In the Laboratory
Thermodynamics of Air: Reinterpreting the Method of Clement and Desormes Frederick M. Hornack Department of Chemistry, University of North Carolina, Wilmington, NC 28403-3297 Early in the 19th century, the heat capacity ratio Cp/C v for gases was measured by means of a disarmingly simple procedure. Judging by papers published in recent years, this historic method continues to be serviceable in the undergraduate physical chemistry laboratory. The theory (1), basic assumptions (2, 3), and innovations in apparatus (4) have been discussed in detail. Although the method of Clement and Desormes seems archaic in comparison with modern electronic instrumentation, the accuracy of the results is satisfying and students can directly observe some of the processes and changes in state which are mentioned so often in abstract lecture presentations. In the approach described here, the experiment and the acquisition of data are done precisely in the traditional manner. However, calculation of the heat capacity ratio is completely sidestepped. Instead, it is recognized that in the case of air, the work done in the rapid adiabatic expansion must equal the heat absorbed when the gas returns to room temperature at constant volume. On this basis, the heat capacity Cv can be directly determined. Students deal with a thermodynamic cycle and have the unusual opportunity to apply the equation ∆E = Q + W in an experimental context. To a close approximation, air behaves like an ideal, diatomic gas. Interest in the experiment might be generated by talking about diatomic oxygen and nitrogen and their theoretical energies. Since air accurately obeys the ideal gas law, the forces between molecules are negligible. The vibrational energy is quite constant over the small temperature range and the kinetic energy of the gas is a function of temperature only. According to the equipartition principle, diatomic molecules have two degrees of rotational and three degrees of translational freedom. The sum of the rotational and translational kinetic energies is 2.5 RT. Since Cv = (∂E/∂T) v, the theoretical heat capacity is 2.5 R. Also, Cp = Cv + R = 3.5 R, which agrees nicely with the literature values for the components of air and with the results obtained experimentally as described below. Since only air is used, the auxiliary apparatus is minimal, the procedure is simplified, and more time becomes available to discuss the relevant thermodynamic principles. We use a 20-L glass carboy bottle enclosed in its original shipping container, and no thermostat bath is needed. The air is pressurized with a squeeze-bulb and the connecting tube is fitted with a pinch-clamp. The apparatus incorporates the neck of a 25-mm test tube, which is sealed with a rubber stopper as shown in Figure 1. In practice, the stopper is raised about two inches above the opening and replaced as rapidly as possible. An investigation with a pressure sensor and an oscilloscope showed that when the stopper is pulled, the gas drops to ambient pressure in about a tenth of a second, followed by dampened oscillations. A dibutyl phthalate (DBP) manometer can be used to measure differential pressure to about 0.5 mm DBP (0.04 mmHg). The air is pumped up to about 20 cm DBP (differential pressure h1) and
equilibrated at room temperature. It is important to make h1 a constant for a series of experiments so that reproducibility can be readily checked. This is done by partially releasing the pinch-clamp and carefully allowing the oil meniscus to settle to the desired mark on the manometer scale. The subsequent rapid adiabatic expansion to atmospheric pressure Pa is followed by isochoric warming to room temperature T r while a new differential pressure h2 is established. The highest consistent values of h2 are chosen. With some prompting from the instructor, students can learn that applying force to molecules causes their energy and temperature to rise. This is evident immediately after the initial work of compression. As h1 is established, a slowly falling meniscus is observed as the gas cools to room temperature. In the rapid expansion step, molecules lift the atmosphere and lose kinetic energy. Again, this effect is betrayed by the behavior of the meniscus. Under the conditions of the experiment, an adiabatic temperature drop of about 1.5 °C is predicted, but a direct measurement of ∆T appears to be elusive. A thermistor probe placed in the expanding gas responded sluggishly and required about 20 seconds to reach equilibrium. A temperature drop of only 0.3 °C was noted. At this point, the customary formulas are bypassed and students use the pressure and temperature data to define the three states of the gas (Pi,V i,Ti), i = 1, 2, 3. Using spreadsheet graphics, the points can be plotted on a PV diagram. More importantly, the values of ∆V and ∆T in the adiabatic expansion can be calculated quite accurately and the results are insensitive to the absolute values of room temperature and pressure. From a more advanced point of view, the experiment could be regarded as a determination of (∂V/∂T)s and the estimation of the heat capacity from Cv = –P(∂V/∂T)s.
Figure 1. Adapter with connecting arms for pressurizing bulb and manometer.
Vol. 73 No. 10 October 1996 • Journal of Chemical Education
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In the Laboratory
The densities of mercury (13.54 g/mL) and DBP (1.045 g/mL) can be used to convert the DBP readings to mmHg. There are two ways to proceed: 1. Employ the simple gas laws and express molar volumes and other variables to at least four decimal places. Students must pay attention to detail and resist the urge to drop “insignificant” digits. This is one of those interesting cases where strict adherence to the rules destroys the accuracy of differences between large quantities. In the order given, one can use: P1 = Pa + h 1 P2 = Pa P3 = Pa + h 2 T1 = Tr T3 = Tr V1 = RT1/P 1 V3 = RT3/P 3 V2 = V3 T2 = T 3P2/P3 (also T2 = P2V2/R) From these, ∆V and ∆T can be calculated to three significant figures. 2. Alternatively, but less straightforwardly, the three equations PiVi = RTi can be used to derive
tained h2 = 49.0 mm DBP, ∆T = –1.45 °C, ∆V = 306 mL, Q = 31.2 J, and Cv = 21.5 J/deg mole. An integration of PdV on an adiabatic path with heat capacity ratio 1.4 shows that the reversible work exceeds Pa∆V by only 0.3 J; this is a challenging exercise. Students are apparently studying a virtually reversible thermodynamic cycle on a small scale. The net change in a state function such as entropy must vanish around any cycle. Going counterclockwise, the entropy changes for the isochoric and isothermal steps are Cv ln(T 3/T 2), and R ln(V1/V2). Again, since small differences are involved, four decimal places should be used in calculations. Various partial differential coefficients can be evaluated—for example, (∂P/∂T)v = R/V = 0.00343 atm/deg, which shows excellent agreement with ∆P/∆T on the experimental isochore. Similarly, total differentials such as ∆P = (∂P/∂T)v ∆T + (∂P/∂V)T∆V check out well. A more complex case is (∂V/∂T) s. From dS = (∂S/∂T) vdT + (∂S/∂V)TdV = 0 and the Maxwell relation (∂S/∂V)T = (∂P/∂T)v there is obtained (∂S/∂T)v + (∂P/∂T)v (∂V/∂T)s = 0
T2 – T 1 = ∆T = T1(P2 – P3)/P3 = –T1h2/P3 (T1 is room temperature) V2 – V1 = ∆V = V1(P1 – P3)/P3 = V1(h1 – h2)/P 3 Here, no problems with the rules of significant figures arise. The work done by the gas (–W) is assumed to be the irreversible value Pa∆V. Since the ideal gas returns to the initial temperature, ∆E = Q + W = 0 and therefore Q = P a∆V and Cv = Q/∆T. Typically, calculated values of Cv are too high by about 5%. For example, with P a = 765 torr, Tr = 295 K, and h 1 = 177.5 mm DBP, there is ob-
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For an ideal gas this becomes (∂V/∂T)s = –V Cv /RT which, under the conditions of the experiment, has a value of about –0.21 L/deg, in good agreement with the observed value of ∆V/∆T. Literature Cited 1. 2. 3. 4.
Buep, A. H.; Czekalski, M.; Baron, M. J. Chem. Educ. 1988, 65, 416–417. Bertrand, G. L.; McDonald, H. O. J. Chem. Educ. 1986, 63, 252–253. Meyer, E. F.; Stewart, G. H. J. Chem. Educ. 1988, 65, 282. Moore, W. M. J. Chem. Educ. 1984, 61, 1119–1120.
Journal of Chemical Education • Vol. 73 No. 10 October 1996
