The Adiabatic Expansion of Gases and the Determination of Heat Capacity katios A Physical Chemistry Experiment William M. Moore Utah State University, Logan, UT 84322 Adiabatic processes involving gases are discussed repeatedly in the development of the Laws of Thermodynamics in physical chemistry courses. However, the experimental aspects of adiabatic processes have been neglected in the Dhvsicd chemistw lahoratorv. There is a eood reason for this heg~ect.The experiment on the adiabaticexpansion of gases has a bad re~utation.The results are inconsistent and usually greatly in error. In our laboratory we have made several modifications to the experiment, and now the procedure can give respectable results which will please even the most critical student group. An excellent summary of methods for determining heat capacities and heat capacity ratios has been given by Partington (I). The method we will be discussing has been atrihuted to Clement and Desormes (I), hut it was developed for over-pressuring by Gay-Lussac and Welter ( l , 2 ) . The basic technique is given in several physical chemistry laboratory textbooks ( 3 , 4 ) .The experiment consists of opening the cap on a slightly over-pressurized vessel and then quickly reolacine the c a n The ooenine of the vessel ~ r o d u c e an s adiabatic expansion accompaniei by a drop in temperature of the eas. Closine the vessel ~roducesan isochoric heatina of the to its original temperature. If the pressure in the enclosed vessel is measured before the opening and then again after thermal equilibrium has heen re-established, the heat capacity ratio can be calculated as will he shown below.
vent to atmosphere
The experimental apparatus is shown in Figure 1.Instead of removing a cap, we have used a 1-in. diameter metal ball valve which can be rapidly opened and closed. The principal modification is the replacement of a liquid manometer with a pressure transducer. With a 1-psi diaphragm, these pressure transducers are extremely sensitive and the output voltage can he either digitized for computer input or chart recorded. The volume of the plastic carboy is approximately 50 1and no thermostating of the container is required. The metal valve and carboy must be firmly held in place, since high torque is developed when turning the valve handle. A recorder with variable voltage ranges, fast response (0.5 sec f a ) , and a minimum chart speed of 10 in. per min is required to record the event. With our apparatus, the pressure differential in made to be 5040 torr. The Dresswe transducer is calibrawd hv atmchine a differential mercury manometer or other pressure gauge 6 the outlet from the hall valve and measurine several Dressures in the vessel. The pressure changes can best he s b o h on the indicator diagram of Figure 2. The initial state (PI) and final state (P3) are on the isotherm a t the ambient temperature of the room. When the valve is opened, the gas adiabatically expands to atmospheric pressure ( P d . Assuming that the processes are reversible, the adiabatic equation (PlVl' = PzV2') can he combined with the isothermal equation (PlV1 = P3Vz) to yield eqn. (1). We have performed this experiment on four different gases. The results shown in the table refute many of the claims of excessive error and inconsistency attributed to the experi-
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L Figure 1. Experimental apparatus for the adiabatic expansion of gases; ( I ) Naigene 50-1 plastic carboy, (2) Whitey I-inch B65FI6 brass ball valve. (3) Vaiidyne DP-l5L variable reluctance differential pressure transducer with l p s i diaphragm.
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50
V(LITERS)
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Figwe 2. indicator diagram far monatomic gas such as argon: (1) (2) adabatic expansion of gas: (2) (3) isochoric heating to original carboy temperature; (1) (3) isotherm at original carboy temperature.
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Volume 61
Number 12 December 1984
1119
Determination of Heat Capacity Ratios and C, at 295 K Heat Capacity at Constant Pressure (J mol-' deg-') Literature (5) Experimental
Heat Capacity Ratio Gas Nirogen
Helium Argon
Carbon dioxide
Literature (5)
Experimental
1.40 1.67 1.67 1.29
1.39 f 0.01 1.50 f 0.05
29.1 ...
29.5 f 0.7
1.63 f 0.03
20.8
1.30 f 0.02
36.9
21.4 f 0.7 35.0 f 2.0
TIME (SEC) Figure 3. Timedependent pressure curves far the adiabatic expansion and the isochwic heating processes.
ment. The principal assumptions which need to he examined are (1)sufficiently low thermal conduction to surroundings, (2) reversihility, and (3) ideality. Although the pressure transducer is not inexpensive, it can find extensive use in a physical chemistry laboratory (6). Even with our modernization of the experiment, values of CdCv for helium would not exceed 1.55. A monatomic gas will have the greatest temperature change during an adiahatic expansion. Since we are using the same vessel for all experiments AT is proportional to the initial temperature gradient dTldx. The heat flow is given by eqn. (21, where is the coefficient of thermal conductivity, and A is the surface area exposed. The K for helium is 9 times greater than that for argon. The results in the table show that the heat capacity ratio for argon can be measured successfully even though the initial tem~eratureeradient is the same as for helium. All gases with the exceptlion of helium and hydrogen, which have hieh values of K . should he amenable to this experimental method. The other aspect of the heat exchange broblem is shown in the inset to Figure 3. The original volume (V,) to be expanded is not the volume of the carboy hut a volume slightly less. The gas that is expelled when the valve is opened acts as a piston, so that Vz is the actual volume of the carboy. The assumption here is that heat transfer to the piston pas is also ~ v . ~ l i g ~during h l e ihe adiabatic expan~ion. Tht! adiahatic expansion can be treated as reversible sincr there is only asmall pressure chanpr inwl\,trl. If the cillrulution is done f
