h = 750 mm 37694.00 2 Under isobaric conditions, a temperature increase results in a Universal clamp 37715.00 2 volume dilatation which can be read from a gas syringe. The Right angle clamp 37697.00 2 molar heat capacities CV and Cp are calculated from the pressure or volume change. Changes in the equipment required for use of the Basic- Unit: Equipment (instead of * above mentioned) Precision manometer 03091.00 1 Cobra3 Basic-Unit, USB 12150.50 1 Barometer/Manometer, hand-held 07136.00 1 Power supply 12 V/ 2 A 12151.99 1 Digital counter, 4 decades 13600.93 * 1 Cobra3 current probe 6 A 12126.00 1 Digital multimeter 07128.00 * 2 Push-button switch 06039.00 1 Mariotte flask, 10 l 02629.00 1 Connecting cord, l = 500 mm, blue 07361.04 4 Gas syringe, 100 ml 02614.00 2 Cobra3 Universal writer software 14504.61 1 Stopcock, 1-way, straight 36705.00 1 PC, Windows® 95 or higher Stopcock, 3-way, T-shaped, capillary 36732.00 1 Rubber stopper, d = 26 / 32 mm, 3 holes 39258.14 1 Tasks Rubber stopper, d = 50.5 / 59.5 mm, 1 hole 39268.01 1 Determine the molar heat capacities of air at constant volume Rubber tubing, di = 6 mm 39282.00 2 CV and at constant pressure Cp. Nickel electrode, d = 3 mm, with socket 45231.00 2 Nickel electrode, 76 x 40 mm 45218.00 1 Set-up and procedure Chrome-nickel wire, d = 0.1 mm, l = 100 m 06109.00 1 Perform the experimental set-up according to Fig. 1.

Fig. 1: Experimental set-up (CV).

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Insert the two nickel electrodes into two holes in the three-hole the measuring series. To achieve this, connect one of the digi- rubber stopper and fix the terminal screws to the lower ends of tal multimeters in series as an ammeter and the other in parallel the electrodes. Screw two pieces of chrome nickel wire, which as a voltmeter. are each about 15 cm long, into the clamps between these two Perform at least 10 measurements. After each measurement, electrodes so that they are electrically connected in parallel. The perform a pressure equalisation with the ambient atmospheric wires must not touch each other. Insert the one-way stopcock pressure by opening the three-way cock. The electrical current into the third hole of the stopper and insert the thus-prepared which flows during the measurements must not be too strong, stopper in the lower opening of the bottle. i.e. it must be sufficiently weak to limit the pressure increase due The 5 V output of the electrical 4-decade digital counter serves to the heating of the gas to a maximum of 1 hPa. as the power source. The electrical circuit is illustrated in Fig. 3.

To determine CV, connect the precision manometer to the bottle with a piece of tubing. To do this, insert the second stopper, Fig. 3: Connecting the counter-timer. which has been equipped with the three-way stopcock, into the upper opening of the bottle. The manometer must be positioned exactly horizontally. Read the pressure increase immediately after cessation of the heating process. The manometer must be filled with the oil which is supplied with the device. The scale is now calibrated in hPa. The riser tube of the manometer must be well wetted before each measurement. Start the measuring procedure by activating the push-button switch. The measuring period should be as short as possible (less than one second). Determine the current which flows during the measurement and the voltage separately at the end of

Fig. 2: Experimental set-up (Cp)

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For this reason it may be necessary to use only one heating Set-up and procedure with Cobra3 wire. Perform the experimental set-up according to Fig. 4. Insert the two nickel electrodes into two holes in the three- In order to be able to determine Cp, replace the manometer with two gas syringes which are connected to the bottle via the hole rubber stopper and fix the terminal screws to the lower three-way stopcock (see Fig. 2). One of the gas syringes is ends of the electrodes. Screw two pieces of chrome nickel mounted horizontally and the other is positioned vertically with wire, which are each about 15 cm long, into the clamps its plunger oriented downwards. While making measurements, between these two electrodes so that they are electrically the three-way cock must be positioned in such a manner that it connected in parallel. The wires must not touch each other. only connects the vertical syringe with the bottle. To increase Insert the one-way stopcock into the third hole of the stopper the plunger's mass, a nickel sheet metal electrode is attached and insert the thus-prepared stopper in the lower opening of to it with two-sided tape. Start the plunger rotating manually the bottle. before the measurement so that it rotates throughout measure- The 5 V output of the Cobra3 Basic-Unit serves as the power ment. In this manner the static friction between the plunger and source. the body of the syringe is minimised and the measured values To determine CV, connect the precision manometer to the bot- are sufficiently exact. If the plunger stops prematurely, the vol- tle with a piece of tubing. To do this, insert the second stop- ume increase (V) read on the vertically mounted syringe is too per, which has been equipped with the three-way stopcock, low. Determine the air pressure, which is required for the calcu- into the upper opening of the bottle (Fig. 4). The manometer lations, with the aid of a digital barometer. For the pressure in must be positioned exactly horizontally. Read the pressure the gas container use a value which lies 14 hPa below the increase immediately after cessation of the heating process. atmospheric pressure due to the weight of the syringe’s plunger. The manometer must be filled with the oil which is supplied For the determination of Cp, also perform at least 10 measure- with the device. The scale is now calibrated in hPa. The riser ments. After each measurement remove air from the system tube of the manometer must be well wetted before each until the vertical syringe again exhibits the initial volume deter- measurement. mined in the first measurement. To do this, turn the three-way Connect the current sensor (A) to the first analog input of the cock in such a manner that both syringes and the bottle are Cobra3 Basic-Unit. To measure the voltage (V) connect the connected with each other. cords to the second analog input.

Fig. 4: Experimental set-up (CV).

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Connect the Cobra3 Basic-Unit to the computer port COM1, COM2 or to USB port (use USB to RS232 Adapter). Start the ‘Measure’ program and select Cobra3 Universal Writer Gauge. Begin recording the measured values using the parameters given in Fig. 6. Start the measuring procedure by activating the push-button switch. The measuring period should be as short as possible (ten seconds). Determine the current which flows during the measurement and the voltage separately at the end of the measuring series. Perform at least 10 measurements. After each measurement, perform a pressure equalisation with the ambient atmospheric pressure by opening the three-way cock. The electrical current which flows during the measurements must not be too strong, i.e. it must be sufficiently weak to limit the pressure increase due to the heating of the gas to a maximum of 1 hPa. For this reason it may be necessary to use only one heating wire. In order to be able to determine Cp replace the manometer with two gas syringes, which are connected to the bottle via the three-way stopcock (see Fig. 5). Fig. 6: Measuring parameters

Fig. 5: Experimental set-up (Cp)

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For the determination of Cp, also perform at least 10 measure- it follows that ments. After each measurement remove air from the system f until the vertical syringe again exhibits the initial volume deter- C R (9) V 2 mined in the first measurement.

and taking equation (6) into consideration: Theory and evaluation f 2 The first law of thermodynamics can be illustrated particularly Cp R (10) well with an ideal gas. This law describes the relationship a 2 b between the change in internal intrinsic energy Ui, the heat exchanged with the surroundings Q and the constant-pres- The number of degrees of freedom of a molecule is a function sure change pdV. of its structure. All particles have 3 degrees of translational freedom. Diatomic molecules have an additional two degrees of dQ = dUi + pdV (1) rotational freedom around the principal axes of inertia. Triatomic molecules have three degrees of rotational freedom. The molar heat capacity C of a substance results from the Air consists primarily of oxygen (approximately 20%) and nitro- amount of absorbed heat and the temperature change per gen (circa 80%). As a first approximation, the following can be mole: assumed to be true for air: 1 dQ C · (2) f =5 n dT CV = 2.5 R -1 -1 n Number of moles CV = 20.8 J · K · mol and One differentiates between the molar heat capacity at constant Cp = 3.5 R volume CV and the molar heat capacity at constant pressure Cp. -1 -1 Cp = 29.1 J · K · mol . According to equations (1) and (2) and under isochoric conditions (V const., dV = 0), the following is true:

Determination of Cp 1 dUi C · (3) The energy Q is supplied to the gas by the electrical heater: V n dT Q = U · I · t (11) and under isobaric conditions (p = const., dp = 0): where U Voltage which is applied to the heater wires 1 dUi dV C p (4) (measured separately) p n a dT dT b I Current which flows through the heater wires (measured separately) Taking the equation of state for ideal gases into consideration: t Period of time in which current flowed through the wires pV = n R T (5) At constant pressure the temperature increase T induces a it follows that the difference between Cp and CV for ideal gases volume increase V. From the equation of state for ideal gases, is equal to the universal gas constant R. it follows that:

– = (6) nR V Cp CV R ¢V ¢T ¢T (12) p T It is obvious from equation (3) that the molar heat capacity CV is a function of the internal intrinsic energy of the gas. The internal and taking equation (2) into consideration, the following results energy can be calculated with the aid of the kinetic gas theory from equations (11) and (12): from the number of degrees of freedom f: 1 U · I ¢t · V 1 Cp · (13) Ui fkB NA T n (7) n ¢V · T 2 The molar volume of a gas at standard pressure p = 1013 hPa where 0 and T0 = 273.2K is: -23 kB = 1.38 · 10 J/K (Boltzmann Constant) 23 -1 V0 = 22.414 l/mol. NA = 6.02 · 10 mol (Avogadro's number) The molar volume is: Through substitution of

p0 V0 T V (14) R = kB NA (8) mol T0 p

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In accordance with the following, the number of moles in vol- The slope of the straight line in Fig. 7 is equal to ume V is: ¢V ml = 5.242 V ¢t s n (15) Vmol with Cp can be calculated using equation (13) under consideration of (14) and (15): U = 4.59 V and I = 0.43 A p V UI ¢t C is obtained with equation (16) 0 0 · · (16) p Cp ¢ T0 p V -1 -1 Cp = 31.29 J · K · mol ±7% The pressure p used in equation (16) is calculated from the atmospheric pressure minus the pressure reduction due to the weight of the syringe's plunger. The pressure reduction is calculated from: Determination of CV Under isochoric conditions, the temperature increase T pro- m · g duces a pressure increase p. The pressure measurement K pK results in an alteration of the volume which must be taken into FK consideration in the calculation: 0.1139 kg · 9.81 ms2 = p V T 7.55 · 104 m2 ¢T ¢V ¢p p ¢V V¢p (17) nR nR pV 1 2 = 1480 kg m-1 s-1 = 14.8 hPa It follows from equations (3) and (1) that: p = pa – pK 1 ¢Q p¢V = 975 hPa – 14.8 hPa C · (18) V n ¢T = 960 hPa and with equations (11) and (17) one obtains: where p Atmospheric pressure minus the pressure reduction due p · V U · I¢t – p¢V to the weight of the syringe's plunger C · (19) n · T p¢V V¢p pK Pressure reduction due to the weight of the plunger p Measured atmospheric pressure (= 0.1139 kg) a The indicator tube in the manometer has a radius of r = 2 mm. mK Mass of the plunger A pressure change of ∆p = 0.147 hPa causes an alteration of g Acceleration of gravity (= 9.81 m · s-2) 1cm in length; the corresponding change in volume is there- -4 2 fore: FK Area of the plunger (= 7.55 · 10 m ) V = a · p (20)

where

1 cm cm3 a = pr2 · · = 0.855 (21) 0.147 hPa hPa Fig. 7: Volume change V as a function of the heat-up time t. U = 4.59 V, I = 0.43 A. thus

pVU · I¢t – ap · ¢p C = (22) V nT · (ap V) · ¢p

Taking equations (14) and (15) into consideration, it follows that:

p0 V0 U · I¢t ap CV = · – (23) T0 a (ap V) · ¢p ap V b

The slope of the straight line in Fig. 8 is equal to

¢p hPa = 0.529 ¢t s

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Fig. 8: Pressure change p as a function of the heat-up time t. As a consequence of heat losses to the surroundings, the U = 4.59 V, I = 0.43 A experimentally determined values for Cp and CV are somewhat larger than the theoretical ones. The difference between the molar heat capacities provides the value for R.

The experimental result is

R = Cp – Cv = 9.27 J · K-1 · mol-1 ±9%

Value taken from literature:

R = 8.3 J · K-1 · mol-1

Note Using this apparatus, other gases (e.g. carbon dioxide or argon) can also be measured. These gases are then introduced through the stopcock on the bottom of the vessel.

Data and Results C can be calculated using equation (23) if equation (21) is taken v Literature values: into consideration. -1 -1 Cp(oxygen) =29.4 J · K · mol With the atmospheric pressure -1 -1 CV(oxygen) =21.1 J · K · mol -1 -1 p = 1011 hPa Cp(nitrogen) =29.1 J · K · mol -1 -1 CV(nitrogen) =20.8 J · K · mol (this part of the experiment was done the next day), a volume of R = 8.314 J · K-1 · mol-1 V = 10 l = 83.14 hPa · l · K-1 · mol-1 U = 4.59 and = 0.43 A Experimental results: the following value for CV is obtained: -1 -1 Cp (air) = 33 J · K · mol -1 -1 -1 -1 CV = 22.02 J · K · mol ±5% Cv (air) = 26 J · K · mol

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